Mathematics of Everyday Life - Math @ McMaster University

PRELIMINARY EDITION (FALL 2021)

99 Numbers: Mathematics of Everyday Life

PART 1 Introduction................................................................................................ 1 Mathematical Reasoning and Numeracy .................................................... 7 Numbers: Quantitative in Quantitative Reasoning ................................... 38 Counting and Number Systems ............................................................... 69 Proportional Relationships...................................................................... 84 Linear and Non-linear Relationships ....................................................... 99 Quantities Changing Exponentially......................................................... 122 Covid-19.................................................................................................. 149 PART 2 Financial Matters ..................................................................................... 168 Cryptocurrencies and Blockchain........................................................... 188 Economy and Social Indicators ............................................................. 200 Human Population ................................................................................. 225 Climate Change .................................................................................... 240 Uncertainty and Probability .................................................................... 261 Working with Probability......................................................................... 282 Gambling................................................................................................. 301 Basics of Statistics and Data Collection ................................................. 312 Working With Data ................................................................................ 328 Important Statistical Principles ............................................................. 348

Chapter 1 Introduction 1

1 Introduction

What is numeracy? What makes a person numerate?

In broadest terms, numeracy (or numeric literacy or quantitative liter-acy) can be viewed as a combination of specific knowledge and skills, which areneeded (but do not suffice) to function in the modern world. What other skills,apart from numeracy, do we need to posses? For instance, social intelligence,cross-cultural competency, and new-media literacy.

Numeracy involves reasoning from, and about, numeric information (data),which can be presented in a variety of ways (such as numeric, graphic, narrative,visual, and dynamic forms). Inspired in part by questions we routinely ask inmathematics (What is this? Why is this true? How do we know?), we expandnumeracy to include critical, evidence-supported thinking, common sense, andlogical reasoning in situations and/or contexts that do not explicitly nor implicitlyinvolve numbers or quantitative information. (Examples follow.)

In our conceptualization of numeracy, and for the purpose of this course, anumerate person is assumed to be a university student, rather than a generalmember of our society. (One reason lies in the fact that we must make certainassumptions about the background, mathematical and otherwise.)

To make this concept of numeracy more transparent, we now illustrate itsaspects in several examples (which are in no way an exhaustive).

In the article After 7,500% rally, cryptocurrency founder sells his coins pub-lished in the Globe and Mail, on 20 December 2017, we read “Litecoin droppedabout 4 per cent to $ 319 at 1:02 p.m. in New York, according to prices onBloomberg. The coin is still up about 75-fold since the end of 2016, according tohttp://coinmarketcap.com prices. The market value was $ 17.5-billion.”

A numerate person is able to understand and work with the percent infor-mation (for instance, they know that the value of the Litecoin, before it dropped4% to $ 319, could not have been $ 500); they understand that 75-fold increasemeans 75 times, i.e., that the Litecoin value at the end of 2016 is multiplied by 75;they can visualize $ 17.5 billion, by comparing, or relating to other quantities, suchas saying that “$ 17.5 billion is the salary of 175 thousand top paid high schoolteachers in Ontario.”

A numerate person can compute their body mass index (from a formula theyrecovered from Wikipedia, for instance), look up its value in the chart to determinewhether they are deemed overweight or not. They can interpret results of theirblood test by relating numbers: if their cholesterol reading is 5.78, and the desiredlevel is less than 5.20, they realize that their cholesterol level is outside normallimits.

A numerate person can reason logically: they understand that while nausea,headache, fever, and vomiting are symptoms of bacterial meningitis, they do notcause meningitis; as well, someone showing these symptoms does not necessarilyhave meningitis. (Thus, one has to be careful how to interpret answers fromonline symptom checkers.) A numerate person understands the difference betweencausation and correlation.

A numerate person has basic understanding of probability and risk. For in-stance, if something is known to occur twice a week, then 4 occurrences in a certainweek might not constitute an epidemic, but rather reflect a probabilistic fact thatunlikely events nevertheless do occur. As well, a numerate person will recognizethat statistics derived from a small sample is very likely worthless, and often not inany way representative of the population from which the sample has been drawn.

2 99 NUMBERS: MATHEMATICS OF EVERYDAY LIFE

A numerate person is aware of their digital footprint in social media and elec-tronic marketplaces, and is concerned about privacy and confidentiality. Mathe-matical algorithms can spy on the interactions within social media, in order manip-ulate the users’ behaviour. For instance, read about the case of Cambridge Analyt-ica at http://www.cbc.ca/news/opinion/cambridge-analytica-opinion-1.4588857).

A numerate person is familiar with basics and biases of search algorithms andtargeted marketing.

A numerate person is able to use graphs to understand, illustrate and illumi-nate concepts, presented in both static and dynamic forms. In the article Thischart shows how bread prices soared during the price-fixing scheme published inMacLeans, 22 December 2017, https://goo.gl/U5CV7A we find the following figure(Source: Statistics Canada, CANSIM Table 326-0021):

We read: “A glance at Statistics Canadas data on food prices and the consumerprice index, which compares the cost of a fixed set of goods and services over time,seems to show the price fixing in action.”

A numerate person is able to look up the meaning of consumer price index(CPI), and by learning that CPI was reset to 100 in 2002, note that the graphsare accurate, as they all cross the CPI of 100 line in 2002. As well, they wouldrealize that indeed the CPI for bread, rolls and buns has been increasing at a fastpace since 2002, and grew substantially larger than other quantities representedin the graph. For instance, they could determine that it is about 50 units largerthan the CPI for all items in 2015.

A numerate person is a critical person, and asks what “ecotourism,” or thelabel “green” on a laundry detergent actually mean. They do not pay for an “allinclusive” vacation before insisting to know what is not included. A numerate per-son reads the small print and, for instance, understands how the annual percentagerate (APR) is used to compute the monthly interest on their credit card.

As well, a numerate person holds certain beliefs and values. They understandand appreciate the importance of mathematical and logical reasoning for living


and making decisions; they accept the fact that numeracy takes time and practiceto achieve and is an important part of life-long learning; they are willing to adoptattitudes, beliefs and work habits in order to overcome potential learning and other(personal) barriers that might exist; they are inquisitive, motivated to learn ontheir own, and are willing to engage with challenging topics and ideas.

Why numeracy?A survey conducted by the Conference Board of Canada in 2012-2014 [1] claims

that about 55% of Canadian adults have inadequate numeracy skills, and that itis a “significant increase from a decade ago.” Based on the first results of theProgramme for the International Assessment of Adult Competencies, StatisticsCanada reports that “Canada ranks below the OECD average in numeracy, andthe proportion of Canadians at the lower level is greater than the OECD average.”[2] (References appear at the end of this chapter.)

First steps ...

Numeracy is about looking critically at information (often involving numbers),and making sense of it. But above all, is it about asking questions, actually,asking good questions, so that we can understand, and based on our under-standing, make good decisions.

To illustrate this, we look at several examples.(1) What does natural in Natural spring water mean? In 100% natural springwater? What does pure in Pure Life premium drinking water mean? In puredrinking water? (All of these were taken from labels of bottled water.)

When we look at food labels on bottled water, we find ingredients such asmagnesium sulfate, potassium chloride, salt, calcium chloride, magnesium chloride,and potassium bicarbonate, together with - of course - purified water (i.e, filteredtap water). Routinely, bottled water companies purify water, and then add someof the ingredients listed above (or some others) back into it. Why? Usually, theyclaim, to give taste to water. So natural is not really natural, and neither is purepure, 100 percent water.(2) The table below shows a sodium content for four beverages, together with thepercent daily value of sodium.

Drink Sodium content % daily value

Monster Rehab Energy Iced Tea 110 mg/240 mL 5

Coca Cola 30 mg/250 mL 1

Starbucks Doubleshot Fortified Coffee Drink 160 mg/444 mL 7

Gatorade Perform Orange Thirst Quencher 250 mg/591 mL 11

So what is the daily value of sodium?Based on Coca Cola, if 30 mg is 1 percent, then the daily value is 3000 mg. If

110 mg represents 5% (Monster), then the daily value is (110/5)*100 = 2200 mg.If 160 mg represents 7% (Starbucks), then the daily value is (160/7)*100 = 2285mg. If 250 mg represents 11% (Gatorade), then the daily value is (250/11)*100 =2272 mg.

According to Health Canada (https://goo.gl/Q4qqjm), adequate intake ofsodium for teens and adults is 1500 mg/day, with upper limit of 2200-2300mg/day.Comments?

(3) Shinerama is Canada’s largest post-secondary fundraiser in support ofCystic Fibrosis Canada (McMaster students routinely participate). What is thereto ask about, isn’t this a good cause?


Yes it definitely is, but – it costs to raise money!Let’s check how Cystic Fibrosis Canada does it. The web page 2017 Charity

100: Grades at https://goo.gl/ED1pA4 gives a ranking of charitable institutions,by evaluating them on important parameters.

The screenshot from the web page shows that, in terms of charity efficiency,Cystic Fibrosis Canada is given the grade of D.

What does it mean? Comparing with other organizations, we see that that’sa fairly low ranking (actually among the worst two in its category FundraisingOrganizations).

Probing further, we look at two important parameters: charity efficiency (de-fined as the percentage of total funds collected that actually go toward the cause),and fundraising efficiency (the amount of money needed to raise $ 100). (Thesedefinitions can be found on https://goo.gl/ED1pA4.)

The figure shows that the charity efficiency of Cystic Fibrosis Canada is 56.0%,and its fundraising efficiency is $ 32.00.

As comparison (data taken from the same web page): the charity efficiency ofTerry Fox Foundation is 84.0%, and its fundraising efficiency is $ 16.00. UnitedWay of Calgary and Area does it even better: its charity efficiency is 89.0%, andthe fundraising efficiency is $ 6.00.

What is Really Important about the Fundraising case Study?The fact that Cystic Fibrosis Canada has the charity efficiency of 56.0%, on

its own, means very little. Is it good, bad, average? It only made sense whencompared with other organizations. Hence the message - numbers by themselvesare not as informative, or powerful. We need to place them into context, such ascomparing to a reference frame obtained by looking at other organizations.

Closer to home example. You got your Math 2UU3 test back, and your gradeis 28/40=70%. That, by itself does not as much as knowing what other people goton the same test. For instance, if the class average was 55%, then you know thatyou did very well, but if the class average was 85%, then it’s quite the opposite.

Can you think of other examples where establishing a reference frame providesa more valuable information?

Question (always ask questions!): is the site we used in this case study reliable?Who created it and who has been and maintaining it?


(4) Consumers looking at the following label in a British supermarket chain Tesco

might conclude that the meat they are buying is from a farm, and (looking at thename) believe that it is a local farm. However, investigations show that althoughWoodside Farm exists, they do not supply the meat to the supermarket in question.As a matter of fact, Woodside Farms is a brand name, and not a farm.

In Guardian, 13 December 2017, Tesco faces legal threat over marketing its foodwith ’fake farm’ names we read “In March 2016 Tesco, the UKs largest retailer,sparked controversy after launching a budget range of seven own-label farm brandsincluding Woodside Farms and Boswell Farms for fruit and veg as well as meatbased on British-sounding but fictitious names. Some foods were imported fromoverseas and given British names to make them sound local.”(5) Many more examples coming, starting with the next section.

Not everything is quantitative, nor quantitative is useful

There are situations when knowing the temperature does not say the wholestory. If it is a few degrees (Celsius) below zero, we might feel cold, very cold, orfreezing (or not at all cold). The size of a plot of land might not be as useful–forinstance for farmers–as the quality of the grass on it (hence the cow index, used inIreland or mother cow index used in parts of the US; both indices “measure” theability of the plot of land to sustain certain number of cattle). To many people thephrase very low risk of side effects seems to be lot more useful than 0.3% chanceof side effects.

We can find many more examples where measuring, using quantitative scales,metres, grams, seconds, etc. do not tell the whole story, or tell it in ways whichare not useful nor meaningful.

The article How we measure without maths published by BBC News (27 July2018, https://bbc.in/2vxBEne) talks about qualitative scales and the need, orpreference for narratives over quantitative data.

We read “These are yardsticks that measure observable, but not necessar-ily numerical, properties and we use them all the time. Qualitative scales aresometimes humorous and often downright bizarre, but they are just as valuable asquantitative scales for imagining relationships between properties and standardis-ing ideas.”

These qualitative assessments/ scales “range from chili pepper heat to mineralhardness to ocean breezes [...] Qualitative scales allow us to label variables withlittle or no quantitative information. These unusual units of measurement areoften colloquial: guesstimations and as-the-crow-flies rules of thumb that allow forquick assessments and comparisons.”


And yet, qualitative descriptions “prove their usefulness time and again. With-out them, we would struggle to conceptualise ideas of pain (a doctor might aska patient to rank his symptoms) or grade the severity of weather conditions (likethe Beaufort Scale does).”

Suggestion: heck Wikipedia for Beaufort Scale; in particular, look at the chart,and note the verbal descriptions. A part of the chart is reproduced here:

(Source: Wikipedia https://bit.ly/1jxYJ0y)

Section references

[1] The Conference Board of Canada (n.d.; the page claims that the data is accurateas of June 2014). Adults With Inadequate Numeracy Skills. https://goo.gl/jx6Jny[2] Statistics Canada. Skills in Canada: First Results from the Programme for theInternational Assessment of Adult Competencies (PIAAC)

Chapter 2 Mathematical Reasoning and Numeracy 7

2 Mathematical Reasoning and Numeracy

In this chapter we learn about building blocks of mathematics (definition, theo-rems, and logical reasoning), and what they represent in numeracy, i.e., how theyrelate to real-life situations.

Definitions

In mathematics, a definition introduces a new object, a new property of ob-jects, or a new relation between objects, based on previously defined or establishedobjects, properties and/or relations. By “established” we mean through acceptedmathematical routines, such as by providing acceptable evidence (“proof”).

A mathematical definition is succinct, clear and precise, and does not leavespace for ambiguity. It answers the question “What?” In mathematics, it is notpossible to work with anything that is not defined (there is a caveat to this; readon).

Consider the following example of a mathematical definition, which introducesa new concept, that of a prime number:

A prime number is a positive integer which has exactly two distinct divisors:the number 1 and itself.

By reading (and re-reading) this definition carefully we can figure out what aprime number is. First of all, only positive integers (e.g., 1, 2, 3, 4, 5, etc.) couldbe prime numbers. Since the number 1 has only one divisor (namely, itself), it isnot a prime number (as the definition requires two distinct divisors). The number2 is divisible by 1 and by 2, and thus, having exactly two distinct divisors, it isa prime number. The number 6 is not a prime number, as it has more than twodistinct divisors: 1, 2, 3, and 6.

There are alternative ways to define a prime number (try to find one!), butthey all have the same meaning. No matter what definition is used, 1 and 6 arenot prime numbers, and 2 and 17 are prime numbers.

Once established, mathematical definitions do not change (for instance, thedefinition of a prime number that we wrote is more than 2300 years old). As well,they do not get adjusted based on culture, language, geography, nor for politicalpressures, nor for any other reason. Outside of mathematics, in other disciplines,definitions can rarely be qualified in the same terms. One can argue that, in reallife, there is nothing that would qualify as a mathematical definition. Real lifedefinitions may depend on authority, history, context, geography, time, and so on.

Let us explore definitions in real life.First of all, we do not routinely use the word definition in everyday language.

Instead of asking “Can you define this for me?” we might say “Can you explain?”“What does it mean?” “What is our agreement on this?” “What is our under-standing about this?” “What is the convention?” and so on.

As our case studies will show, it is a challenge (and often not possible) tocreate a statement that would share the properties of a mathematical definitionwhen dealing with non-mathematical contexts. In some cases (can you suggestsome?), it is even desirable to keep things unclear and vague.

In mathematics, there is no ambiguity about the meanings of the words used(not just in definitions, but everywhere), as all of them are precisely defined.

In contrast, in real life, our understandings are often implicit. A “square of achocolate” is not a square, but a three-dimensional shape. As a two-dimensionalfigure, a square has no thickness, so we cannot bite into it (actually nothing onour planet is two-dimensional). Red wine is not red, and white wine is not white.


A “red herring” does not have to be red, nor a herring. So, when it’s not a driedsmoked herring, turned red by smoke, what is a “red herring”?

Case study What is a Planet?

After the revision of the definition of a planet at the meeting of the Interna-tional Astronomical Union in August 2006, our Solar system “lost” its most distantplanet, Pluto. The new definition (see Why is Pluto no longer a planet? BBCNews, 13 July 2015 https://goo.gl/vxpMUw) states that a planet is a celestialbody that

(a) is in orbit around the Sun(b) has sufficient mass for its self-gravity to overcome rigid body forces so that

it assumes a hydrostatic equilibrium (nearly round) shape, and(c) has cleared the neighbourhood around its orbit.

In the BBC article, we read: “Pluto met the first two of the these criteria, butthe last one proved pivotal. ‘Clearing the neighbourhood’ means that the planethas either ‘vacuumed up’ or ejected other large objects in its vicinity of space. Inother words, it has achieved gravitational dominance.”

Case study What is Climate Change?

Everyone seems to be talking about it – but what is a good definition (con-vention, understanding) of the term “climate change”?

The David Suzuki Foundation is a Canadian non-profit organization withheadquarters in Vancouver (and offices in Toronto and Montreal), working onprotecting the natural resources and environment, and on creating a sustainableliving for all Canadians. On the web page https://goo.gl/T3Jnno, under Whatis Climate Change? we read: “In a nutshell, climate change occurs when long-term weather patterns are altered – for example, through human activity. Globalwarming is one measure of climate change, [as] is a rise in the average globaltemperature.”

NASA (National Aeronautics and Space Administration) is a U. S. agency incharge of space programs, and aeronautics and aerospace research. In its answerto What is Climate Change?, at https://goo.gl/T3Jnno, we find NASA’s under-standing: “Climate change is a change in the usual weather found in a place. Thiscould be a change in how much rain a place usually gets in a year. Or it could bea change in a place’s usual temperature for a month or season. Climate changeis also a change in Earth’s climate. This could be a change in Earth’s usual tem-perature. Or it could be a change in where rain and snow usually fall on Earth.Weather can change in just a few hours. Climate takes hundreds or even millionsof years to change.”

The Australian Academy of Science (at https://goo.gl/cfthcz) defines climatechange as “a change in the pattern of weather, and related changes in oceans, landsurfaces and ice sheets, occurring over time scales of decades or longer.”


As we can see, there are similarities, but also differences in how these orga-nizations conceptualize climate change. Clarity and sharpness of a mathematicaldefinition are no longer there.

Case study ADHD and Social Anxiety Disorders

Unlike mathematical definitions, the definition of ADHD (Attention-Deficit/Hyperactivity Disorder) has undergone numerous revisions, and the most recentversion appears in 2013 in the fifth edition of Diagnostic and Statistical Manualof Mental Disorders [1].

It is not just revisions and updates; sometimes, people (experts, researchers)do not agree on a definition. As soon as the Diagnostic and Statistical Manual(Third Edition), in 1980, introduced the term “social phobia”, there has beenconfusion as to whether or not shyness is a mental disorder (later, “social phobia”was replaced by a new category “social anxiety disorder” or SAD).

In the article When does benign shyness become social anxiety, a treatabledisorder? [2] which aimed at clarifying the issue, we read: “While many peoplewith social anxiety disorder are shy, shyness is not a pre-requisite for social anxietydisorder.”

Note: the sentence “While many people with social anxiety disorder are shy,shyness is not a pre-requisite for social anxiety disorder” can be represented visu-ally (these kinds of representations will became very useful), as in this figure.

shy people

people with SAD

Shy people are represented by the blue oval, and SAD people by a yellow oval.The overlap represents the fact that “many people with social anxiety disorder areshy.” The part of the yellow oval which is outside the blue oval in the diagramrepresents the fact that one can suffer from SAD without being shy.

The adjustments to the definitions of SAD have had serious consequences. Intheir online article Shyness ... Or Social Anxiety Disorder? Social Anxiety Insti-tute https://goo.gl/F8UFpf states: “The definition of ‘social anxiety disorder’ hasshifted over the past thirty years as the seriousness of the situation became clearer,and government epidemiological data consistently showed a larger percentage ofthe general population suffering from social anxiety symptoms.”

Not everyone agrees. “In Shyness: How Normal Behavior Became a Sickness(Yale University Press, October 2007, check https://goo.gl/WkkSML), Northwest-ern’s Christopher Lane chronicles the ‘highly unscientific and often arbitrary way’in which widespread revisions were made to ‘The Diagnostic and Statistical Man-ual of Mental Disorders’ (DSM), a publication known as the bible of psychiatrythat is consulted daily by insurance companies, courts, prisons and schools as wellas by physicians and mental health workers.”

Of course, there is potential agenda to this. The article continues: “By labelingshyness and other human traits as dysfunctions with a biological cause, the doors


were opened wide to a pharmaceutical industry ready to provide a pill for everyalleged chemical imbalance or biological problem, he adds.”

In Summer 2018, we found ads in Hamilton (HSR) buses that solicited vol-unteers to participate in a McMaster University “research study evaluating aninvestigational medication in Social Anxiety Disorder.”

Case study How to Define the Colour Red?

In physics, the coulour red is defined as “the color at the end of the visiblespectrum of light, next to orange and opposite violet. It has a dominant wavelengthof approximately 625–740 nanometres.” (Wikipedia, https://goo.gl/9Uyzh6; ananometre is one-billionth of a metre).

For numerous tasks that computers do, for the use in software, or on theinternet (say, background of a web page), the colour red is defined using the stringFF0000 (which represents the mix of red, green and blue pigments); alternatively,this string can be represented as the RGB (red, green, blue) triple (255, 0, 0).

Left: colour picker as found, for instance, in Adobe Illustrator or Photoshop.Right: picking a red colour (Adobe Dreamweaver interface, and HTML code be-low).

One can make a definition precise, but then its use might be impractical, ormight not make much sense in routine real-life interactions. We will not be buyingan FF0000 coloured car any time soon, nor order online a t-shirt in the 625–740nanometres spectrum. In real-life we might describe red as the colour of fire orthe colour of blood.


Investigate Further Examples of Definitions in Real Life; Hints and Leads

(1) Within sciences, many terms are misunderstood, and/or used in a number ofways, such as: uncertainty, energy, genetic, trauma, minerals, and so on. Readabout it at https://bit.ly/2N2TYhr.(2) In biology, the basic term “species” is not defined. What constitutes acceptableevidence (“proof”) is unclear.(3) After a long and challenging process, in January 2017, McMaster Universityproduced (updated in late 2019) and made public a policy document address-ing sexual assault, as mandated by the Liberal Government. Read about it athttps://goo.gl/yGNanX. In your view, is it easy to produce such a document?What can you say about definitions?(4) Gender and Olympic games. Read https://goo.gl/60TgMZ)(5) What does “driving under influence” (DUI) mean in Ontario? Is it the same inall Canadian provinces? How does it compare to U.S., or elsewhere in the world?What is the difference between DWI (driving while intoxicated) and DUI?(6) What does it mean that a yoghurt is 88% fat free? What is the precise meaningof the term “fat free”? What is the definition of a green product?(7) What is “ecotourism”? (Is this an example of a red herring?)(8) Are we truly accepting of diverse opinions if we label some opinions as notdesired, and block people from expressing them? If we claim to be tolerant, thenshould we accept those who are not tolerant?(9) What is milk? Read about why it matters: https://goo.gl/9d8WUi.(10) Apex car rental in New Zealand sells insurance that “has you fully covered.”However, on closer inspection, you realize that you are not covered at all if youdrive intoxicated, or if you drive on the wrong side of the road and get into anaccident. Moreover, if your pour diesel instead of gasoline in the tank and damagethe engine, you have to pay for the repairs. If you drive on unsealed roads anddamage the undercarriage, or a flying stone hits a windshield (windscreen in NewZealand) and cracks it, you pay for that. Any damage to the roof of the car is notcovered by insurance. So much for full coverage.(11) Pick an ad for an “all inclusive” vacation and try to figure out what it doesnot include.(12) Definition of overweight depends on geography (which is affected by otherthings). According to North American and European Union standards (check, forinstance, Centres for Disease and Prevention https://goo.gl/z8KGwE) a person iscalled obese if their body mass index (BMI) is larger than 30. In Japan, a personwhose BMI is larger than 25 is declared to be obese (source: New Criteria for’Obesity Disease’ in Japan, https://goo.gl/YqbXKf).

Recall that BMI is computed by dividing one’s mass (measured in kg) by thesquare of their height (height measured in metres); thus, the units of BMI arekg/m2.(13) What does “no sodium” mean? No dandruff? [class slides]

Remember Question We Must Always Ask

What?That’s the question! Important information about a product we are buying,

or on financial documents (such as mortgages, loans, or investments), legal andother contracts, software download agreements, etc. is contained in “small print,”and often there are pages and pages of it. Routinely, we skip it (and the otherside knows that!), and do not think much about it.

How many of us actually check all information about privacy as we updatesoftware on our laptop, or upload an app to our phone? Do we know exactly what


information about us, and how, will be widely shared and used? How many of uscheck every detail of a car rental agreement when renting a car?

We should never sign a car loan, a mortgage, or any document,paper or online, without knowing what exactly we are signing, andwhat the ramifications are.

The following examples serve as warning of things that could happen if wedon’t do it.(1) In the online article Major pet insurer says dog injuries from ’jumping, run-ning, slipping, tripping or playing’ not covered published on 3 October 2016 athttps://goo.gl/uWqivx, we read about a pet owner (Ms Richardson) who boughtdog insurance policy from the company called Petsecure. One day, while running,her dog injured its hind leg. The owner claimed that is a common injury, as dog’spaws, while running, can get caught in a rabbit or fox hole, and the resultingmomentum forces the knee to twist, causing injuries to ligaments or muscles.

However, when the owner filed a claim to get reimboursed for the hospitaliza-tion of her dog, the company denied her claim. According to the article, “Petsecurepointed to a clause in her policy denying coverage if a dog is injured while ‘jumping,running, slipping, tripping or playing.’ ”

If Ms Richardson, the dog owner, had carefully read the policy in the firstplace, she would have realized that the policy was completely worthless (and weadd highly unethical; what is dog supposed to do, if not to jump, run, slip, trip orplay?). She should not have bought the policy in the first place. (The good newsit that, in the end, under the pressure from media, Petsecure did honour her claimand reimboursed 80% of her expenses).

In the same article, we read: “A Vancouver lawyer who specializes in animalrights law says Richardson’s policy has ‘one of the craziest clauses she’s ever comeacross. Basically, what that policy says is, the dog can’t be a dog.’ ” The lawyersays that she “gets a lot of complaints from pet owners about insurance policiesnot covering things they expected would be covered” and ends with importantmessage “Regardless of what that glossy brochure says ... always, always read thefine print. And not just the front page, or the first page, but the entire policy tomake sure that what you think you’re getting, is what you’re actually getting.”(2) On 1 September 2017, CBC News (https://goo.gl/9iri68) published the articleCalgary man warns Cuba travellers about fine print after paying 10X cost of dam-aged TV. Long story short: a couple goes on an all-inclusive vacation to Varadero,Cuba. One day the husband, accidentally, broke the TV set in their hotel room.The couple fully admitted fault for the damage, but were shocked to realize thatthey were to pay about $ 5000, i.e., 10 times the value of the broken TV set.

Was that a scam?Not really - Sunwing, the company they booked the vacation through, states on

their webpage that the “Rule of 10 will be in place, established by local authorities.In the case of damaged items, customers will be charged the value of the itemmultiplied by 10.”

Although they tried to help the couple, Sunwing representatives were notsuccessful. We read: “the property [the hotel] deferred to their published policywhich reads ‘when damages caused by a break or loss of property, whether classifiedas fixed or useful assets, are the result of an intentional act of the clients or arelinked to vandalism, the responsible person will be charged ten (10) times thevalue of the purchase price of the asset broken or lost.’ ” In the end, the hotelmanagement reserved their right to apply the full penalty charge as per the statedpolicy.


Case study What is Radio Frequency Exposure?

Initially, when cell phones came out, their makers strongly denied any effectscell phone radiation (also known as radio frequency (RF) exposure) could possiblyhave on human health.

The article Mother who kept her phone in her bra every day for 10 years isconvinced it caused her terminal breast cancer in U.K.’s Daily Mail (9 September2015; https://goo.gl/qBLWm5) reports about the case of a 51-year old womanwho was diagnosed with an aggressive form of breast cancer. The woman believedthat her cancer was caused by the radiation from her phone.

What is cell phone radiation?We can find some information about RF exposure burried deep in our phones.

On iPhone 5, following the sequence Settings > General > About > Legal > RFExposure, we find the following:

This carefully crafted narrative (written, no doubt, by lawyers) seem to beimplying that it’s a good idea to keep our phone away from our body (and thereis definitely no need to keep it close to us when we sleep).

In Measurements of Radiofrequency Radiation with a Body-Borne Exposimeterin Swedish Schools with Wi-Fi published in “Frontiers of Public Health” [3], theauthors report on measuring the intenisty of RF radiation in school classroomsin Sweden. Introducing the topic, they write: “Lately, it has been discussed ifradiofrequency (RF) radiation can have long-term adverse effects on children’shealth” and later: “Exposure to RF radiation was classified as a possible humancarcinogen, Group 2B, by the International Agency for Research on Cancer (IARC)at WHO in 2011. The decision was mainly based on case-control human studies onuse of wireless phones by the Hardell group from Sweden and the IARC Interphonestudy, which showed increased risk for brain tumors, i.e., glioma and acousticneuroma” (see references in [3]).

To read more about RF exposure, and about some attempts at determiningwhether or not the RF radiation is harmful, follow the link (American CancerSociety) https://goo.gl/HLfVBp

In the article The inconvenient truth about cancer and mobile phones publishedby The Guardian (14 July 2018, https://bit.ly/2La9RD4) we read “We (average


citizens, op.a.) dismiss claims about mobiles being bad for our health but isthat because studies showing a link to cancer have been cast into doubt by theindustry?” However, the paper claims that “On 28 March this year, the scientificpeer review of a landmark United States government study concluded that thereis clear evidence that radiation from mobile phones causes cancer, specifically, aheart tissue cancer in rats that is too rare to be explained as random occurrence.”

Note: Relate the danger of RF exposure to the Internet of Things, which willmultiply the number of devices connected (wirelessly) to the internet.

Investigate What Did the Artist Want to Say?

In this painting, Belgian surrealist artist Rene Magritte drew a pipe. The textbelow the pipe says “This is not a pipe.”

Why is this not a pipe? If it is not a pipe, what is it?

Theorems and Causation

A theorem in mathematics is a statement that establishes a new relationshipbetween previously defined mathematical objects, relations or properties. In orderto be accepted, a theorem needs to be supported by (mathematically) acceptableevidence, i.e., a proof.

A theorem, applied to real life, suggests that we ask how and why (‘how’ issometimes answered by an algorithm).

The key word is evidence, and this is where mathematics differs from anyother discipline, or any other real-life situation. To be accepted, mathematicalproof must be based on previously established facts and results, and must followrigid rules of logical reasoning. In principle, a proof can be traced all the wayback to the beginning of a mathematics discipline, whose foundations consist offacts and statements that must be taken for granted (these statements are calledaxioms).

Are there axioms (i.e., statements whose validity we take at face value) in reallife? Perhaps religious beliefs (often qualified as ‘truth’), can be taken as axioms?For instance

What is acceptable evidence outside of mathematics?In most sciences, repeated experiments (with – as much as possible – identical

starting/initial conditions) that produce identical, or very similar outcomes, areaccepted as evidence. How many experiments? In general, the more, the better(however, easier said than done).


Evidence-based medicine (EBM) is an approach to evaluating medical prac-tices by requiring solid, strong evidence, based on well-conducted and well-designedresearch. (Although medicine is a very old discipline, EBM is a relatively new phe-nomenon, as it emerged in the 1960s).

EBM will support a strong recommendation for a certain practice if it comesfrom a randomized control trial (there are other possible designs, such as meta-analysis; read more about EBM on Wikipedia, https://goo.gl/TpAFkK.) A ran-domized control trial involves a statistical analysis of the (possible) differencesin the behaviour or characteristics between the treatment and the control groups.Meta-analysis combines results of multiple studies; systematic review is acomplete, exhaustive summary of current evidence, based on literature search andsecondary data.

A mathematician would not accept repeated experiments as evidence, unlessthey cover absolutely all possible cases, and all yield identical results. GoldbachConjecture states that every even number greater than 2 can be written as a sumof exactly two prime numbers. For instance, 4 = 2 + 2, 6 = 3 + 3, 18 = 13 + 5,24 = 11 + 13, and so on. Although the experiment of checking for this propertyhas been successfully performed for the first 2 billion billion even numbers (i.e.,repeated 2 · 1018 times) all with identical outcomes (i.e., even number = sum ofexactly two primes), this is not accepted as proof that the conjecture is true for alleven numbers. Formulated in 1742, Goldbach Conjecture has become the oldestand the most famous unsolved (as of today) mathematical problem.

Math theory cannot be built on conjectures, hypotheses or other statementswhich have not been rigorously proven. This is why the development of some maththeories “waits” until certain results are proven (for example, certain subareas ofnumber theory need the Riemann hypothesis, which has not been proved true (orfalse) yet).

Needless to say, in real life (physics, chemistry, etc.) this kind of evidence(billions of repeated experiments with identical outcomes) would definitely be ac-cepted, and, if appropriate, would be viewed as a solid foundation for a theory.

Here is an anecdote that illustrates the difference in the way we might thinkof acceptable evidence. Persons A, B, and C are on the train in Ireland, and seea cow in the field.

Person A says: “All cows in Ireland are black.” Person B says: “There is a blackcow in Ireland.” Person C says: “There is a cow in Ireland one of whose sides isblack.” Which one is a mathematician?

Repeated experiments are not always possible. For instance, cosmology (studyof our universe) is based on one experiment, i.e., our Universe, as we do not knowof (i.e., are not aware of) any other universes.

When exactly repeated experiments yielding similar outcomes are accepted assignificant is formalized by a statistical approach which we will discuss later.


Case study How Much Water?

We have all heard about how drinking 6 to 8 glasses of water daily benefitsour health and well-being. What evidence supports this claim?

The article 8 glasses of water a day ‘an urban myth’ published by CBC News(10 June 2012, https://bit.ly/1AI1j9I) re-states the view that drinking water isbeneficial. However, it seems that this particular quantity of 6-8 glasses per dayhas no scientific evidence. The article provides a more appropriate, common-senseadvice - listen to your body! When you’re thirsty, drink some water, when youare not - don’t!

The article What drove us to drink 2 litres of water a day? published inAustralian and New Zealand Journal of Public Health (https://bit.ly/2PEXwGj)gives a revealing historic background on the 6-8 glasses of water suggestion, goingback to 19th century advice for healthy living. As well, it emphasizes that there isa difference between thirst and dehydration. (Note: know the definitions! Thirstand dehydration are two different things; so are sadness and depression.) We read“In today’s western society there is an accepted popular view that the moment onefeels thirsty, one is dehydrated. Consequently, the only way to avoid this highrisksituation is to consume copious amounts of water. Supporters of this view believeconsuming beverages other than water will only lead to further dehydration. ”

The article concludes with “Water is important for health; however, the rec-ommendation of 8 glasses of pure water per day appears an overestimation ofrequirements. All fluids are important in meeting requirements and water shouldnot be singled out. We should be educating the general public that beverages liketea and coffee, despite their caffeine content, do not lead to dehydration and willcontribute to a person’s fluid needs, something worth considering when discussingfluid requirements.”

Sinister forces? In The truth about sports drinks (New Zealand Herald, 16November 2016, https://bit.ly/2f4iJGa) we read “One of industry’s greatest suc-cesses was to pass off the idea that the body’s natural thirst system is not a perfectmechanism for detecting and responding to dehydration [Note: mixing up thirstwith dehydration!] These include claims that: ‘The human thirst mechanism is aninaccurate short-term indicator of fluid needs ... Unfortunately, there is no clearphysiological signal that dehydration is occurring.’ ” Consequently, “healthcareorganisations routinely give advice to ignore your natural thirst mechanism. Dia-betes UK, for example, advises: ‘Drink small amounts frequently, even if you arenot thirsty – approximately 150 ml of fluid every 15 minutes – because dehydrationdramatically affects performance.’ ”

People tend to exaggerate: one can have serious consequences by consum-ing too much water. We learn of one such case in Hiker Fatality From Se-vere Hyponatremia in Grand Canyon National Park published in Wilderness andEnvironmental Medicine (September 2015, Volume 26, Issue 3, Pages 371-374,https://bit.ly/2wmZv9u. The authors, T. M. Myers and M. D. Hoffman, write:“We present the case of a hiker who died of severe hyponatremia at Grand CanyonNational Park. The woman collapsed on the rim shortly after finishing a 5-hourhike into the Canyon during which she was reported to have consumed large quan-tities of water.”

Furthermore, in Drinking too much water can be fatal to athletes published inScience Daily (14 Sept 2014, https://bit.ly/2Lu8Txw) we read “The recent deathsof two high school football players illustrate the dangers of drinking too much waterand sports drinks, according to Loyola University Medical Center sports medicinephysician Dr. James Winger.” In particular, “Over-hydration by athletes is calledexercise-associated hyponatremia. It occurs when athletes drink even when theyare not thirsty. Drinking too much during exercise can overwhelm the body’sability to remove water. The sodium content of blood is diluted to abnormallylow levels. Cells absorb excess water, which can cause swelling – most dangerously


in the brain.” This is a serious condition, as hyponatremia “can cause musclecramps, nausea, vomiting, seizures, unconsciousness, and, in rare cases, death.”

The article continues “Georgia football player Zyrees Oliver reportedly drank2 gallons of water and 2 gallons of a sports drink. He collapsed at home afterfootball practice, and died later at a hospital. In Mississippi, Walker Wilbankwas taken to the hospital during the second half of a game after vomiting andcomplaining of a leg cramp. He had a seizure in the emergency room and laterdied. A doctor confirmed he had exercise-associated hyponatremia.”

Case study Sports drinks

What kind of evidence is there?In Sports drinks are selling consumers a myth that is slowing them down (Busi-

ness Insider, 1 November 2016, https://read.bi/2MwelVW) we read “From eightglasses of water a day to protein shakes, we’re bombarded with messages about[what] we should drink and when, especially during exercise. But these drinkingdogmas are relatively new. For example, in the 1970s, marathon runners were dis-couraged from drinking fluids for fear that it would slow them down. Now we’reobsessed with staying hydrated when we exercise, not just with water but withspecialist drinks that claim to do a better job of preventing dehydration and evenimprove athletic performance.”

The authors claim that the reason “behind this huge rise in sports drinks liesin the coupling of science with creative marketing. An investigation by the BritishMedical Journal has found that drinks companies started sponsoring scientists tocarry out research on hydration, which spawned a whole new area of science.”

As well, “These same scientists advise influential sports medicine organisa-tions, developing guidelines that have filtered down to health advice from bodiessuch as the European Food Safety Authority and the International Olympic Com-mittee. Such advice has helped spread fear about the dangers of dehydration.”

Punchline: “One of industry’s greatest successes was to pass off the idea thatthe body’s natural thirst system is not a perfect mechanism for detecting andresponding to dehydration.”

Is there evidence? We read “Many of the claims about sports drinks are oftenrepeated without reference to any evidence. A British Medical Journal reviewscreened 1,035 web pages on sports drinks and identified 431 claims they enhancedathletic performance for a total of 104 different products. More than half the sitesdid not provide any references - and of the references that were given, they wereunable to systematically identify strengths and weaknesses. Of the remaining half,84% referred to studies judged to be at high risk of bias, only three were judgedhigh quality and none referred to systematic reviews, which give the strongestform of evidence.”

However, “the current evidence is not good enough to inform the public aboutthe benefits and harms of sports products. What we can be almost sure about isthat sports drinks are not helping turn casual runners into Olympic athletes. Infact, if they avoided these sugar-laden drinks they would probably be slimmer andfaster.”

Investigate Examples of Evidence and “Evidence”

(1) [Coincidence; Ear infections in children] The article The new ear infectionrules published at https://bit.ly/2dZlgG7 discusses the effectiveness of antibioticsin the treatment of ear infections in children. “Evidence” that antibiotics work issometimes based on coincidence – untreated, the infection would disappear on its


own. Thus the new ear infection rule, in many cases, suggests: give no antibiotics,just wait!

So, to provide good evidence we must: show that intervention (antibiotic)indeed has an effect (ear infection gone) AND show that when there is no inter-vention (no antibiotic given), there is no effect (ear infection persists).(2) [Personal bias] The article Europeans greatly overestimate Muslim population,poll shows (The Guardian, 13 December 2016, https://bit.ly/2hKOmGy) showshow personal bias affects one’s understanding of data. So, when we poll people onall kinds of things, are we collecting good, accurate evidence?(3) [Sounds good] Just because it sounds good, it does not have to be good. In Self-help ’makes you feel worse’ (BBC News, 3 July 2009, https://bbc.in/2BY4GCJ)we read “Canadian researchers found those with low self-esteem actually felt worseafter repeating positive statements about themselves. They said phrases such as’I am a lovable person’ only helped people with high self-esteem.”(4) [Authority, celebrity] Evidence coming from authority, celebrity, popular cul-ture, media influencers. The case of health and wellness strategy called GOOP:Dont blame Gwyneth Paltrow (MacLean’s 30 June 2017, https://bit.ly/2oocS5v).There is more: Gwyneth Paltrow’s Goop expanding to Canada - and some medicalexperts aren’t happy (Global News, 24 August 2018, https://bit.ly/2NxPr4H).(5) [Evidence exists, yet not trusted] Evidence against using BMI (body mass in-dex): The risks of a quick fix: a case against mandatory body mass index reportinglaws. (Eat Disord. 2008 Jan-Feb; 16(1):2-13. doi: 10.1080/10640260701771664.https://bit.ly/2LsLvAr) From the abstract, we learn “As the United States ad-dresses obesity, a number of state legislatures are considering laws that requireschools to track and report students’ body mass index (BMI), a measurement ofbody weight (weight/height squared). This article describes the state level activ-ity on mandatory BMI reporting, offers numerous arguments against this practice,and suggests an alternative approach to promoting health in youth. MandatoryBMI reporting laws place a new and inappropriate responsibility on the schools.Proponents of such laws imply that BMI reporting will have positive outcomes,yet there is virtually no independent research to support this assumption. Theauthors argue that these laws could do significant harm, including an increasedrisk for children to develop eating disorder symptoms.”

Related: School: 66-pound girl is ’overweight’ report says “A mom is outragedwhen her 9-year-old daughter is classified as overweight by a school health programin New York” (CNN News, https://cnn.it/TNfjO7).(6) The Guardian, 3 January 2018, published the article Watchdog bans advert’sclaim eHarmony is ‘scientifically proven’ in which they report that “The advertis-ing regulator has banned the online dating service eHarmony from claiming it hasa ‘scientifically proven matching system’. Upholding a complaint about a billboardad on London Underground, the Advertising Standards Authority (ASA) said theclaim was misleading because eHarmony could not prove its service provided agreater chance of finding lasting love.” Check https://bit.ly/2M6do5i.

Causation and Implication

The logical structure of a theorem is given by implication (outside of math, moreoften referred to as causation).

In narrative form (symbols A and B represent two statements), an implicationcan be expressed in many different forms, such as:

“if A then B”“assume A, then B (follows)”


“A causes B”“B is caused by A”“B follows from A”“B is a result of A”“B is a consequence of A”“in case A, then B”“A is/are B”“if assumption(s) A then conclusion(s) B”

For instance, in the statement “A cat is a mammal”, ‘cat’ is the assumption (or thecause) and ‘mammal’ is the conclusion (or the effect). In a formal math language,we write “if cat then mammal,” or (using A ⇒ B to symbolize the implication “ifA then B”), we write cat ⇒ mammal.

Another example: the statement “Shortness of breath is a symptom of anallergy” establishes allergy as the cause of the shortness of breath. Thus, we write“if allergy then shortness of breath,” or allergy ⇒ shortness of breath.

Remember that “medical condition (disease) implies symptoms”, i.e., medicalcondition (disease) ⇒ symptoms.

An implication “if A then B” can be visualized using so-called Venn diagrams:

A

B

cats

mammals

We think of these as boxes: whatever is in A (smaller box) must be in B(larger box), since it contains the smaller box. The implication cat ⇒ mammal isillustrated in the picture on the right: a cat are in the smaller box (where all catsare), which is contained in a larger box, where all mammals are. This diagramsuggests that not all mammals are cats, as there are animals in the larger box(dogs, rats, elephants, etc.) which are not in the smaller box. Thus, we concludethat the implication mammal ⇒ cat is not true. (More about this soon.)

Likewise, other interpretations of the implication (causation) A ⇒ B can bevisualized as well:

A

B

cause

effect symptoms

disease

How do we use, i.e., correctly interpret an implication (causation) A ⇒ B?We need to check that all assumptions hold (i.e., check that everything listed

under A holds). If so, then the conclusion (or conclusions) hold(s). If one or moreassumptions do not hold, then the implication cannot be used. However, what isclaimed (conclusion/effect) may or may not be true.

Going back to the implication cat ⇒ mammal – we hold an animal in ourhands, and verify that it is a cat. Thus, we conclude (from the given implication)that it is a mammal. If we hold a raccoon in our hands, we cannot apply theimplication (a raccoon is not a cat, so the assumption is not true). However, theconclusion (raccoon is a mammal) is true. Now imagine that we hold a spider inour hands. The assumption is not true (a spider is not a cat), so we cannot use


the implication. In this case, the conclusion does not hold either (a spider is nota mammal).

Important to remember: cause implies effect, but the absence of the causedoes not say anything about the effect. Another illustration: strong earthquakecauses destruction. However, in the absence of a strong earthquake there could beno destruction, or destruction could occur due to other causes (such as tsunamior war or a medium-strength earthquake).

From an implication A ⇒ B, we can derive a converse statement, B ⇒ A (inwords, we swap cause and effect).

There is no relation between the truthfullness of a statement and its converse.For instance, we know that prolonged UV exposure causes sunburn and skin dam-age (written more formally, prolonged UV exposure ⇒ sunburn and skin damage).However, sunburn and skin damage do not cause prolonged UV exposure.

Or: the converse of “If the last digit of a number is 6, then it is even” is thestatement “If a number is even then its last digit is 6” which is not true (as aneven number could end in 0, 2, 4, or 8 as well). “A cat is a mammal” is true, butits converse “A mammal is a cat” is not.

It could happen that the converse of a true statement is also true. For instance,for a student at McMaster University, both “If GPA is 12.0, then A+ in all courses”and its converse “If A+ in all courses, then GPA is 12.0” are true.

If both A ⇒ B and B ⇒ A are true, we say that A and B are equivalent,and write A ⇔ B.

Example 2.1 Equivalent Statements

In the opinion piece Smash the Wellness Industry published in New York Times (8June 2019, at https://nyti.ms/2F1MYgZ) U.S. novelist Jessica Knoll writes “Thiswas before I could recognize wellness culture for what it was - a dangerous conthat seduces smart women with pseudoscientific claims of increasing energy, re-ducing inflammation, lowering the risk of cancer and healing skin, gut and fertilityproblems. But at its core, ‘wellness’ is about weight loss. It demonizes caloricallydense and delicious foods, preserving a vicious fallacy: Thin is healthy and healthyis thin.”

We recognize the vicious fallacy at the end of the paragraph as the equivalencehealthy ⇔ thin.

Can you find more examples of equivalent statements? (Note: not at all easyin real life; it’s easier if you think of abstract concepts, such as in math.)

From any statement A we can form its negative notA (often denoted by ¬A).If A=“cat,” then notA=“not a cat,” i.e., anything but a cat (could be a mouse,cruse ship, happiness, math test, universe, etc.)

Another useful modification of an implication A ⇒ B is a contrapositivestatement notB ⇒ notA. A statement and its contrapositive have the same truthvalue, i.e., they are either both true or both false.

For instance, the contrapositive of “If cat then mammal” is “If not mammalthen not cat.” Or, the contrapositive of “If you drive, then do not drink” is “Ifyou drink, then do not drive.” Makes sense!

Using Venn diagrams, we can visualise contrapositive statements:


A

B

A

B

implication contrapositive

The diagram on the left represents A ⇒ B: the brown dot is in box A, and asbox A is in the larger box (box B), the brown dot is also in box B.

The diagram on the right: the brown dot is not in the larger box (so notB),and as the larger box contains the smaller box, the brown dot cannot be in thesmaller box (thus notA). In other words, notB ⇒ notA.

In real life, there are very few (if any?) “mathematically strong” implications(meaning that they indeed apply to all cases).

Mathematically speaking, the implication meningitis (disease) ⇒ nausea, vom-iting, and headaches (symptoms) means that everyone who has meningitis experi-ences these symptoms. In reality, however, it could be that someone has meningitisbut experiences none of the three symptoms (i.e., likely has other symptoms, orno symptoms at all).

The implication “Magnitude 5 earthquake causes destruction of property and/or life” is true in most cases, but not always. UV exposure causes skin damagein most people, but likely there is a person who is not negatively affected by UVexposure. Researchers have figured out that reversing the habit of eating breakfast(either from having breakfast to not having it, or from not having breakfast tohaving it) leads to initial weight loss (but again, not in every person).

Thus, in real life, an implication (causation) often comes with a qualifier, suchas “very likely,” “probably,” “quite often,” and so on; i.e., instead of A impliesB, we find statements such as “A might imply B,” “A probably implies B,” “Aimplies B with 75% chance,” etc. Nevertheless, the reasoning that comes with ourunderstanding of how implications work is very useful.

A group of friends decides that “If the weather is nice, then we go for a hike”.However, the weather turned not to be nice, so they should not go for a hike. Isthis a logically sound conclusion?

Example 2.2 Math Implication vs Real Life

The message on this street sign could be interpreted as the implication “If it isbetween 8am and 6pm, then I am not allowed to park.”

What is our common understanding of the remaining time interval, from 6pmto 8am? How does it differ from a mathematician’s interpretation?


Correlation

Cause and effect is only one possible relationship between two things. By ‘things’we mean objects, statements, events, variables, measurements, etc., for instancewhatever A and B in A ⇒ B represent.

A correlation is a mutual connection, or a relationship between two or morethings. (Later, we will see that there is a way to quantify the strength of thatrelationship). The easiest way to understand what correlation is is to look atexamples.

In the article How the smartphone affected an entire generation of kids pub-lished in MacLeans (22 August 2017, at https://bit.ly/2okfP7i Jean M. Twenge,Professor of Psychology at San Diego State University, writes “I wondered if thesetrends - changes in how teens were spending their free time and their deterioratingmental health - might be connected. Sure enough, I found that teens who spendmore time on screens are less happy and more depressed, and those who spendmore time with friends in person are happier and less depressed.”

Note that Prof. Twenge did not use the word ‘cause’, but instead ‘connection/connected’. So, there is a relationship between the way teens spend their free timeand their mental health, but there is no evidence of causality. And she is careful topoint that out: “Of course, correlation doesnt prove causation: Maybe unhappypeople use screen devices more.”

This last sentence is important, so we repeat it – Correlation is weaker thancausation, and in many cases correlation does not imply causation.

Case Study 2.3 Screen Time, Smart Phones and Mental Health

In Limiting children’s screen time linked to better cognition (27 September 2018,at https://bbc.in/2OXS6p7), BBC News Health Reporter Alex Therrien writes“Children aged eight to 11 who used screens for fun for less than two hours a dayperformed better in tests of mental ability, a study found. Combining this withnine to 11 hours of sleep a night was found to be best for performance. Researcherssaid more work was now needed to better understand the effects of different typesof screen use. However, they acknowledge that their observational study showsonly an association between screen time and cognition and cannot prove a causallink. And it did not look at how children were using their screen time, be it towatch television, play videogames or use social media.”

Note how the author used the word ‘linked’, and points out that more researchis needed to figure out the effects (thus, to establish a causation) of different typesof screen use.

In Social media, but not video games, linked to depression in teens, accord-ing to Montreal study, Kate McKenna, a CBC News reporter (15 July 2019, athttps://bit.ly/2JynMkE), opens by saying that the “study [to be reported on]examined mental-health implications of high levels of screen time.”

Read the article to determine if ‘implication’ is meant in a mathematical sense,i.e., if the study refers to causation or not.

The first three paragraphs are: “Screen time - and social media in particular- is linked to an increase in depressive symptoms in teenagers, according to a newstudy by researchers at Montreal’s Sainte-Justine Hospital.

The research team, led by Patricia Conrod, investigated the relationship be-tween depression and exposure to different forms of screen time in adolescents overa four-year period.


‘What we found over and over was that the effects of social media were muchlarger than any of the other effects for the other types of digital screen time,’ saidConrod, a professor of psychiatry at the University of Montreal.”

Identify all words in these paragraphs that refer to a link between depressionand exposure to screens. Do these words suggest correlation or causation?

In an opinion piece by a psychology professor Dr. Dennis-Tiwary Taking Awaythe Phones Wont Solve Our Teenagers Problems published in New York Times (14July 2018, at https://nyti.ms/2KUP47c) we read that “Although some researchdoes show that excessive and compulsive smartphone use is correlated with anxietyand depression, there is a lack of direct evidence that devices actually cause mentalhealth problems.”

Does this is any way contradict the conclusions of the previous two articles?What can you conclude after reading these three pieces?

Example 2.4 Causation or Correlation?

In each case, figure out whether the paragraphs quoted are about causation(s) orcorrelation(s), and identify them (i.e., state which variables are related). Thenread the entire article to find (if any) further causations or correlations.

(1) In the article Bad trip from smoking pot? It could be a sign of mental illnesspublished in Toronto Star (21 July 2018, at https://bit.ly/3ljpBoK), we read: “Inparticular, research in Denmark has discovered heavy cannabis users are substan-tially associated with the development of schizophrenia and bipolarism. In fact,of those who were hospitalized with a pot-related mental condition, almost 50per cent were diagnosed with schizophrenia or bipolarism later on in life. Therisks increase the younger a person starts using. Experts have not yet determinedwhether cannabis causes schizophrenia or bi-polar disorder, or whether it simplytriggers a first psychotic episode.”

(2) WebMD article Child Leukemia Again Linked to Power Lines (2 June 2005,https://wb.md/2MEetmv) claims that “Living near high-voltage power lines raiseschildren’s risk of leukemia by 69%, a British study shows. That doesn’t prove thatpower lines cause the deadly blood cancer, the study’s authors are quick to pointout. Despite 30 years of research, scientists still can’t come up with a plausiblereason why the weak magnetic fields near power lines might cause leukemia.”

(3) The Conversation (Academic rigour, journalistic flair) of June 17, 2019 pub-lished an analysis of teenage behaviours titled Teenage sexting linked to increasedsexual behaviour, drug use and poor mental health https://bit.ly/2Yetlga We read:“While sexting is linked to sexual behaviour and mental health factors, correla-tional studies do not provide evidence to suggest that sexting is in any way thecause of risky behaviour or poor mental health.”

Example 2.5 Does Smoking Cause Cancer?

Lung cancer information (Mayo Clinic, https://mayocl.in/2C1nfBl) states that“Smoking causes the majority of lung cancers - both in smokers and in peopleexposed to secondhand smoke. But lung cancer also occurs in people who neversmoked and in those who never had prolonged exposure to secondhand smoke. Inthese cases, there may be no clear cause of lung cancer.”

Think about what kind of experiment would convince people that smokingactually causes cancer. Would such an experiment be feasible, allowed? If not,then why do we believe that smoking causes cancer, isn’t it just a correlation?


In many situations it might make sense to describe a correlation more precisely.If an increase in A is followed by an increase in B, or a decrease in A is followedby a decrease in B, then A and B are said to be positively correlated. Forinstance, exposure to screens (phone, laptop, TV) has been found to be positivelycorrelated with depression.

If A increases as B decreases and A decreases as B increases, then A and Bare said to be negatively correlated. Example: the number of trees on a hill isnegatively correlated to the chance of a landslide.

In the case the variables are presented as graphs, we could easily spot the signof a correlation:

positively correlated negatively correlated

Note that the blue graph follows the trend of the brown graph: it increasesroughly where the brown graph increases, and decreases where the brown graphdecreases. The green graph does the opposite: it decreases roughly where thebrown graph increases, and increases where the brown graph decreases.

Examine the three pairs of variables (A and B, A and C, and B and C) givenin this table for the sign (positive or negative) of their correlation:

Example 2.6 Positive or Negative Correlation?

Identify each case as a positive or negative correlation.(a) Car speed and travel time to a destination(b) Air temperature and ice cream sales(c) Amount of food eaten and hunger(d) Salary and spending(e) Cigarette smoking and years of life remaining(f) Snowfall and the number of people driving(g) Hair length and shampoo use.

How do correlations occur? For instance, A and B can be correlated because:A implies B or B implies A (so causation implies correlation)A is equivalent to B

A third thing C causes both A and B

A implies C and C implies B

Coincidence (no connection between A and B)In the following exercise we explore examples of these situations.


Example 2.7 Correlations

Looking at the above list, identify how each of these are correlated:(a) Children with larger feet spell better than children with smaller feet(b) The more gasoline we put in our car, the more the car loses in its resale

value(c) As a student’s study time increases, they are happier(d) On a sunny day, UV index increases as the temperature increases(e) As we climb higher and higher, the air temperature decreases(f) The number of fire engines at a site and the extent of the damage to the

property(g) The more margarine people consume, the higher the divirce rate in the

U.S. state of Maine:

(Source: Spurious correlations, http://www.tylervigen.com).(h) As the consumption of ice cream increases, so does the number of drowning

deaths(i) As the demand for a product increases, so does its price.

Axioms

If a fact (formula, statement, property) in mathematics depends on previouslyestablished facts, and these facts depend on previously established facts, wheredoes it all start?

It all starts with axioms – every theory in mathematics is built from a set ofstatements whose validity is taken for granted, i.e., they cannot be proven to betrue.

For instance, the fact that x + y = y + x for all real numbers is an axiom.Of course, we can convince ourselves that this statement is true by examiningexamples: 3 + 4 = 7, and since 4 + 3 = 7 we conclude that 3 + 4 = 4 + 3; etc. Butthe axiom states that x + y = y + x is true for all (infinitely many) choices forx and y. An axiom tells us that multiplication by 1 does not change the number(x ·1 = x); nor does a number change if we add zero to it (x+0 = x). In Euclideangeometry (i.e., the geometry that we learn at school and which applies to our dailylives), an axiom states that a line is determined by two points. Check Wikipedia(under Historic Development) for a complete list of axioms for Euclidean geometry.

As well, we need to have a “starting” object in math, as math does not allowfor circular definitions (i.e., we cannot define A in terms of B, and then B in termsof A). Usually that “starting” object is a set, i.e., we do not define what a set is.We determine a set by listing its elements (for instance, we can consider the set ofall people who live in Canada today).


In real life, a good way to characterize axioms is to say that they representa set of beliefs (and that philosophy and religion are well-positioned to determinewhat these axioms could be). One way or another, axioms (even in mathematics)are statements (facts) that we decide to believe in (thus, every mathematician andevery scientist must be a believer!).

Other disciplines have axioms, as well. For instance in astronomy and cosmol-ogy, we assume (i.e., take for granted, believe) that the same laws of physics applyat all locations in universe (except at singularities, such as black holes). As well,we assume that the Universe is homogeneous, i.e., that it looks about the same inall directions.

Axioms change with new discoveries: in physics, Newton’s axioms were re-placed by the axioms of Einstein’s special relativity, and then those were replacedby the axioms of general relativity.

Are there axioms in biology? What would, or what do they look like? Whatabout axioms in social sciences? Philosophy? What is the (original) meaning ofthe word “dogma”?

Algorithms and Formulas

Algorithms and formulas give us means of calculating quantities that we areinterested in. For instance to compute someone’s body mass index (BMI), wemeasure their height h (in metres) and mass m (in kg) and then use the formula

BMI =mass

height2=

m

h2

For instance, a person of mass m = 68 kg and height h = 1.7 m has the body massindex of

BMI =m

h2=

681.72

≈ 23.53

The symbol ≈ denotes the fact that the value of the fraction in not exactly 23.53;instead the value was rounded off to two decimal places. (How we round off anumber depends on the context; more about it later).

The formula

a =a1 + a2 + a3 + · · · + an

n

calculates the average value (the mean) of the numbers a1, a2, a3, . . . , an. Thus, iffour houses in our neighbourhood were sold for $ 376 000, $ 412 000, $ 398 000, and$ 564 000, then the average price (denote if by P ) of a house sold is

P =376 000 + 412 000 + 398 000 + 564 000

4= $ 437 500

We will meet many formulas in this textbook. An example of an algorithm isprovided in your course outline: it tells you how your course grade is calculated.

Case study Data Sets and Algorithms

One of a major strengths of algorithms these days is their ability to extractpatterns from large data sets. Often referred to as AI (artificial intelligence),algorithms are aggressively present in our lives, sifting through huge, deeply con-cerning amounts of data collected, legally or illegally (through our use of certainweb pages, smart phones, etc.) about every aspect of our lives.

On 16 February 2012, Forbes magazine (https://goo.gl/46Lf2X) publishedthe article How Target Figured Out A Teen Girl Was Pregnant Before Her FatherDid. We read “Every time you go shopping, you share intimate details about yourconsumption patterns with retailers. And many of those retailers are studying


those details to figure out what you like, what you need, and which coupons aremost likely to make you happy.”

Target’s statistician Andrew Pole discusses how Target uses customer data tolearn about them, their needs, spending patterns, and so on. For instance Pole andhis analysis looked at historic buying data of women who signed up for Target babyregistries and “before long some useful patterns emerged. Lotions, for example.Lots of people buy lotion, but one of Pole’s colleagues noticed that women onthe baby registry were buying larger quantities of unscented lotion around thebeginning of their second trimester. Another analyst noted that sometime in thefirst 20 weeks, pregnant women loaded up on supplements like calcium, magnesiumand zinc. Many shoppers purchase soap and cotton balls, but when someonesuddenly starts buying lots of scent-free soap and extra-big bags of cotton balls, inaddition to hand sanitizers and washcloths, it signals they could be getting closeto their delivery date.”

And now the scary part: “As Pole’s computers crawled through the data, hewas able to identify about 25 products that, when analyzed together, allowed himto assign each shopper a ‘pregnancy prediction’ score. More important, he couldalso estimate her due date to within a small window, so Target could send couponstimed to very specific stages of her pregnancy.”

Note: the words algebra and algorithm come from the same person: Abu AbdAllah Muhammad Ibn Musa al-Khwarizmi (ca 780- 850 CE), Persian mathemati-cian, astronomer, astrologer, geographer and author, who worked in Baghdad.Al-Khwarizmi wrote the famous Book of Completion and Balancing in which hedefines algebra (‘al-jabr’ means subtracting a quantity from one side of the equa-tion and adding it to another). The word algorithm is derived from the latinizationof Al-Khwarizmi’s name, Algoritmi.

Investigate Formulas

What is wind chill, and what measurements are used to calculate its values?What about humidex? What is UV index?

In the article This Is How Many Calories You Burn While Walking (source:https://bit.ly/2EhUG3i, 15 March 2016) we read “Once they had identified thecomponents needed for an accurate prediction, the researchers created a new equa-tion that accounted for both height and body mass. To put it to use, measurevelocity (distance divided by time) in meters per second, and height in meters:”

Kcal means kilocalories; so 1 Kcal = 1000 calories, but not when food is concerned:in that case, 1 Kcal = 1 (food) calorie. Be aware when travelling:

.


When we write the formula using math symbols, we need to make sure toidentify all quantities that are used, and their units (for instance, if we use B forbody mass, then we say that B represents a person’s body mass in kg, etc.)

The formula below gives the energy expenditure when we run (of course, it isan approximation)

Calories per minute = 0.035B + 0.029BV 2

Hwhere B is the body mass in kilograms, V represents the velocity in metres persecond and H is the height in metres.

The above article states: “Height essentially economizes walking, so the talleryou are, the less energy it takes per pound to walk a mile, or the slower the rateyou burn calories while walking at the same speed compared to someone who isshorter.” How is this reflected in the formula?

Note: Reasoning about fractionsAB if A increases and B does not change, then the fraction increasesAB if A decreases and B does not change, then the fraction decreasesAB if A does not change and B decreases, then the fraction increasesAB if A does not change and B increases, then the fraction decreases

Investigate Algorithms

(1) The article When technology discriminates: How algorithmic bias can makean impact published by CBC News on 10 August 2017, https://bit.ly/2vWUbvz,states that “Research has shown that algorithms can actually perpetuate–evenaccentuate–social inequality.” Explain what is the main concern, related to howalgorithms work, presented in this article.

What was the scandal involving Facebook and Cambridge Analytica in 2018about? You can start your investigation by reading https://bit.ly/2pCFpEJ(2) ‘It’s ridiculous. It’s Picasso’: Facebook reviewing anti-nudity policy after block-ing Montreal museum ad (CBC News, 2 August 2018, https://bit.ly/2OFhwrS)Montreal’s fine arts museum complained after Facebook blocked ads featuring cu-bist nude paintings. Here is the offending painting:

In Here’s Why Facebook Removing That Vietnam War Photo Is So Important(fortune.com, 9 September 2016 https://for.tn/2c5mu2A) we read: “In the latestcontroversy involving the giant social networks news judgement, Facebook (fb,+0.40%) removed an iconic photo from the Vietnam War: A picture of a young


Kim Phuc running naked down a road after her village was hit by napalm.” Aswell: “When a Norwegian newspaper editor–who posted the photo as part of aseries on war photography–tried to re-post it, along with a response from Phucherself, his account was suspended.”

In New York Times article of 9 September 2016 (https://nyti.ms/2AzgXNg)Facebook Restores Iconic Vietnam War Photo It Censored for Nudity we read “Theimage is iconic: A naked, 9-year-old girl fleeing napalm bombs during the VietnamWar, tears streaming down her face. The picture from 1972, which went on to winthe Pulitzer Prize for spot news photography, has since been used countless timesto illustrate the horrors of modern warfare.”

We read: “But for Facebook, the image of the girl, Phan Thi Kim Phuc, wasone that violated its standards about nudity on the social network. So after aNorwegian author posted images about the terror of war with the photo to Face-book, the company removed it.” And finally, “The move triggered a backlash overhow Facebook was censoring images. When a Norwegian newspaper, Aftenposten,cried foul over the takedown of the picture, thousands of people globally respondedon Friday with an act of virtual civil disobedience by posting the image of Ms.Phuc on their Facebook pages and, in some cases, daring the company to act.Hours after the pushback, Facebook reinstated the photo across its site.”(3) In Heat health risk prompts change to Environment Canada warning system(CBC News, Jul 27, 2018, https://bit.ly/2JZxWrF) we learn that the heat warn-ings algorithm needs to be adjusted by regions (the article also mentions humidex,which is another formula/algorithm). How to define a heat wave is important, aspeople could (and do) die from it.(4) Algorithms we know from math: how to solve an equation (“move termsfrom one side to another”) finding extreme values (or peaks, tipping points) of afunction, and so on.

To learn more about algorithms and how they work (or don’t), read the articleWhen algorithms go bad: Online failures show humans are still needed publishedby CBC News (1 October 2017, at https://bit.ly/2M3jFyF).

Highlights: “We’ve grown ‘dependent on algorithms to deliver relevant searchresults, the ability to intuit news stories or entertainment we might like,’ saysMichael Geist, a professor at University of Ottawa and Canada Research Chair ininternet and e-commerce law.


These formulas, or automated rule sets, have also become essential in manag-ing the sheer quantity of posts, content and users, as platforms like Facebook andAmazon have grown to mammoth global scales. In the case of Amazon, whichhas over 300 million product pages on its U.S. site alone, algorithms are necessaryto monitor and update recommendations effectively, because it’s just too muchcontent for humans to process, and stay on top of, on a daily basis.

But as Geist notes, the lack of transparency associated with these algorithmscan lead to the problematic scenarios we’re witnessing.”

Math Language and Real Life Language

Sometimes it becomes necessary to distinguish the language used in mathematics(we will refer to it as ‘math language’) and the language that we use in everydaysituations (‘English language’ or ‘everyday language’).

Since we want everyone who reads or listens to the same piece of math toreceive the same message, we have to make sure that math language is clear,precise and unambiguous. As well, the context — which could be explicitly stated,or suggested but not explicitly stated — needs to be made clear.

The sentence “The sum of two odd numbers is an even number” is an exampleof a math statement. Previous to this statement, concepts of the sum, as wellas even and odd numbers were defined. So, the statement above establishes onepossible relationship between known concepts and/or known objects. From thedevelopment of the material, it is clear (but not specified in the sentence above)that we are talking about integers.

In everyday language, we use words whose meaning is often vague, unclear ordepends on our experience. For instance, the word “warm” in “It is warm outside”does not have the same meaning for everyone. Thus, “It is warm outside” is nota math statement. A few more examples contrast the two languages:

Math Language English (everyday) Language

14 is larger than 2. The roof is kind of red.

... is equal to 80 About 80 people showed up

This is a square. This car is cheap.

... the chance is 45% It might rain today

... the tax is 13% ... somewhat larger

We are not saying that everyday language is not informative, or makes nosense. As a matter of fact, within a certain context, most people might agreewith a statement (such as “This car is cheap”). As well, we know that “about 80people” probably means 78 or 83 people, but not 47 or 150.

In our discussion of probability we will see that “the chance is 45%” can begiven a precise mathematical meaning, although it involves randomness.

In certain situations, we use math terms but not in their precise mathematicalsense. For instance, “a square of chocolate” is not a square. To say “Earth is asphere” is not correct: Earth is a three-dimensional solid, whereas a sphere is atwo-dimensional surface. When we say “area of a circle” (many school textbooksuse this phrase) we mean the area of the region (i.e., the disk) enclosed by thecircle. The circle is a curve and so has length, but not area.

Note: We need to be careful when using the word “between”. How many wholenumbers (integers) are there between 2 and 8? If we subtract, we get 8 − 2 = 6which may or may not make sense.


If “between” 2 and 8 means “between and including” 2 and 8, then the answeris 7. However, If “between” 2 and 8 means “between but not including” 2 and 8,then the answer is 5.

Quantifiers and Rules of Formal Logic

When we want to specify the quantity of things or objects which possess certainproperty, we use quantifiers. Two commonly used quantifiers are the univer-sal quantifier, meaning “for all,” or “any,” or “for every,” and the existentialquantifier, meaning “for some,” or “there are,” or “there exists.”

In real life, we often use variations (relaxed versions) of these, such as “almostall,” “many,” “a few,” “barely any,” and so on. However, it is not really possibleto develop formal logic with these quantifiers. (Actually there have been attemptsat it, but that’s not relevant for what we’re trying to do here.)

Note: in working with logical statements, we focus on their structure - we willnot worry much about whether or not we are writing good (correct style, grammar,etc.) English sentences.

To state that every single cat in Hamilton is black, we use the universal quanti-fier, and say “All cats in Hamilton are black” or “Every cat in Hamilton is black.”The math symbol for the universal quantifier is ∀. Denoting cats by x and theproperty of being black by A, we write ∀xA (might look weird, but is useful!).

The sentences “There is a black cat in Hamilton” or “There exists a blackcat in Hamilton” use the existential quantifier, and mean that there is at leastone black cat in Hamilton (of course, there could be two, or 10 or 295 black catsin Hamilton). Math symbol for the existential quantifier is ∃. Using the notationjust introduced, we write ∃xA (where, as before x represents cat(s) and A is theproperty of being black).

How do we work with statements that involve universal and existential quan-tifiers?

Consider the statement involving the universal quantifier “All cats in Hamiltonare black.” If we wish to prove that this statement is true, then finding one cat,or two or twenty two cats in Hamilton, and verifying they are all black will notdo. We have to locate every single cat in Hamilton and verify that it is black.

However, if we wish to prove that the claim “All cats in Hamilton are black” isfalse, all we need is to find one cat in Hamilton that is not black (it does not matterwhat colour it is, as long as it is not black). That cat is called a counterexample.

By this reasoning, we reached an important conclusion: In order to prove thata statement involving a universal quantifier is true, we have to prove that it holdsfor each and every object that is implicated. To show that a statement involvinga universal quantifier is false, we must find a counterexample, i.e., one case forwhich the statement does not hold. This latter case can be described by a mathformula:

¬ (∀xA) = ∃x (¬A)

We read this formula as: negative of (i.e., we are proving that the universal state-ment is false) all objects have some property A is there exists an object whichdoes not have the property A. Remember that we cannot use an example to provethat a statement involving a universal quantifier is true. However, to disprove suchstatement, we can use an example (that’s what we call a counterexample).

Now, consider the following statement which expresses an existential property“There is a black cat in Hamilton.”

Finding one black cat in Hamilton will prove that this statement is true.However, if we wish to prove that the statement is false, we have to show that not


a single cat in Hamilton is black. So, we have to find all cats in Hamilton, andcheck that none is black.

Thus, we conclude that in order to prove that a statement involving an ex-istential quantifier is true, we have to show that it holds for at least one objectthat is implicated. To disprove a statement involving a universal quantifier (i.e.,to show it is false), we must show that it does not hold for all of the objectsimplicated. In terms of symbols,

¬ (∃xA) = ∀x (¬A)

Back to our example: the left side ¬ (∃xA) reads “not (exists cat black)” and theright side ∀x (¬A) reads “all cats (not black)”. Put this into good English, andwe’re done.

Example 2.8 Working with Quantifiers

In each case, identify x (i.e., the objects involved) and A (the property that relatesto the objects). Then say what you would have to do to prove that the statementis true. Finally, say what is needed to prove that the statement is false (i.e., thatits negative is true).

(a) “Every child with asthma has allergies”(b) “Some humanities students take Math 2UU3”(c) “Every investment generates profit”(d “There is a bird which can fly as high as 500 metres above ground”(e) “All McMaster students have to buy a Hamilton bus pass”(f) “There is a bird living on Earth that cannot fly.”

Solution for (f): the object involved is bird(s), so that’s x. The property A is‘cannot fly.’ Using the symbols we introduced, we write the given statement as∃xA.

To prove that “There is a bird living on Earth that cannot fly” is true, weneed to find at least one such bird. Perhaps the most famous example is a kiwibird, a nocturnal animal that lives in New Zealand. (Given that kiwi birds do notfly and New Zealand is an island, it’s not at all clear how they got there.) Anotherbird which cannot fly is takahe, which is native to Australia and New Zealand:

To disprove the given statement, we interpret the right side in the formula¬ (∃xA) = ∀x (¬A) Thus, we have to show that all birds (∀x) on Earth have theproperty ¬A = negative of cannot fly = can fly. In short, we have to show thatall birds on Earth can fly.

Given two properties A and B,we can combine them to create new properties,A and B (denoted by A&B and formally called conjunction) and A or B (de-noted by A∨B and formally called disjunction). The ‘or’ used is the inclusive or.Thus, “apples or oranges” includes three possibilities: apples only, oranges only,and apples and oranges.


To prove a statement that involves ‘and’ we need to prove that both propertiesare satisfied. Thus, to prove “There is a cat in Hamilton which is black and hungry”we have to identify a cat in Hamilton which is both black and is hungry.

To prove a statement that involves ‘or’ we need to prove that at least oneproperty is satisfied. Thus, to prove “There is a cat in Hamilton which is blackor hungry” we have to identify a cat in Hamilton which is either black (but nothungry), or a cat which is hungry (but not black), or a cat which is both blackand hungry.

How about disproving such statements?How can a cat not be black and hungry? For instance, if it’s a brown cat (so

not black) but hungry, or if it’s a black cat with full stomach (so not hungry), or ifit’s a brown cat (not black) with a full stomach (not hungry). Using the symbolswe introduced, we write

¬ (A&B) = ¬A ∨¬B

Likewise,¬ (A ∨B) = ¬A&¬B

(Justify this formula with an example.)

Note that we are doing math with sentences!

Example 2.9 Putting it All Together

In each case, explain how you would prove and disprove each statement.(a) “Everyone who has flu has fever and runny nose”(b) “There is a person who smokes both traditional cigarettes and e-cigarettes”(c) “All women in Ireland have red hair and green eyes.”(d) “Some women in Ireland have red hair and green eyes.”(e) “Some women in Ireland have red hair or green eyes.”(f) “All women in Ireland have red hair or green eyes.”

Solution for (c): To prove this statement, we would have to show that every singlewoman in Ireland has red hair and green eyes.

To disprove this statement, first write it in terms of symbols:

∀x (A&B)

where x represents a woman in Ireland, A represents red hair and B representsgreen eyes (check that it makes sense).

Now use the math that we learned to compute

¬ (∀x (A&B))

First, the negative of the universal quantifier is the existential quantifier

¬ (∀x (A&B)) = ∃x (¬ (A&B))

Now use the formula for the negative of ‘and’

¬ (A&B) = ¬A ∨¬B

and put it together:

¬ (∀x (A&B)) = ∃x (¬ (A&B)) = ∃x (¬A ∨¬B)

Interpret the expression on the right: there exists a woman in Ireland who doesnot have red hair or does not have green eyes. Done!

Solution for (e): To prove this statement, we would have to show find onewoman in Ireland who has red hair or green eyes (meaning she has red hair (andeyes which are not green), or green eyes (and not red hair), or has both red hairand green eyes).


To disprove, we proceed as above, by combining the formulas we discussed:with the same meanings for x, A, and B,, the given statement can be written as

∃x (A ∨B)

Its negative is:

¬ (∃x (A ∨B)) = ∀x (¬ (A ∨B)) = ∀x (¬A&¬B)

Thus, to disprove the given statement, we have to show that all women in Irelanddo not have red hair and do not have green eyes.

Now that we have seen how formulas work, try reasoning without writing themdown. Keep in mind that:

(i) the negative of the existential quantifier is the universal quantifier, and viceversa: the negative of the universal quantifier is the existential quantifier

(ii) the negative of ‘and’ is ‘or’ between the negatives of the statements(iii) the negative of ‘or’ is ‘and’ between the negatives of the statements

One more thing and we are done. Consider the implication A ⇒ B, i.e., cause ⇒effect. As mentioned before, this means that if the cause is present, the effect willhappen.

But how do we prove that an implication is not true? Consider the statement“If you use your phone continuously for 8 hours, its battery will die.” To disproveit, we have to use our phone continuously for more than 8 hours and show thatits battery is not dead (there is charge left).

Thus, in order to prove that cause ⇒ effect is not true, we need to demonstratethat the cause is present but the effect is absent. In symbols,

¬ (A ⇒ B) = A&¬B

Example 2.10 Proving Things Wrong

Say what you would have to do to disprove each statement.(a) Singing opera causes hair loss(b) If Ann takes two Tylenol tablets, then her headache will be gone in two

hours(c) Solving all homework questions guarantees the grade of A+ in a course(d) People who sleep 6 hours or less at night experience fatigue and periodic

loss of concentration during the day.

Solution for (a): To disprove this statement, we have to identify one person whosings opera but is not experiencing hair loss.

Solution for (c): To disprove this statement, we have to identify one studentwho indeed solved all homework questions, yet they did not end up with A+ in acourse.

What does it mean that A does not imply B? Consider the statement “It has beendemonstrated that drinking 6-8 glasses of water every day does not cause kidneyproblems.”

Look at the diagrams below. The diagram on the left represents the statementA implies B (the box representing A is contained in the box representing B). Sothe arrangement of the two boxes where A is not contained in B (diagrams in themiddle and right) represents A does not imply B.


A

B

A B

A

B

A implies B A does not imply B A does not imply B

Thus, A does not imply B could mean two things: in some cases A impliesB but in some cases it does not (middle diagram) or A does not imply B at all(diagram on the right).

So the correct way to interpret “It has been demonstrated that drinking 6-8glasses of water every day does not cause kidney problems” is to say that drinking6-8 glasses of water every day might, or might not cause kidney problems.

However, usually this statement is interpreted as “If you drink 6-8 glasses ofwater every day, you will not have kidney problems.”

Note the subtle difference between disproving the statement A implies B andthe statement A does not imply B.

Evidence and “Evidence”, Further Cases

[1] In the article Gwyneth Paltrows new Goop Lab is an infomercial for her pseu-doscience business (The Conversation, 12 January 2020, https://bit.ly/2tSzi54)we read: “Last week, Netflix dropped the trailer for Gwyneth Paltrow’s new showThe Goop Lab. It is a six-episode docuseries launching on Jan. 24 that, accordingto the trailers, focuses on approaches to wellness that are ‘out there,’ ‘unregulated’and ‘dangerous.’ (Read: science-free.)”

“The backlash by health-care professionals and science advocates was immedi-ate and widespread. And for good reason. As noted by my friend, obstetrician andgynecologist Dr. Jen Gunter in Bustle magazine, the trailer is classic Goop: ‘Somefine information presented alongside unscientific, unproven, potentially harmfultherapies”’

“We know the spread of this kind of health misinformation can have a signif-icant and detrimental impact on a range of health behaviours and beliefs. Thisis the age of misinformation and this show seems likely to add to the noise andpublic confusion about how to live a healthy lifestyle.”

[2] In the CTV News report ‘Totally bummed’ Nobel Prize winner admits a ‘painful’mistake (3 January 2020, https://bit.ly/2Rcr4Ng) we read: “A Nobel Prize winnerhas admitted she ‘did not do her job well’ after a scientific paper published lastyear was retracted. American Frances Arnold won the Nobel Prize for chemistryin 2018 ‘for the directed evolution of enzymes.’ In a series of tweets on January2, she revealed a different paper, published May 2019, has been pulled from thehighly-respected ‘Science’ magazine.”

The article continues to say: “‘For my first work-related tweet of 2020, I amtotally bummed to announce that we have retracted last year’s paper on enzymaticsynthesis of beta-lactams,’ Arnold tweeted. ‘The work has not been reproducible.It is painful to admit, but important to do so. I apologize to all.’ ”

[3] Does drinking milk increase in the risk of breast cancer in women?In the CTV News report One cup of milk per day associated with up to 50 per

cent increase in breast cancer risk: study (https://bit.ly/2QPXReY, 25 February2020) we read “New evidence suggests that women who drink as little as one cup ofdairy milk per day could increase their risk of developing breast cancer by up to 50per cent. Researchers from the Loma Linda University Adventist Health Sciences


Center in California say the observational study gives ‘fairly strong evidence’ thatdairy milk or factors closely related to the consumption of dairy milk is linked tothe development of breast cancer in women. ‘Consuming as little as one-quarterto one-third of a cup of dairy milk per day was associated with an increased riskof breast cancer of 30 per cent,’ study author Dr. Gary Fraser said in a pressrelease.”

However, the blog No, Milk Is Not Going To Give You Breast Cancer. TheLatest Study, Decoded. (AbbyLanger Nutrition, https://bit.ly/3vPe5Ur) does notagree. After presenting details about the study presented in the CTV news piece,the author says: “I also want to take this opportunity to remind you that corre-lation does not equal causation, meaning that just because a result seems to belinked to a behavior, doesn’t mean that the behavior is the CAUSE of the result.Especially in the case of nutrition, there are so many confounders that could creepin. Here are a few: This study didn’t control for diet over 8 years. Participantscompleted one diet assessment, and results were collected 8 years later. What iftheir diets changed? Has your diet changed over the past 8 years?”

“There were no fruit or vegetable intakes measured. Those who consumedmore soy likely consumed more fruits and vegetables overall. That’s my guess,but unfortunately the study didn’t bother reporting that metric (if they measuredit at all). Diet quality wasn’t measured, either. The study did show that thewomen who consumed milk were also heavier, ate more processed red meat, andwere less active. Obesity is a suspected risk factor for breast cancer.”

[4] How can being mindful be bad for us? In the BBC News article How toomuch mindfulness can spike anxiety (https://bbc.in/3zVjv2d 4 February 2021) weread: “I had assumed that I was just uniquely bad at taming my thoughts. Yet agrowing body of research suggests that such stories may be surprisingly common,with one study from 2019 showing that at least 25% of regular meditators haveexperienced adverse events, from panic attacks and depression to an unsettlingsense of ‘dissociation.’ Given these reports, one researcher has even founded a non-profit organisation, Cheetah House, that offers support to meditators in distress.‘We had more that 20,000 people contact us in the year 2020,’ says WilloughbyBritton, who is an assistant professor in psychiatry and human behaviour at BrownUniversity. ‘This is a big problem.’ ”

Notes and Further Reading

[1] Online journal The Conversation published a piece YouTubes algorithms mightradicalise people but the real problem is we’ve no idea how they work (21 January2020, at https://bit.ly/3dx8o4J) in which they argue that “maximising watchtimeis the whole point of YouTube’s algorithms, and this encourages video creators tofight for attention in any way possible. The company’s sheer lack of transparencyabout exactly how this works makes it nearly impossible to fight radicalisation onthe site. After all, without transparency, it is hard to know what can be changedto improve the situation.”

[2] Related to [1]: in The Conversation, on 24 June 2020, How fake accountsconstantly manipulate what you see on social media and what you can do aboutit (at https://bit.ly/3dETfNZ) we read: “Social media platforms don’t simplyfeed you the posts from the accounts you follow. They use algorithms to curatewhat you see based in part on ‘likes’ or ‘votes.’ A post is shown to some users,and the more those people react - positively or negatively - the more it will behighlighted to others. Sadly, lies and extreme content often garner more reactionsand so spread quickly and widely.”[3] More about bias in algorithms. The Washington Post article Internet Culture:Googles algorithm shows prestigious job ads to men, but not to women. Heres


why that should worry you. (6 July 2015, at https://wapo.st/2BPpj4g) reportsthat “Fresh off the revelation that Google image searches for ‘CEO’ only turn uppictures of white men, there’s new evidence that algorithmic bias is, alas, at itagain. In a paper published in April, a team of researchers from Carnegie MellonUniversity claim Google displays far fewer ads for high-paying executive jobs ... ifyou’re a woman.”

As well, “this isn’t the first time that algorithm systems have appeared to besexist - or racist, for that matter. When Flickr debuted image recognition tools inMay, users noticed the tool sometimes tagged black people as ‘apes’ or ‘animals.’A landmark study at Harvard previously found serious discrimination in onlinead delivery, like when searching ethnic names on Google turned up more resultsaround arrest records. Algorithms have hired by voice inflection. The list goes onand on.”

Section references:

[1] Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition. 2013.American Psychiatric Association.[2] Dalrymple, K. L., and Zimmerman, M. (2013, November). When does benignshyness become social anxiety, a treatable disorder? Current Psychiatry, 12(11),21-38.[3] Lena K. Hedendahl, Michael Carlberg, Tarmo Koppel, and Lennart HardellMeasurements of Radiofrequency Radiation with a Body-Borne Exposimeter inSwedish Schools with Wi-Fi Frontiers in Public Health. 2017; 5: 279. Publishedonline 2017 Nov 20. doi: 10.3389/fpubh.2017.00279


3 Numbers: Quantitative in Quantitative Reasoning

A number is a concept (often labeled as mathematical concept) that we use tocount, measure, label, calculate with, and so on.

Every known society or civilisation that has existed on Earth has, in one wayor another, developed and used number systems. One of the first things a childlearns is to count (moreover, there is evidence that children as old as a few monthscan distinguish between one, two and three; and so can dogs, cows and some otheranimals).

In this chapter we discuss concepts related to numbers, as they are needed inreal life situations.

Numbers

Natural numbers are numbers used to count: one, two, three, four, five, and soon. The symbols that represent numbers, such as 1, 2, 3, 4, 5, and so on are callednumerals. The system most people use today is based on the late 14th centuryHindu-Arabic numeral system. It is a positional number system (i.e., the valueof a numeral depends on its location within a number) based on the powers of 10(and thus called a decimal system). For instance, the first (leftmost) occurrenceof 7 in 17074 contributes 7000 to the value of the number, whereas the second 7contributes 70 to its value.

By expanding natural numbers by adding zero and negative (counting) num-bers we obtain integers. By forming ratios of integers (fractions) we obtain ra-tional numbers.

By division, rational numbers can be converted to decimal numbers. Thesedecimal numbers have either a finite number of decimals (such as 1/8 = 0.125),or an infinite number of decimals, which become periodic (such as 1/3 = 0.3333...(3 repeated), 3/11 = 0.27272727... (27 repeated), 8/21 = 0.380952380952380952...(380952 repeated)), and 1/28 = 0.03571428571428... (571428 repeated). The num-bers whose decimal representation has an infinite number of decimals which donot become periodic are called irrational numbers. The most famous irrationalnumber is π = 3.1415926535.... Rational and irrational numbers, put together,form real numbers.

Although called real numbers, there is not much real about them. In oureveryday lives, we most often use natural numbers, integers and rational numbers,together with their decimal representations.

When we measure something, we attach units to numbers. Of the amazingarray of the ways people measure things, we will stick to those which are mostcommonly used. Many belong to the SI system (or the metric system) of units,such as the metre, the second and the kilogram. However, there are numerousexceptions: instead of the SI unit Kelvin (for temperature) we use degrees Celsius(or Fahrenheit). As well, we use pounds, ounces, pints, as well as feet and inches(for instance, building standards in North America use these units).

Exact Values and Approximations

Some things are exact in the sense that we know their precise, unique value. Forinstance, the convention we accept about the time states that 1 hour = 60 minutesand 1 minute = 60 seconds (thus, there are exactly 24 · 60 · 60 = 86, 400 seconds ina day), or (again, by convention) 100 degrees Celsius = 212 degrees Fahrenheit.

Chapter 3 Numbers: Quantitative in Quantitative Reasoning 39

Money and counting (smaller quantities) often result in exact values: one canbuy 6 eggs, 4 bananas and two jars of peanut butter and pay $ 15.45. Largercounts (such as a number of people at an open air concert, population of Que-bec, or the number of students attending colleges) are estimates. As well, largeramounts of money (say, government debt) are approximate, rounded-off amounts.For instance, Canadian National debt, i.e., the amount of money the Canadiangovernment owes to holders of Canadian Treasury security, such as treasury billsand bonds, is about 700 billion dollars (check http://www.debtclock.ca/).

It is not possible to exactly measure anything – it is inevitable that instru-ment(s) used and the people who use them will generate different readings for Thus,every measurement comes with a measurement error, and can be expressed as

measured value ± measurement error

as in 3.15 ± 0.05. This means that the true value is believed to lie in the rangefrom 3.10 to 3.20; using interval notation, we write [3.10, 3.20].

NASA’s Global Change Climate keeps track of the rising levels of sea waterat their site https://climate.nasa.gov/vital-signs/sea-level/

The measurement 91(±4) mm is the increase in the sea level since 1993. Thus,the sea level increased between 87 mm and 95 mm.

The causes of sea level increase are related to climate change: the added watercomes from melting ice (glaciers), and from the thermal expansion of seawater. Thegraph below tracks sea level changes, with the baseline (Sea Height Variation =0) taken to be the sea level on 1 January 1993.


Below is a manufacturing blueprint for some metal piece, specifying its di-mensions (called d1, d2, d3, and h). Note that instead of exact values, in eachcase a range of values is given, expressed using the minimum and the maxi-mum allowed values. For instance d1 has the minimum of 1.7mm, and the maxi-mum of 1.84mm, which can be written as an interval (range) [1.7, 1.84]. (Source:https://bit.ly/2JJ26T9)

A measurement error can be expressed as percent, as in “[...] the value is 12.7,with the measurement error of 1%.” This means that the interval (range) wherethe true measurement lies is [12.7 − 0.127, 12.7 + 0.127].

Quite often, even though there are errors involved, a measurement is presentedas a single number. For instance, the dimensions of the table from the Ikea onlinecatalogue (https://bit.ly/2SvqjPj) are given as exact values (in inches, with metricunits in parentheses).

The dimensions are not presented as ranges, but as a single number. Is thelength of the table exactly 220cm? It is not, however it might be confusing for anaverage shopper to read that, for instance, the length of the table is in the range[219.85,220.15] centimetres. But more important – do we really care if the tablewe’re contemplating to buy is 1.5mm longer or shorter than 220cm?

Note that the measurements in inches are given as fractions (which is com-mon practice in things related to construction, building standards, furniture, ap-pliances, etc., where fractions with denominators 2, 4, 8, 16, 32 (powers of 2) areused).

Similarly, tech specifications for the size and the weight of a MacBook Prolaptop (Apple Store, at https://apple.co/1hLVobV) use ‘exact’ values, rather thanincorporating a measurement error

(Note that this time, measurements in the U.S. system of units are written asdecimal numbers.)


Notes and comments:(1) The result of measuring something might depend on the device we use tomeasure. For an example, read How long is the coast of Great Britain? It dependshow you measure it at https://bit.ly/2Y5yFTX(2) Heisenberg’s uncertainty principle (also known as the uncertainty prin-ciple) gives a limit to the precision with which certain quantities can be measured.In 1927, Heisenberg proved that the more precisely we measure the position of some(atomic) particle, the less precise our measurement of its momentum is (and viceversa).(3) In math we make a distinction between exact values, when we use the equalssign = and approximate values, when we use the approximately equal sign ≈.However, often we get sloppy and write π = 3.14 when we should say π ≈ 3.14.In real life, the symbols = and ≈ are not often used, but rather narrated, usingwords such as ‘is/are’ (as in “A table is 220 cm long”), ‘roughly’ (as in “That plotof land is roughly 15 metres wide”), ‘approximately,’ ‘about,’ ‘close to,’ and so on.(4) A common context where approximations appear is in recipes for preparingmedications: instead of using interval notation such as [3.3, 3.5] milligrams, inthese situations, we see ‘3.4 mg ±0.1 mg,’ or ‘minimum 3.3 mg and maximum 3.5mg’.

Case Study 3.1 Approximations Everywhere

(1) In the news article (NBC News, 18 June 2019, at https://nbcnews.to/2IqhX8v)More than 16 tons of cocaine worth up to $ 1B [1 billion] seized in massive bustin Philadelphia both numbers in the title are approximations. (By the way, thereporter is inconsistent when it comes to the actual amount of cocaine seized.)(2) On 20 July 2019, Iowa State Daily reports that “A press release from the citysaid a ‘transmission line fault’ caused the entire Ames electric service territory tolose power at 12:35 pm Saturday, causing ‘approximately’ 26,000 of their customersto go without power.” (https://bit.ly/2Y8jPXQ)(3) CBC News report How tiny homes could provide a path to security for newCanadians (21 July 2019, at https://bit.ly/30MR9W6) states that “In St. John’s,approximately 12,100 households live in unaffordable housing, of which nearly 65per cent rent.”(4) In the report Woman arrested for stealing approximately $ 1,500 of items fromacquaintance Indiana Daily Student (13 June 2019, at https://bit.ly/2LBVFDs)reported that “[...] suspect took a black RCA tablet, a 32-inch television, awatch, a speaker, Bluetooth headphones and clothing items valued at approxi-mately $ 1,500.”(5) On 20 July 2019 in Halifax Pride parade 2019 takes over downtown HalifaxCBC News reports that “Parade officials said there were approximately 150 paradeentrants in the 2019 event.” (https://bit.ly/2YZvhq9)

Rounding Numbers

We start this section with a simple calculation – what is 7.2·2.40.8 ? If we round off

all numbers to the nearest integer, we obtain7.2 · 2.4

0.8≈ 7 · 2

1= 14

If we calculate this expression without rounding off we obtain (cancel 2.4 and 0.8)7.2 · 2.4

0.8= 7.2 · 3 = 21.6

which, rounded off to the nearest integer, gives 21.6 ≈ 22. Big difference!


This illustrates a well known fact in numeric mathematics, namely that theround off errors (i.,e., replacing a true value of a number with some approximation)accumulate, rather than cancel each other. Hence, an important rule:

We perform all calculations at maximum precision (which depends on whatwe use: a hand calculator, software, or an online calculator) and then round off toa desired precision once at the end.

How do we round off a number? The most common rule, and the one we willuse in this textbook, is to base the decision on the first digit (from the left) thatwill be dropped.♦ If the first digit to be dropped is 0, 1, 2, 3, or 4, then we round down, i.e.,we drop that digit and also all digits to the right of it. For instance, 5.2374299rounded off to one decimal place is 5.2, and rounded off to three decimal placesis 5.237 (so rounding down refers to the fact that the rounded off value is smallerthan the original value).♦ If the first digit to be dropped is 5, 6, 7, 8, or 9, then we round up, i.e., wedrop that digit and also all digits to the right of it, and add 1 to the number that’sleft. For instance, 5.2875299 rounded off to one decimal place is 5.3, rounded offto three decimal places is 5.288, and rounded off to six decimal places is 5.287530(so rounding up refers to the fact that the rounded off value is larger than theoriginal value).

There is a variation of this rule (discussed below), as well as other rules, de-pending on the context where rounding off is used. Check, for instance, Cashrounding on Wikipedia (https://bit.ly/20v1Wib) and read about Swedish round-ing.

Let’s go back to the rule we adopted for this course (digits 0-4 are roundeddown, digits 5-9 are rounded up). Rounding off changes the value of a number,except if the digit to be rounded off is zero. Thus, of the 9 cases when the numberchanges when rounded off, in 4 cases it is rounded down, and in 5 cases it isrounded up. This means that, in a large data set (say, 900,000 numbers) about400,000 numbers would be rounded down, and about 500,000 numbers would berounded up. Thus, the average of all numbers after rounding off would be largerthan before rounding (and in many cases that’s not acceptable).

To fix the situation, we adjust the rule for rounding off when digit 5 is involved:♦ [same as before] If the first digit to be dropped is 0, 1, 2, 3, or 4, then we rounddown, i.e., we drop that digit and also all digits to the right of it.♦ [same as before, except for the digit 5] If the first digit to be dropped is 6, 7, 8,or 9, then we round up, i.e., we drop that digit and also all digits to the right ofit, and add 1 to the number that’s left.♦ [new rule] If the first digit to be dropped is 5, then we round it so that the digitto be rounded off is even.

Example for the last rule: round off 3.257 to one decimal place. If we roundup, we get 3.3, and if we round down we get 3.2. The first decimal (the digit to berounded) needs to be even, so it’s 3.2. Now round off 3.357 to one decimal place.If we round up, we get 3.4, and if we round down we get 3.3. The first decimal(the digit to be rounded) needs to be even, so it’s 3.4.

This rule is good to know, as it is useful in some situations.

Notes and Comments:(1) In his work on numeric mathematics, John Von Neumann proved that roundoff errors propagate (i.e., do not cancel each other, no matter how we round off)and their accumulation (in numeric procedures involving a large number of calcu-lations) can lead to serious problems.


(2) Measurements and calculations in engineering, often require a high level ofprecission (and numbers are routinely rounded off to six, or nine, or ten decimalplaces).

Example 3.2 Rounding off

Round off each number to the requested number of decimal places, as shown.

number one two three four

3.72658 3.7 3.73 3.727 3.7266

0.246809

1.55555

44.050607

0.111111

0.999999

Rounding to nearest integer values is done analogously, and here we can visu-alize it–the question is what number is it closer to?

Rounding $ 1.67 to the nearest dollar gives $ 2.00 (since 1.67 is closer to 2 thanit is to 1). As well, 547 rounded to the nearest hundred is 500 (since 547 is closerto 500 than to 600). 1,500 rounded off to the nearest thousand is 2,000 (recall thatwhen the digit in question is 5, we round up). The number 6, 946.58

rounded to one decimal place: 6, 946.6rounded to the nearest integer: 6, 947rounded to the nearest ten: 6, 950rounded to the nearest hundred: 6, 900rounded to the nearest thousand: 7, 000.

Note that certain decimal numbers are always rounded off, such as periodic infinitenumbers 0.333... (decimal 3 repeated) 0.405405... (group 405 repeated). The goodnews is we can express them as fractions (1/3 and 15/37 respectively) and thusrecord their exact value. However, as irrational numbers cannot be expressedas fractions, their values are always rounded off, and thus approximated. Forinstance, 3.14, 3.14159, and 3.14159266, are approximations of π.

There are cases when the context suggests that we deviate from the rules forrounding off, or the way we round off is not relevant.

For instance, in a mathematical modelling of deer population in AlgonquinNational Park, an estimate of 1346.4 deer was obtained. For people working withdeer, it’s irrelevant whether it is 1346 or 1347 deer. As well, they might be happywith an estimate of ‘about 1350,’ or even more rough, ‘about 13 hundred.’

You are in charge of a school trip for 342 students and their teachers, andneed to organize bus transportation. As this is a longer trip noone is allowed tostand, i.e., every person must have a seat. Given that a bus has 55 seats, howmany buses do you need to order?

Dividing, you get 342/55 = 6.21, rounded to the nearest integer gives 6. Butof course, you will not order 6, but 7 buses.


Estimation

An estimation is a calculation or a judgement of the value of a quantity, usuallydone when relevant pieces of information are not available, or might not be known,or are not stable (i.e., change all the time) or when their values are not knownprecisely (so that an exact calculation is impossible).

For instance, it is almost impossible to find out exactly how many paper coffeecups are used (and discarded) on McMaster University’s campus in one day. Wedo not know (but can estimate) how much plastic ends up in Lake Ontario everyday.

Whenever we work on an estimate, we need to keep track of all data used,and assumptions made. For instance, to estimate a number of human heartbeatsin a day, we need to know the heartbeat rate. This information is unstable (as itcould change from person to person; as well, it depends on the type of activity:sleeping, sitting, walking, running, etc.).

Consulting the American Heart Association web page https://bit.ly/2LNnjy1we find “For adults 18 and older, a normal resting heart rate is between 60 and100 beats per minute (bpm), depending on the person’s physical condition andage.”

So, assuming that the heartbeat rate is 80 beats per minute (the average ofthe two extremes given), we obtain

80 · 60 · 24 = 115, 200

heartbeats in a day.Of course, with a different assumption, we would get a different estimate (thus,

when we estimate, there are no unique answers). For instance:60 heartbeats per minute yields 60 · 60 · 24 = 86, 400 heartbeats in a day72 heartbeats per minute yields 72 · 60 · 24 = 103, 680 heartbeats in a day84 heartbeats per minute yields 84 · 60 · 24 = 120, 960 heartbeats in a day100 heartbeats per minute yields 100 · 60 · 24 = 144, 000 heartbeats in a day

Note that we made a further (unreasonable!) assumption that one’s heartbeat rateremains constant throughout a 24-hour period (and equal to the resting rate).

In reality, noone knows exactly how many times a human heart beats in oneday (or even in one hour, or in one minute – unless we monitor and count).However, from the above estimates we can assert that very likely a human heartbeats between 86,400 and 144,000 times in a day.

Note: Remember to state all assumptions you made when calculating an estimate.

Example 3.3 How Many Seconds are There is a Year?

One thing is easy – we have already figured out that there are exactly 60 ·60 ·24 =86, 400 seconds in one day.

Now a more difficult question: how many days are there in a year? Could be365 or 366, depending on whether it’s a leap year or not. Computing the average(365 + 366)/2 = 365.5 is not appropriate. Why?

Given that there is 1 leap year in 4 years, the average length of a year is365 + 365 + 365 + 366

4=

14614

= 365.25

days. Thus, assuming that a year has 365.25 days, the number of seconds in ayear is

60 · 60 · 24 · 365.25 = 31, 557, 600


The fact that there are 365.25 days in a year was used to define the Julian Calendar,which was in effect until the 16th century (in Europe; other parts of the world useddifferent calendars, or used the Julian calendar longer)

In the 16th century, a reform of the Julian calendar (as it was shown not tobe precise; see the next example) created the Gregorian calendar, according towhich certain leap years were eliminated (those divisible by 100 but not by 400;thus, 1700, 1800 and 1900 were not leap years, but 2000 was a leap year). In otherwords, in a span of 400 years, there are 97 leap years (366 days) and 303 non-leapyears (365 days). (According to the Julian calendar, there were 100 leap years ina span of 400 years.) This makes the average length of a year

365 · 303 + 366 · 97400

= 365.2425

days. With this number in mind, the number of seconds in a year is computed tobe

60 · 60 · 24 · 365.2425 = 31, 556, 952

Of course, scientists being scientists, they did not stop there. Further estimateshave been obtained, such as the one callled mean tropical year, whose length is365.242189 or 365.24217 days (one day being equal to 86,400 seconds). Read moreabout it at https://bit.ly/2JKj4Ak.

Example 3.4 Calendar Trouble, or Why Precision Matters

Let us compare the Julian length of the year (365.25 days) with the more accurate(and shorter) Gregorian length of the year (365.2425 days), based on astronomicobservations of the length of the solar year.

The difference between the two lengths is 365.25− 365.2425 = 0.0075 days, or0.0075 · 86, 400 = 648 seconds, or 10.8 minutes. Thus, the first year according tothe Julian calendar was 10.8 minutes longer than it should have been.

start of the Julian calendar end of year 1 according to the Julian calendar

end of year 1 according to the solar year difference = 10.8 minutes

Not a big deal. However, over a hundred years, that discrepancy grew to10.8 · 100 = 1080 minutes, or 18 hours! In other words, people celebrating thearrival of a new year 100 years after the adoption of the Julian calendar would becelebrating it 18 hours into the new year!

start of the Julian calendar end of year 100 according to the Julian calendar

end of year 100 according to the solar year difference = 18 hours

The Julian calendar was introduced in 45 BCE. For every 100 years of its use,it was off by another 18 hours. Thus, in the year 1555 CE (so 1600 years after itsadoption) the Julian calendar was ahead of the solar year by 16 · 18/24 = 12 days!

At the time, people did not need these calculations–they realized that thesummer solstice (which can easily be observed) did not agree with 21 June. In1555, the day of the summer solstice was 9 June). To be in tune with the solaryear, two things needed to be done: fix the length of the year, and removed acertain number of days from the present (Julian) calendar. Thus, the Gregoriancalendar was born (with length of a year = 365.2425 days). In 1582 it was adoptedby several countries in Europe. However, instead of 12, 10 days were dropped fromthe calendar, in month of October; the calendar looked like


It took about 300 years for most of the world to adopt the new Gregoriancalendar, and as there were doing it they needed to remove more days to synchro-nize with those who already modified their calendars. For instance, Canada, U.S.,and U.K. adopted the Gregorian calendar in 1752, and removed 11 days; Japanadopted it in 1872/73, and removed 12 days; Turkey adopted it in 1926/27, andremoved 13 days. So different countries followed different calendars for quite sometime, and consequently used different rules for the leap years (i.e., it was all a bigmess).

In 2021, the Julian calendar is roughly 13 days behind the Gregorian calendar.Thus, the start of the Fall term, 7 September 2021 (Gregorian), is 25 August 2021(Julian).

Note: There is an amazing variety in the ways people have been keeping trackof time; some of these do not care about leap years, nor the precise length of ayear. For instance, Mayan (Maya = Mesoamerican culture) long count calendarkeeps track of the number of days from the creation of universe (corresponds to11 August 3114 BCE). According to this calendar, 7 September 2021 is the day1,875,182.

Note: Calendar correction is just one situation where small differences in numbersmatter. Over time, small numbers could (and do) accumulate to something large.Perhaps the best example are banking fees: we might feel that individual fees arenot large (say, paying $ 2 for some transaction). However, given the huge numberof transactions, it is known that these ‘small’ fees contribute billions of dollars tothe profits of banks and other financial institutions.

Case Study 3.5 Raw Count (Enumeration) vs Rough Estimate: Number of Sex Partners

How do we arrive at estimates in everyday life?In New York Times article The Myth, the Math, the Sex (12 August 2007,

at https://nyti.ms/2GDyHpe) we read “One survey, recently reported by the fed-eral government, concluded that men had a median of seven female sex partners.Women had a median of four male sex partners. Another study, by British re-searchers, stated that men had 12.7 heterosexual partners in their lifetimes andwomen had 6.5.”

These findings echo an earlier report Why Men Report More Sex Partners thanWomen published in LiveScience (February 17, 2006 at https://bit.ly/2Jmmmrt)“Psychologist Norman R. Brown at the University of Michigan has done severalstudies on the apparent flaw in these surveys. The latest was a web-based surveyof 2,065 heterosexual non-virgins with a median age in their late 40s. The womenreported on average 8.6 lifetime sexual partners. The men claimed 31.9.”

Can these reports be true?Let’s try to figure it out. These reports are about heterosexual sex, so we

assume that there are six women and six men (representing the fact that thepopulation is roughly fifty-fifty; these numbers do not matter, as you can checkyourself by picking, say 10 women and 10 men). Each segment in the diagramrepresents one sexual relation between a male and a female, and the number ofsexual partners for every person is shown.


(2) W

(2) W (3) W(3) W

(0) W M (3)M (0)

M (1)M (3)M (2)

M (4) (3) W

Now we compute the averages: for women it is2 + 3 + 0 + 3 + 3 + 2

6=

136

and for men4 + 0 + 3 + 1 + 3 + 2

6=

136

Exactly the same, as it should be!Of course, this is an ideal situation (for instance, it could be that a woman

from this group had a sexual relation with a man not included here). But thisalone does not account for the discrepancy from the reports.

In the LiveScience report, we find a more plausible explanation: “ Womenrely on a raw count, a method Brown says is known to result in underestimation.They tend to say, ‘I just know,’ and if you ask them to explain how they know,they say, ‘Well, there was John, Tom, etc.’ Men also rely on a flawed strategy:[they] are twice as likely to use rough approximation to answer the question.And rough approximation is a strategy known to produce over-estimation.”

For further statistics, read Women’s Health magazine article The AverageNumber Of Sexual Partners For Men And Women Revealed (30 September 2015,at https://bit.ly/2GV5J7J)

Small, Human Scale, and Large Numbers

Numbers come in all sizes, from very small to very large. How do we write them,compare them and reason with them? How do we internalize, ‘get a feel for,’numbers – for instance, how do we make ourselves understand that one billion isa thousand times larger than a million, or that it takes one billion nanometres toform one metre?

Here is an example, illustrating why this is important.

Example 3.6 Coffee, Anyone?

The article British university ‘sorry’ after wrongly giving students 300 coffee cupsworth of caffeine published on SI News web page https://bit.ly/2HgRoAx on26 January 2017 tells a story about a medical experiment at the University ofNorthumbria in England that went terribly wrong.

In March 2015, two sports students volunteered to participate in an experimentabout the effects of caffeine on exercise. They were supposed to be given a cup ofcoffee with 300 mg = 0.3 grams of caffeine and start an exercise routine.

How much is 300 mg of caffeine? The article claims that a shot of espressohas about 65 mg of caffeine. Starbucks’ single espresso has 75 mg of caffeine(https://bit.ly/2q9P1I8) whereas Tim Hortons’ small espresso has 45 mg of caf-feine (https://bit.ly/2H1yXBZ). Thus, 300 mg of caffeine is equivalent to roughlyfour shots of Starbucks’ espresso or a bit less than seven shots of Tim Hortons’small espresso.

Or, consider large drinks: Starbucks’ Grande Pike Place Brewed Coffee con-tains 310 mg of caffeine; Tim Hortons’ Large Coffee has 270 mg, and XLarge Coffee


has 330 mg of caffeine (see above links). Thus, 300 mg of caffeine is contained inabout one large cup of coffee.

By mistake, the two students were given 30 grams (30,000 milligrams) ofcaffeine (in powder form), which is one hundred times larger than what they weresupposed to be given – hence the equivalent of having one hundred coffees. (Note:in British media it was reported as 300 cups of coffee, assuming about 100 mg ofcaffeine per coffee cup–although that is not at all clear from the article.)

The university explained that the researchers involved “had used a mobilephone to calculate the caffeine dosage, resulting in the decimal point being in thewrong place.”

The good news is that both students recovered, one after a temporary short-term memory loss. However, the question remains – how did it happen that theresearchers failed to distinguish between a dose and the one which is hundred timeslarger?

Soon, we will learn that instead of saying “one hundred times larger” we couldsay “two orders of magnitude larger.”

‘Human scale’ numbers we can easily work with, relate to our experiences, andmake meaningful comparisons. For instance, we know that by adding three hun-dred and two thousand we cannot get a hundred thousand. Without multiplying,we know that the product of 16 and 81 is larger than one hundred and smallerthan ten thousand.

We can easily visualize ten (e.g., ten books), or fifty (e.g., the length of anolympic-size pool is 50 metres), or one million (as this is how much money (ormore) we need to buy an average house in Toronto), or one tenth (1 millimetre isone tenth of a centimetre).

If we have 100 dollars, we know that we can buy ten things which cost 10dollars each. We know that we cannot buy a new car for 10,000 dollars, but wecan pay a yearly tuition to study at McMaster with this money.

Example 3.7 Small(er) Numbers

(1) As part of the analysis The price of making a plastic bottle The Economist (15November 2014, https://econ.st/2fxm1Eb) published this diagram showing howcheap it is to produce a plastic bottle–and hence the reason why there are somany of them filling our recycling bins, landfills, and so many large cargo shipstrying to dispose of them in some poor country.

(Note: the above link is not available without subscription to The Economist, butyou can find other sources confirming this.)


(2) How cheap (or expensive) is drinking water?Depends on who is buying. In Why Nestle’s Aberfoyle well matters so much to

Guelph, Ont., residents (CBC News, 26 September 2016 https://bit.ly/2ONa65s)we read: “A showdown between water advocates and politicians is expected Mon-day night in Guelph, Ont., over the divisive issue of Nestle Waters Canada seekingrenewal of a water-taking permit for its bottling plant in nearby Aberfoyle. Thepermit has become a flashpoint in a battle between environmentalists, communityleaders and corporate interests, after it was revealed that the province only chargesbottled water companies $ 3.71 per million litres of water.”

$ 3.71 per million litres of water amounts to 371/1, 000, 000 = 0.000371 centsper litre.

How much do citizens of Hamilton, Ontario pay for water? As a recent bill(from 2019) shows, they are charged at the rate of 78 cents per cubic metre ofwater (1 cubic metre =1000 litres), which amounts to 0.078 cents per litre.

We can compare these numbers roughly: in scientific notation, 0.000371 = 3.71 ·10−4, and 0.078 = 7.8 ·10−2. Thus, we pay 2 orders of magnitude more than NestleWaters Canada.

A precise comparison 0.078/0.000371 = 210.24 tells us that we pay 210 timesmore for water than Nestle Waters Canada!

To add insult to injury, we then buy bottled water at very high prices. Put (1)and (2) together to calculate how much does it actually cost to produce a bottleof water.

(Sold at Rogers Centre during a Toronto Blue Jays game. Smart? Really?)

There are numbers that are beyond human scale and defy our intuition andimagination. For example:(1) A new iMac computer from Apple comes with a 1 TB (= one terabite) harddrive. One terabite is one thousand billion bites.(2) Nanotechnology involves working with matter whose size is about 1 to 100nanometers. One nanometre is one billionth of a metre (imagine cutting a stickone metre long into one billion pieces).


(3) According to the World Bank, in 2018, the gross domestic product (GDP) ofCanada was 1,709.3 billion US dollars (which is 1.7093 trillion US dollars). Thediagram below shows the changes in the GDP of Canada from 2009 to 2018 (source:Trading Economics, https://tradingeconomics.com/canada/gdp).

(4) The human brain contains about one hundred billion neurons. Written out,that number is 100,000,000,000.(5) The average volume of a human red blood cell is about 100 micrometres cubed.A micrometre is one millionth of a metre.(6) One light year (the distance that light travels in vacuum in one year) is about94,607,304,725,808,000 metres.(7) The mass of one grain of salt is about 0.00000005 kilograms. Thus, one kilo-gram of salt contains about 1/0.00000005, which is about 20 million grains ofsalt.(8) The total volume of water contained in Earth’s oceans is estimated to be about1,350,000,000,000,000,000,000 litres.

(9) The diameter of a proton is approximately 0.000000000001 millimetres. Ifwe could manage to align 1/0.000000000001 = 1, 000, 000, 000, 000 (one thousandbillion) protons next to each other on a line, they would cover the distance of 1millimetre.(10) At one point the rate of inflation in Zimbabwe was so high (called hyper-inflation) that the currency was quickly becoming worthless. As a result, ban-knotes of astronomically high value were printed, such as the one from 2019 (fromWikipedia):


Zimbabwe has since introduced a new currency; see Why does Zimbabwe havea new currency? (BBC News, 22 July 2019, https://bbc.in/2JV2k8p). We read:“The economy is no longer in its extreme inflationary spiral, but the country hascontinued to suffer from severe shortages of food, medicine and fuel. Last month,the Zimbabwean authorities reintroduced the Zimbabwean dollar as the country’ssole legal tender.”

On 20 May 2018, a blog EconomicPolicyJournal.com under Venezuela is Ex-periencing Hyperinflation,Turkey May Be Next shows a picture of a banana andthe amount of money needed to buy it. (https://bit.ly/2YkcBA4)

(11) Pornhub is a “Canadian pornographic video sharing and pornography siteon the Internet. It was launched in Montreal, providing professional and amateurpornography since 2007” (Wikipedia).

In its 2018 Year in Review, published on 11 December 2018 Pornhub sharesmajor statistics about its site (https://bit.ly/2zUYKHr). In particular, they claimthat they generated 4,403 petabytes of traffic (What is a petabyte? Read on).

Example 3.8 When Youtube Counter Broke

The article Gangnam Style music video ‘broke’ YouTube view limit (BBC News,4 December 2014 https://bbc.in/2LKS5aj reported on the fact that the counterfor the video reached its limit (and thus would not increase with new views).We read: “YouTube said the video – its most watched ever – has been viewedmore than 2,147,483,647 times. It has now changed the maximum view limit to9,223,372,036,854,775,808, or more than nine quintillion.”

The screenshot below was taken in July 2019, with the counter showing over3.3 billion views.


The article continues: “How do you say 9,223,372,036,854,775,808? Nine quin-tillion, two hundred and twenty-three quadrillion, three hundred and seventy-twotrillion, thirty-six billion, eight hundred and fifty-four million, seven hundred andseventy-five thousand, eight hundred and eight.”

Note: for technical details about how it happened, read The Economist HowGangnam Style broke YouTube’s counter https://econ.st/310OmZG

How do we make sense of these, and other (very) small and large numbers? Tostart, we need a productive way of writing them down–and that is accomplishedby using powers of 10. Recall that 100 = 1

101 = 10102 = 10 · 10 = 100103 = 10 · 10 · 10 = 1000

and, in general,10n = 10 · 10 · 10 · · · · · 10

︸︷︷︸

n

= 100 . . . 00, where 1 is followed by n zeroes .

Using division, we define negative powers:

10−1 =110

= 0.1

10−2 =1

102=

1100

= 0.01

10−3 =1

103=

11000

= 0.001

and, in general,

10−n =1

10n= 0.00 . . .01 where the total number of decimals is n (thus, there

are n − 1 zeroes before the 1 at the end).

Example 3.9 Working With Powers of 10

Recall the multiplication formula

10m · 10n = 10m+n

(i.e., 10m has m zeroes, 10n has n zeroes; when multiplied, we get a number withm + n zeroes, that is, 10m+n). For example, 102 · 1011 = 1013, 10 · 107 = 108, etc.The division formula for the powers is

10m

10n= 10m−n


The number on the top has m zeros, and the one on the bottom has n zeroes.Consider examples:

108

103=

100, 000, 0001, 000

= 100, 000 = 105

(thus 108/103 = 108−3 = 105) and

105

1011=

100, 000100, 000, 000, 000

=1

1, 000, 000=

1106

= 10−6

(in short, 105/1011 = 105−11 = 10−6).

Example 3.10 Multiplying Decimal Numbers and Powers of 10

Recall the rule: to multiply a decimal number by 10n, where n is positive, we movethe decimal point to the right (thus making the number larger). To multiply adecimal number by 10n, where n is negative (thus, dividing), we move the decimalpoint to the left (which makes the number smaller). For instance:

0.003509 · 10 = 0.035090.003509 · 102 = 0.35090.003509 · 103 = 3.5090.003509 · 105 = 350.90.003509 · 107 = 35, 0900.003509 · 1010 = 35, 090, 000

As well,26.193 · 10−1 = 2.619326.193 · 10−2 = 0.2619326.193 · 10−3 = 0.02619326.193 · 10−5 = 0.00026193

A number in scientific notation is written as

x = D.dd . . . d · 10A

where D is a single non-zero digit, dd . . . d are decimals and A is an integer (pos-itive, negative, or zero). The number A is called the order of magnitude thenumber x. As we will see, these are useful notions–the order of magnitude is usedfor rough comparisons between numbers.

In order to tell the order of magnitude of a number we have to write it inscientific notation. Examples:

5.56789 = 5.56789 · 100 (order of magnitude: 0)55.6789 = 5.56789 · 101 (order of magnitude: 1)556.789 = 5.56789 · 102 (order of magnitude: 2)5567.89 = 5.56789 · 103 (order of magnitude: 3)556789 = 5.56789 · 105 (order of magnitude: 5)

We can think of one order of magnitude as one power of ten (i.e., ten-fold). For apositive integer, the order of magnitude is the number of digits minus 1. (Later,we will characterize it in a different way.) Thus, the order of magnitude of 15,000is 4, and the order of magnitude of 3 million is 6. As well,

0.556789 = 5.56789 · 10−1 (order of magnitude: −1)0.0556789 = 5.56789 · 10−2 (order of magnitude: −2)0.00556789 = 5.56789 · 10−3 (order of magnitude: −3)0.0000556789 = 5.56789 · 10−5 (order of magnitude: −5)


Example 3.11 Order of Magnitude

In Electric vehicles powered by fuel-cells get a second look (The Economist, 25 Sept2017, at https://econ.st/2ZkmsaG) we read: “Since then, the cost of fuel-cells hascome down by at least an order of magnitude, as researchers have learned how,among other things, to use less platinum in the catalyst.”

Roughly, by how much has the cost of fuel-cells come down?

Example 3.12 Comparing Quantities Using Order of Magnitude

Since the order of magnitude of 7,491 is 3, and the order of magnitude of 12 millionis 7, we say that ‘12 million is 4 orders of magnitude larger than 7,491’ (or that7,491 is 4 orders of magnitude smaller than 12 million).

Assume that the mass of a house mouse is about 0.02 kg, the mass of a dogabout 20 kilograms, the mass of a cow about 500 kg, the mass of an elephantabout 4500 kg, and the mass of a blue whale about 150,000 kg.

What are the orders of magnitude of these numbers?We can say that a blue whale is three orders of magnitude heavier than a

cow, four orders of magnitude heavier than a dog and seven orders of magnitudeheavier than a house mouse.

A house mouse is three orders of magnitude lighter than a dog. A cow is oneorder of magnitude heavier than a dog and one order of magnitude lighter thanan elephant.

Images are not to scale!

Example 3.13 Free University Tuition for Everyone

Suppose that our government decides that education in Canada should be free,and thus will pay tuition for all university students. How much would that costthe government?

Since we do not know the exact number of university students in Canada, northe amount of tuition they pay, we can only make an estimate.

How many students are there in Canadian universities? We pick a number,and increase it by the order of magnitude, and pick the one that looks right.For instance: 15, 150, 1,500, 15,000, 150,000, 1,500,000, 15,000,000. What is themost likely number of university students in canada? 150,000 is too small, whenwe think of the fact that just two universities nearby, McMaster (about 27,000students) and University of Toronto (over 60,000 students) combined, yield morethan half of 150,000. As well, Canada cannot have 15 million students, as it wouldbe too many (given the total population of 37 million). Thus, of all numbers listed,1,500,000 is the most likely.


What is an average yearly tuition? Apply the same principle, starting with,say, 40: 40, 400, 4,000, 40,000, 400,000, etc. By elimination (400 is too low, 40,000is too high) we conclude that it is 4,000.

Multiplying the number of students (1,500,000) by the average tuition ($ 4,000)we obtain $ 6,000,000,000, i.e., 6 billion dollars.

What if we pick different numbers? Assume that there are 1 million universitystudents in Canada, and the average yearly tuition is about $ 5,000. In this case,we obtain

1 million · 5, 000 = 5, 000, 000, 000 = 5 billion dollars

Not the same amount, but the same order of magnitude! We can try other num-bers, and we’re sure that we will either obtain a number of the same order ofmagnitude, or perhaps one order of magnitude larger or smaller – but definitelynot three orders of magnitude lager or smaller.

Let us compare our estimate with the Universities Canada data (check theweb page https://bit.ly/2nSwoYf)

The average tuition for Canadian students enrolled in canadian universitiesis about $ 6,800 (read the article How Much Does it Cost to Study in Canada?posted at https://bit.ly/2m6A9Hs).

With these numbers, we obtain

1.7 million · 6, 800 = 11, 560, 000, 000 = 11.56 billion dollars

which is one order of magnitude higher than our previous estimates.

The kind of calculation done in the previous example (figuring out, roughly, thetotal tuition for all university students in Canada) is sometimes referred to as aback of an envelope calculation.

Certain powers of 10 are given names and prefixes. Large numbers:

Name/prefix Symbol Numeric English Order of magnitude

kilo K 103 thousand 3

mega M 106 million 6

giga G 109 billion 9

tera T 1012 trillion 12

peta P 1015 quadrillion 15

exa E 1018 quintillion 18


Small numbers:

Name/prefix Symbol Numeric English Order of magnitude

deci d 10−1 tenth -1

centi c 10−2 hundredth -2

milli m 10−3 thousandth -3

micro μ 10−6 millionth -6

nano n 10−9 billionth -9

pico p 10−12 trillionth -12

femto f 10−15 quadrillionth -15

Example 3.14 Nanoparticles and Tattoos

On 13 Sept 2017 New Zealand Hedrald published the piece Why tattoos couldgive you cancer https://bit.ly/2Mt62ZI reporting on a study pointing at potentialdangers of having tattoos. The author of the study said that they: “[...] alreadyknew that pigments from tattoos would travel to the lymph nodes because of visualevidence: the lymph nodes become tinted with the colour of the tattoo. It is theresponse of the body to clean the site of entrance of the tattoo. What we didn’tknow is that they do it in a nano form, which implies that they may not have thesame behaviour as the particles at a micro level. And that is the problem: wedon’t know how nanoparticles react.”

Note: There are two very large numbers which have special names. A googolis written as 1 followed by 100 zeroes, i.e., 10100. A googolplex is defined as10googol which is 1 followed by googol zeroes. Neither number has a meaningin math or in physics. For further names and symbols consult Wikipedia, underOrder of Magnitude.

Going back to the list of large/small numbers (pages 47-49): In (2), 1 nanometreis 0.000000001 = 10−9 metres. One hundred million neurons (4) is 100, 000, 000 =108 neorons. One micrometre (5) μm is 10−6 metres.

One light year (6) is 9,460,730,472,580,800 metres, which is about 9.461 · 1015

metres. The mass in (7) can be written as 0.00000005 = 5.0 · 10−8, and thus itsorder of magnitude is −8.

The number in (8) is 1, 350, 000, 000, 000, 000, 000, 000 = 1.35 · 1021 litres. Inwords, the order of magnitude of the volume of water in our oceans, in litres, is21. Petabyte in (11) means 1015 bytes, i.e., 1015 = 1, 000, 000, 000, 000, 000 bytesor, in words, a million billion bytes.

Example 3.15 Kilobytes, Megabytes, Gigabytes, ...

How large are data files?The smallest unit of data that carries information is called a bit, and has only

two values, which are often denoted by 0 and 1. A byte is a group of 8 bits, andis the smallest unit that can be addressed (accessed) on a device that stores data.For instance, uppercase ‘A’ is represented as 01000001, lowercase ‘a’ is 01100001,and the number 7 is 00110111.

One kilobyte (KB) is not exactly 1000 bytes, but 210 = 1024 bytes (thisdifference is often neglected, and, for practical purposes, we say that 1 KB is equalto 1,000 bytes. 1024 kilobytes form a megabyte (MB), 1024 megabytes form agigabyte (GB), and 1024 gigabytes form a terabyte (TB).


Based on this (and thinking of 1024 as 1000) we say that 1TB has 1000 GB,or 1,000,000 MB, one GB has one billion bytes, and so on.

The size of an average text-only email (so no embedded pictures or media, andno attachments) is about 10-20 kilobytes (KB). A single photograph, taken with aphone, could require anywhere between 1-10 megabytes of space (could be more,for instance if we use a professional camera; for the argument below, assume it’s6 MB). A full-length movie, depending on the resolution (standard definition vshigh definition), could have the size of 2-6 GB (gigabytes). (Recall that 1 MB =1 million (106) bytes and 1 GB = 1 billion (109) bytes.)

For instance, this screenshot from iTunes shows that the movie Grizzlies re-quires about 4 GB in high definition (HD) and 1.6 GB in standard definition(SD).

Thus the storage required for a movie is three orders of magnitude larger thanthat for a photograph, and 5 orders of magnitude larger than the size of a text-onlyemail.

Example 3.16 Roaming Charges

Why do we have to know about megabytes and gigabytes?One reason is roaming charges that companies charge when we are outside

of our calling zones. These charges can be substantial. For instance, if we do nothave a roaming (data plan) with Fido, roaming charges if we use our phone in theU.S. are $ 7.99 fpr 50 MB of data.

Suppose that we watch The Grizzlies from our hotel room in Los Angeles.In standard definition (there is no reason why we should use high definion on a


laptop) the movie is 1.6 GB = 1,600 MB. So, the roaming charge will be1, 600

50· 7.99 = 32 · 7.99 = 255.68

With 13% HST, it’s about $ 289 !!!If we were watching the same movie from Morocco

we would accumulate roaming charges of1, 600

3· 9.99 = 5, 328

dollars, which amounts to about $ 6,020 after taxes.

This is not an academic discussion only; these things happen when people arenot careful.

On March 2013 in the article Dad gets $ 22,000 data roaming “shock” fromFido (https://bit.ly/2JUz9DT) CBC News reports on an 11-year old who streamedYouTube videos in Mexico. We read “A B.C. dad is accusing Rogers of pricegouging, after his 11-year-old son mistakenly racked up $ 22,000 worth of datacharges on his fathers phone, during a family trip to Mexico. [...] his son got asunburn and was allowed to spend time playing video games in the family’s hotelroom over the course of three days. He had games installed on the phone, but alsostreamed several hours of video. The company later told him his son had used$ 22,000 worth, approximately 700 megabytes. According to Rogers website, thatsabout 12 hours of YouTube video streaming.”

Multiplication Principle

One way to generate large numbers (which can be quite useful) is to combineobjects, each of which has several (or many) variations.

Multiplication principle states that if X happens in m ways and Y happens inn ways, then both X and Y happen in m · n ways. This principle applies to anynumber of things: if X1 happens in m1 ways, X2 happens in m2 ways, X3 happensin m3 ways, ..., and Xk happens in mk ways, then X1, X2, X3, . . . , Xk happen inm1 · m2 · m3 · . . . · mk ways.

Example 3.17 How Many Coffees?

You are in a coffee shop and are faced with options: you need to pick the sizeof your drink (small, medium, large, extra large), choose a flavour (plain, vanilla,cinnamon), and make a decision about dairy (milk, cream, no dairy). How manydifferent coffees can you order?To get a feel, start by writing down options in an organized way:

small, plain, milksmall, plain, cream


small, plain, no dairysmall, vanilla, milksmall, vanilla, creamsmall, vanilla, no dairysmall, cinnamon, milksmall, cinnamon, creamsmall, cinnamon, no dairy

There are no other options which involve a small size coffee. Note that there are 9options in this list, where 9 is the product of 3 (options for flavour) and 3 (optionsfor dairy). We can now copy this list and replace ‘small’ by ‘medium’, to get 9 newoptions. Then we replace ‘small’ by ‘large’ for 9 more options and replace ‘small’by ‘extra large’ for the final 9 options. Thus, the answer is 9 + 9 + 9 + 9 = 36.

Using the multiplication principle, we see that there are 4 · 3 · 3 = 36 differentoptions for our drink.

Example 3.18 Credit Card Numbers

Credit card numbers vary in size from 13 to 19 digits, with the most commonlyused cards having 15 or 16 digits. The first six digits from the left are not random:the first digit (or the first two digits) identify a major card network that the cardbelongs to: Visa (4), Mastercard (5), American Express (34, 37), Diners Club (36,38), and so on. The remaining digits in this initial group of 6 digits identify thebank which issued the card, and its type (cash back, tied to a loyalty program,travel rewards, no annual fee, low interest rates, and so on). For instance, TDAeroplan Visa Credit Card numbers start with 4520 88, Bank of America GoldVisa cards start with 4800 11, and Citibank Platinum mastercard cards start with5424 18.

The last digit is not random either – it is called a check digit, and is calculatedfrom the remaining digits. Assuming that the card has 16 digits (as all Visa andMastercard cards do) we see that only 9 digits are left for the actual credit cardnumber.

4 5check digit

80 82

Into each of the nine empty places we can put any digit 0, 1, 2, ..., 9, i.e., wehave 10 choices for each place. Using the mulitplication principle, we see that wecan generate 10 · 10 · . . . · 10 = 109 (one billion) numbers.

In reality, not all numbers are valid credit card numbers (the same digit ap-pearing in all nine places is not acceptable, for instance). As well, Luhn’s algorithm(the algorithm used to check whether a given number is a valid credit card num-ber) blocks additional candidates – if we transpose two neighbouring digits, wewill not get a valid credit card number. As well, if we replace a single digit withanother digit, we obtain a non-valid card number.


Example 3.19 Short URLs

You have noticed that many links in this book look like https://bit.ly/2JUz9DT.They are shorter versions (and thus more convenient to type and use) of theoriginal URLs, produced by the free URL shortener https://bitly.com/. This URLshortener (as any other URL shortener) maintains a large dictionary where a shortURL is tied to the original URL, which is how this works.

For instance, the URL of Internet Live Stats web page (which provides dataon the size of the Internet) is https://www.internetlivestats.com/total-number-of-websites/ and its shortened version is https://bit.ly/2KW7dC9.

How many different web pages can this shortener organize? A shortened URLis uniquely identified by the 8-letter string following https://bit.ly/. A simpleexperiment convinces us that this string is case-sensitive. Thus, in each of the 8places in the string we can put a lowercase letter (26 options), or an uppercaseletter (26 options), or a digit 0-9 (10 options)–thus a total of 62 options.

The multiplication principle tells us that the total number of web pages thatcan have bit.ly shortened URL is equal to 62 multiplied by itself 8 times, which is

628 = 218, 340, 105, 584, 896

approximately 2.18 · 1014 (or 218 million billion). This is several orders of magni-tude larger than all indexed (accessible) web pages on Internet (see Internet LiveStats).

Example 3.20 Licence Plates, IP Addresses and Postal Codes

(1) What is the largest number of car licence plates for vehicles registered inOntario, assuming that the format is LLLL DDD (i.e., four letters followed bythree digits). Assume that the leftmost letter is either A, B, or C. State a reasonwhy the actual number of plates is smaller than the number you obtained.

(2) An IP address is a numeric code which is assigned to every device (computer,phone, router, smart fridge, car, etc.) connected to the Internet (IP stands forInternet Protocol). Until a few years ago, the standard was Internet Protocol 4(IPv4), introduced in 1982, with addresses of the form

DDD.DDD.DDD.DDD

where D represents a single digit (with leading zeros removed). For instance, theauthor of this text was writing it in a coffee shop, and his laptop was assigned theaddress

In theory, what is the maximum number of IPv4 addresses?In practice, the internet faced running out of IPv4 addresses, so since 2010s a

new addressing mechanism, called IPv6, has been introduced. The format is

HHHH:HHHH:HHHH:HHHH:HHHH:HHHH:HHHH:HHHH

(8 groups of 4 characters) where H could be a digit 0-9 or a letter a, b, c, d, e orf. The adoption is still in progress, and it will take time until all devices switch toIPv6 addressing.

Checking for an IPv6 address for his laptop connected through the caffe’s wifi,the author reached the site https://test-ipv6.com/ and obtained the following reply


Anyway, here is an example (the address is not case sensitive):

FE80:0A90:100B:2300:0202:B3FF:FB1E:8146

In theory, what is the maximum number of IPv6 addresses?

(3) Postal Codes in Canada consist of 6 alphanumeric characters, written in theform letter-digit-letter space digit-letter-digit.

L8S 4K1forward sortation

arealocal delivery

unit

The leftmost letter is the postal district, which, outside of Ontario and Quebec,represents an entire province or territory (A for Newfoundland and Labrador, Bfor Nova Scotia, etc.) Two largest metropolitan areas have their own letter (M forToronto (GTA) and H for the Montreal region).

The digit in the second place is used to indicate the type of habitation: 0 isfor rural, and 1-9 designate urban regions.

The local delivery unit “specific single address or range of addresses, whichcan correspond to an entire small town, a significant part of a medium-sized town,a single side of a city block in larger cities, a single large building or a portion ofa very large one, a single (large) institution such as a university or a hospital, ora business that receives large volumes of mail on a regular basis” (Wikipedia).

Postal codes do not include the letters D, F, I, O, Q or U.What is the theoretical maximum of postal codes in Toronto? In Alberta?

(The actual maximum is smaller, since Canada Post reserves some codes for specialpurposes, including H0H 0H0 for Santa Claus).

Note: To think of really really large numbers, consider the number of possiblechess games. It is known that after three moves of both players, there are about121 million possible board setups/games.

In the article FYI: How Many Different Ways Can a Chess Game Unfold?(Popular Science, https://bit.ly/2JVWHYM) we read “According to JonathanSchaeffer, a computer scientist at the University of Alberta who demonstrates A.I.using games, ‘The possible number of chess games is so huge that no one willinvest the effort to calculate the exact number.’ Some have estimated it at around10100,000. Out of those, 10120 games are typical: about 40 moves long with anaverage of 30 choices per move.”

Compared to chess, the standard (3×3×3) Rubik cube has a much smallernumber of configurations: 43,252,003,274,489,856,000, i.e., about 4.32 · 1019. Ifinterested, visit https://www.therubikzone.com/number-of-combinations/.


Absolute and Relative Numbers

Numeric information can be presented in absolute form, i.e., as a number, withno reference to anything else (any other number) (such as “I bought bananas andpaid $ 1.38”). When some kind of reference is included, we call such numericinformation relative (as in “I bought 2 pounds of bananas and paid $ 1.38”). So,price on its own is an absolute number, whereas unit price ($ 1.38/2 = $ 0.69 perpound) is a relative number.

Example 3.21 Absolute and Relative Numbers

Canadian Cancer Society publishes breast cancer statistics (incidence, i.e., thenumber of new cases, and mortality rates) on its web page https://bit.ly/1rSTusm

Classify each number in the table as providing either relative or absolute in-formation.

On the Government of Canada information page about Invasive meningococ-cal disease (IMD) https://bit.ly/2T8xEb8 we read: “Between 2006 and 2011, anaverage of 196 cases of IMD was reported annually in Canada, with an averageincidence of 0.58 cases per 100,000 population.”

The phrase “an average of 196 cases” gives an absolute number, whereas “av-erage incidence of 0.58 cases per 100,000 population” is relative information.

Examples 3.22 Absolute and Relative

(1) You are given a job offer in Charlottetown, P.E.I., and the letter of offer statesthat your starting salary will be $ 45, 000. You are not happy with this information(why?) What kind of information would you prefer to have?

(2) Starbucks’ Grande (16 fl oz ≈ 473 millilitres, which is about 473 grams) Blonderoast has 360 mg of caffeine. (Source: Caffeineinformer, https://bit.ly/2q9P1I8.)The calculation 0.360/473 = 0.00076 shows that about 0.076 % of the coffee isactually caffeine.

Identify absolute and relative information, and comment on their usefulness.

(3) Looking at online prices at Loblaws (https://bit.ly/313FsKZ) in 2019, we find:President’s Choice Rosemary (40 g) for $ 2.99Brussel Sprouts $ 0.44 / 50 g


King Oyster Mushrooms $ 0.72/25 gRadishes (454 g) $ 1.99Minced Garlic Jar (128 g) $ 2.99

Which of these items is the cheapest (per gram), and which is the most expensive?

Assume that a quantity changes from A to B. How can we describe by how muchit changed?

We can calculate the absolute change B−A (i.e., new value minus old value),or relate the absolute change to the starting value A to compute the relativechange

B − A

A

Consider the following situation: on island X there are 40 monkeys, and onisland Y there are 400 monkeys. A year later, the counts were 50 monkeys onisland X and 410 monkeys on island Y.

Note that the absolute changes on both islands are the same: on island X :50 − 40 = 10, and on island Y : 410 − 400 = 10. However, we feel that thesenumbers do not tell the whole story, so we put them into context: the relativechange on island X is

50 − 4040

=1040

= 0.25 = 25%

and on island Y it is410 − 400

400=

10400

= 0.025 = 2.5%

Thus, island X experienced a much dramatic change in its population of monkeys.(In this case, this is obvious if we just look at the numbers – but things can bemore complicated.)

Notes: Information given as percent (percentage) is relative information. When-ever we see precent, our first question must be–percent of what? In the abovecase, it is 25% of the initial population (40) of monkeys on island X and 2.5% ofthe initial population (400) of monkeys on island Y .

A fraction (representing division) can be viewed as percent (of course, a deci-mal number needs to be converted into percent), and is thus relative as well:

IfA

B= p, then A is p percent of B

For instance, from 1/4 = 0.25 we conclude that 1 is 25 percent of 4. Or, wecan interpret 40/400 = 0.1 by saying that the initial population of monkeys onisland X was equal to 10 percent of the initial population of monkeys on island Y.

Sensitivity

We have seen that rounding numbers off does not change the value of a numberby much. But what happens if we round off numbers which enter calculations?

We will answer a slightly different – but related – question: can a small changein a number produce a large change in the value of a number?

We use examples to investigate what happens when we change one numberinvolved in a calculation by 5 percent.Addition: adding 458 and 5100 we obtain 458 + 5100 = 5558. If we increase 5100by 5%, we obtain 5100 · 1.05 = 5355 and this time, the sum is 458 + 5355 = 5813.The two sums are of the same order of magnitude; from 5813/5558 = 1.0458798we conclude that the sum increased about 4.59%.


Multiplication: pick any two numbers x and y, and multiply them to get xy. Whenwe increase x by 5%, we get 1.05x, and this time, the product is 1.05xy. Compare:1.05xy/xy = 1.05, i.e., the product increased by 5% as well.Division: when we increase the numerator in x/y by 5%, we get 1.05x/y. Dividingthe two numbers, we get

1.05x/y

x/y=

1.05x

y· y

x= 1.05

Thus, the quotient mimics the change in the numerator.When we increase the denominator in x/y by 5%, we get x/(1.05y). Dividing

the two numbers, we getx/(1.05y)

x/y=

x

1.05y· y

x=

11.05

= 0.95239

Thus, the value of the fraction decreases by about (1− 0.95239 = 0.04761) 4.76%.Powers: consider 108. If we increase the base by 5%, we obtain 10.58. From

10.58

108= 1.47745

we conclude that the increase is significant - about 47.75%.If we increase the exponent by 5%, we obtain 108.4. From

108.4

108= 2.15119

we conclude that the increase is very large - about 151.19%.

To summarize: a 5% change in a number produces the same, or about thesame change in the result of addition, multiplication and division. However, whenthe powers are concerned, the changes are much larger.

Visualizing Numbers

We can often visualize, and develop good intuition, about numbers which are onthe ‘human scale’ (what exactly that means is not clear, and is not possible tospecify). When we see a group of people we can immediately tell whether thereare 30 or 100 people in it. Driving a car, we can feel the difference between movingat 20 km/h or at 70 km/h. We can tell when we carry 2 kg of bananas, comparedto 10 kg of bananas.

But what about small and large numbers? Let’s look at a few examples.

Example 3.23 Visualizing Large and Small Numbers.

(1) On 2 July 2016, The Guardian (https://bit.ly/2hGTueV) published the articleWe don’t know if your baby’s a boy or a girl: growing up intersex about a childnamed Jack who was diagnosed with mixed gonadal dysgenesis. As a consequence,Jack’s sex couldn’t be determined at birth, and doctors needed time to assign it.We read: “Jack’s specific diagnosis is rare, but being born with a blend of femaleand male characteristics is surprisingly common: worldwide, up to 1.7% of peoplehave intersex traits, roughly the same proportion of the population who have redhair, according to the Office of the United Nations High Commissioner for HumanRights. ”

(2) In the piece The North and the great Canadian lie published in Maclean’s (11Sept 2016, at https://bit.ly/2OpKYWH) the author, attempting to describe thesize of the Canadian population living in the North, writes: “How many Canadiansactually live up north? Approximately 118,000. That’s one-third of one per centof the national population. To put it another way, about as many Canadianslive in Australia as live in Nunavut. If the entire population of the Northwest


Territories decided to attend an Edmonton Eskimos [now called Edmonton Elks]game, Commonwealth Stadium would still have 10,000 empty seats.”

(3) This clock, taken from Wikipedia (https://bit.ly/1VkJrbO) shows geologicaldevelopment of our Earth classified by eons and eras, from its creation (about 4.6billion years ago) until today. In the picture, Ga (giga) = billion yrs ago and Ma(mega) = million yrs ago.

(4) On 12 Dec 2016, CBC News reported on a large amount of raw sewagethat flows into Canadian rivers, lakes and oceans. The article, titled Billionsof litres of raw sewage, untreated waste water pouring into Canadian waterways(https://bit.ly/2YsjEXN) states that “[...] the amount of untreated waste water,which includes raw sewage and rain and snow runoff, that flowed into Canadianrivers and oceans last year would fill 82,255 Olympic-size swimming pools - anincrease of 1.9 per cent over 2014.”

(5) The chance of dying in a plane crash has been estimated to be about 1 in 7million, i.e., 0.00000014 (see [1], page 117). This means that if a person picks one(random) flight every day it would be 19,000 years before they could expect to diein a plane crash.

(6) In Humans Have Bogged Down the Earth with 30 Trillion Metric Tons of Stuff,Study Finds (Smithsonian.com, 9 December 2016, https://bit.ly/2ZgULj5) we read“Humans have produced a lot of stuff since the mid-20th century. From America’sinterstate highway system to worldwide suburbanization to our mountains of trashand debris, we have made a physical mark on the Earth that is sure to last foreons. Now a new study seeks to sum up the global totality of this prodigioushuman output, from skyscrapers to computers to used tissues. That number, the


researchers estimate, is around 30 trillion metric tons, or 5 million times the massof the Great Pyramid of Giza. And you thought you owned a lot of crap.”

Examples 3.24 Scaling

Scaling is a powerful way of visualizing data. Here we explore several examples.(1) You are making a scale model of our Solar System and placed the Sun intoHamilton Hall and Earth into MUSC (= McMaster University Student Centre;assume that the distance between Hamilton Hall and MUSC is 150 metres). Howfar from Hamilton Hall would you have to place our (newly declared) dwarf planetPluto?

????

(2) In [2] we find estimates on the number of species (described and undescribed)on Earth, by group: chordates (80,500) plants (390,800) fungi (1,500,000) inverte-brates (6,755,830). In a 50-minute lecture, you need to talk about the four groupsin such a way that the time that you lecture about a group is proportional to thesize of that group. For each group, compute how long (in minutes and seconds)you will be lecturing about it.


(3) There are 640 students registered in a course. Assuming that the students inthe course have been selected to reflect the demographics of the entire human pop-ulation, how many Canadians would be in the class? How many Indians (citizensof India)?

(4) You decide to paint the floor in your room (of area 15 m2) in a combinationof blue and brown, where blue represents oceans and brown represents land onEarth. If you were to paint it to reflect the actual ratio of oceans to land whatarea of your floor would be painted brown?

Notes and Comments[1] People usually have more problems with small numbers than with large num-bers. One reason lies in the way we write numbers: for large numbers, we usecommas to separate digits into groups of three, but there is no such separationfor small numbers. Consequently, we can easily see read 15, 000, 000 as 15 million,but need to think when reading 0.000015 (so we count zeroes in 0.000001 to seethat it is one millionth, and so 0.000015 is 15 millionths).

Another difficulty stems from the fact that small numbers suggest division (asin 0.000015 = 15/1, 000, 000) and, of all algebraic operations, division is the leastintuitive.[2] Small quantities, by repetition/multiplication, can generate large quantities,and thus have significant effects: think of the pollution coming from one car, andthen multiply by one billion to estimate the pollution coming from an estimated 1billion cars on our planet. One car does not cause a traffic jam, but 10 thousandcars do. One student refusing to use a plastic fork for the lunch they boughtdoes virtually nothing to reduce plastic waste. However, all university students inCanada routinely refusing to use plastic cutlery (and, for instance, bringing theirown cutlery) would make a difference.[3] Recall that the multiplication principle can produce huge numbers by combiningquantities which are fairly small.

Two more examples: there are 2,235,197,406,895,366,368,301,560,000 = 2.235·1027 ways to divide a standard deck of 52 cards among four players. There are120 permutations (orderings) of five objects (if we label them by A, B, C, D, andE, then ABCDE is a permutation, and so are ACBDE, DECAB, CEABD, andso on). There are over 3.6 million permutations which involve 10 objects, about2.4 · 1018 permutations of 20 objects, about 3.0 · 1064 permutations of 50 objects,and about 9.3 ·10157 permutations of 100 objects (for comparison, there are about3.28 · 1080 particles (electrons, quarks, etc.) in the entire Universe).(4) Sometimes, small cannot affect big, but does affect another small quantity: if160 thousand people emigrate from China, it would not affect the Chinese popu-lation in any major way. However, the same number of emigants would halve thepopulation of Iceland.


[1] Check out the CBC News report The race to exascale is on - while Canadawatches from the sidelines (27 December 2019, at https://bit.ly/2QQafrg). Amongother things we find the following: “So what exactly is an exascale computer? It’sa supercomputer capable of performing more than a billion billion calculations persecond - or 1 exaflops.”

“‘Exa’ is the metric system prefix for such grandiose numbers, and ‘flops’ isan abbreviation of ‘floating-point operations per second’ [floating-point means real


numbers]. For comparison, my laptop computer is capable of about 124 gigaflops,or 124 billion calculations per second, which sounds fast. According to the TOP500list, today’s fastest supercomputer is Oak Ridge National Laboratory’s Summit,which tops out at a measured 148.6 petaflops - about one million times faster thanmy laptop.”

[2] Large numbers are mentioned in The Guardian article Electricity needed tomine bitcoin is more than used by ’entire countries’ https://bit.ly/3u47m8w on 27February 2021. We read that Bitcoin mining, i.e., “the process in which a Bitcoinis awarded to a computer that solves a complex series of algorithm” requires hugeamounts of energy. “Cambridge’s Centre for Alternative Finances estimates thatbitcoin’s annualised electricity consumption hovers just above 115 terawatt-hours(TWh) while Digiconomist’s closely tracked index puts it closer to 80 TWh.”

As well: “A single transaction of bitcoin has the same carbon footprint as680,000 Visa transactions or 51,210 hours of watching YouTube, according tothe site. A paper from 2018 from the Oak Ridge Institute in Ohio found thatone dollar’s worth of bitcoin took 17 megajoules of energy, more than double theamount of energy it took to mine one dollar’s worth of copper, gold and platinum.”

[3] Leap years were introduced to synchronize our calendars to astronomic events.For smaller-scale modifications astronomers have introduced leap seconds. Readabout it in What Is a Leap Second? at https://bit.ly/3pL3ozd. We learn thatleap seconds are added “from time to time to ensure our clocks reflect the Earth’srotation speed as accurately as possible. The speed at which our planet rotatesaround its axis fluctuates daily, and it slows down very slightly over time. Byadding an extra second to the time count, we effectively stop our clocks for thatsecond to give Earth the opportunity to catch up.”

When are leap seconds added? “Latest update: In July 2020, the InternationalEarth Rotation and Reference Systems Service (IERS) announced that there wouldbe no leap second in December 2020. There was no leap second added in June 2020either. The next possible date is June 30, 2021. However, the Earth’s rotationhas been speeding up lately, so it is becoming increasingly unlikely that any leapseconds will be added in 2021.”

Chapter references[1] Niederman D. and Boyum, D. (2003). What the Numbers Say. New York:Broadway Books.

[2] Chapman, A. D. (2009). Numbers of Living Species in Australia and the World(2nd ed.). Canberra: Australian Biological Resources Study. pp. 180.

Chapter 4 Counting and Number Systems 69

4 Counting and Number Systems

We are so used to our decimal system that we tend to forget that the way we countand write numbers are conventions, based on historic choices. In this chapterwe unpack the decimal system, and then discuss other number systems, some ofwhich are used today in major ways (deep down, computers do not work with thedecimal system; computer software, Internet, security, et.c all depend on binaryand hexadecimal systems (more about these soon).

Why does an hour have 60 minutes, and not 100 minutes? Why is the rightangle equal to 90 degrees, and not to some power of 10, such as 10 or 100? Whydozen eggs, and not 10 or 15?

The decimal system is a positional number system based on powers of 10.Positional means that the location of a digit within a number determines itsvalue: for instance, the first (leftmost) occurrence of 7 in 17074 contributes 7000to the value of the number, whereas the second 7 contributes 70 to its value.

Roman numeral system is non-positional: a digit is worth the same, nomatter where it is located within a number. Recall that M represents 1000, D is500, C is 100, L is 50, X is 10, V is 5 and I is 1. For example, MMCCCXXXVIIIrepresents the number 2000 + 300 + 30 + 5 + 3 = 2338. The digits are writtenfrom the largest value to the smallest value. There is one exception to this rule:if a smaller value is to the left of a larger value, then it is subtracted: IX is10 − 1 = 9, IV is 5 − 1 = 4 (exception to the rule that the same symbol cannotappear more than three times in a group, 4 is sometimes written as IIII), and CMis 1000− 100 = 900.

Historically, there have been many non-positional systems (for instance usedin ancient Greece and Egypt), as well as positional systems (Maya in CentralAmerica, India, etc.). Non-positional systems become inconvenient when one triesto do algebra (imagine multiplying CCIX by LXXXV); however, the are still used,mostly as labels:

Decimal Number System

To understand how this system works, we deconstruct a number:

38, 912 = 30, 000 + 8, 000 + 900 + 10 + 2= 3 · 10, 000 + 8 · 1, 000 + 9 · 100 + 1 · 10 + 2 · 1= 3 · 104 + 8 · 103 + 9 · 102 + 1 · 101 + 2 · 100


Because of the last line, we say that the decimal system is based on the powersof 10; and the numbers multiplying these powers are the digits of the originalnumber. (Keep in mind that 100 = 1 and 101 = 10.) One more example:

509 = 500 + 9 = 5 · 100 + 0 · 10 + 9 · 1 = 5 · 102 + 0 · 101 + 9 · 100

Note that zero does not have a value, however, it is important that it is there asit serves as a placeholder–without it, we would have a different number, 59.

Thus, i order to write a number in decimal system, we use ten digits: 0, 1, 2,3, 4, 5, 6, 7, 8, and 9. All numbers can be written based on these digits, and withthe positioning based on the powers of 10.

A decimal number can be analyzed in the same way:

7.830056 = 7 + 0.8 + 0.03 + 0.00005 + 0.000006= 7 · 1 + 8 · 0.1 + 3 · 0.01 + 5 · 0.00001 + 6 · 0.000001= 7 · 100 + 8 · 10−1 + 3 · 10−2 + 0 · 10−3 + 0 · 10−4 + 5 · 10−5 + 6 · 10−6

To count in the decimal system, we go: 0, 1, 2, 3, . . . , 8, 9; we reached ten, forwhich we do not have a single digit, but instead we introduce a new position tothe left (tens): 10, 11, 12, . . . , 18, 19; we do not have a new symbol for twenty;instead, we increase the tens digit: 20, 21, . . . , 97, 98, 99 (which is the largest two-digit number); to write one hundred, we introduce a new position to the left(hundreds): 100, 101, 102, and so on.

The decimal sysytem is also called base 10 system and when we need todistinguish between different number bases, we will write (38, 912)10. However,we will stick to aconvention that a number written without indication of its baseis a base 10 number.

Number Systems

What does numer in base 6 look like? (Base 6 counting has been used by Indige-nous peoples in parts of Canada, as well as in Papua New Guinea.)

Base ten numbers need 10 digits–base six numbers need only six: 0, 1, 2, 3, 4,and 5. Thus, 3402 is a valid base 6 number, and we will write (3402)6; however,3476 is not a valid base 6 number (as it contains 7 and 6 which are not base 6digits).

The value (i.e., the decimal equivalent) of a base 6 number is determied anal-ogously to the base 10 , but this time, it’s the powers of 6:

(2315)6 = 2 · 63 + 3 · 62 + 1 · 61 + 5 · 60

= 2 · 216 + 3 · 36 + 1 · 6 + 5 · 1 = 551 = (551)10Here is a convenient way to do this: to convert (235041)6 into a decimal number,write the powers of six above its digits, starting from zero, and going from rightto left:

Now write the number using the powers of 6:

(235041)6 = 2 · 65 + 3 · 64 + 5 · 63 + 0 · 62 + 4 · 61 + 1 · 60

= 2 · 7776 + 3 · 1296 + 5 · 216 + 0 · 36 + 4 · 6 + 1 · 1 == 20, 545 = (20, 545)10

How do we count in base 6?We start as usual: (0)6, (1)6, (2)6, (3)6, (4)6, (5)6; the next number has a

decimal value of 6, but in base 6, (6)10 = (10)6. (Note: in the decimal system,when we cannot go any further with single digits, we introduce a new place, i.e.,


we go from 9 to 10; in base 6, the last single digit is (5)6 and the first next digitis (10)6 – the same thing!)

So, we count: (0)6, (1)6, (2)6, (3)6, (4)6, (5)6, (10)6, (11)6, (12)6, (13)6, (14)6,(15)6; (now think about what happens when we reach 19 in the decimal system)(20)6, (21)6, (22)6, (23)6, (24)6, (25)6; then (30)6, (31)6, and so on, until we reachthe largest two-digit number (it’s 99 in the decimal system, so it must be) (55)6.The next number, and the smallest three-digit number on base 6 is (100)6. Thedecimal value of this number is

(100)6 = 1 · 62 + 04 · 61 + 0 · 60 = 36.

Continue counting:(100)6, (101)6, (102)6, . . . , (105)6, (110)6, (111)6, (112)6, . . . , (115)6,(120)6, (121)6, (122)6, . . . , (155)6,(200)6, (201)6, (202)6, . . . , (205)6, (210)6, . . . , (555)6,(1000)6, (1001)6, and so on.

How do we do algebra in base 6? We thinkin bas 10, but write in base 6. Inthis example we add (42351)6 and (3041)6. Write the numbers one below the other

Add the rightmost digits: 1+1 = 2, and write 2 as the rightmost digit of the sum:

Next, 5 + 4 = 9, but we need to write 9 is base 6: (9)10 = (13)6. (Recall how wecounted in base 6.) So in the second (from right) position we enter 3 and carryover 1:

Now 1+3+0 = 4 (decimal value) which is also 4 in base 6. So the third digit is 4.

Adding the remaining digits, we obtain the answer:

So, after all, this is not so much different from the decimal system.


Exercises 4.1 Practice in Base 6

(1) Convince yourself that (100)6 = (36)10, (1000)6 = (216)10, and (10000)6 =(1296)10.

What is the decimal value of (100 . . .00︸︷︷︸

n

)6 if n is a positive number?

(2) Convert each number to find its decimal value: (204)6, (55)6, (1320)6, (555)6,and (10101)6.

(3) Convince yourself that the following calculation (addition) is correct:

(4) Add the following numbers, and write their sum as a base 6 number:(204)6 + (1231)6(3333)6 + (4444)6(2)6 + (55555)6(10352)6 + (43205)6

(5) What is the decimal equivalent of the largest 6-digit number in base 6?

Partial answers:

(1) Expand(10000)6 = 1 · 64 + (all other terms are zero) = 64 = (1296)10.

(2) (10101)6 = 1 · 64 + 0 · 63 + 1 · 62 + 0 · 61 + 1 · 60 = 1296 + 36 + 1 = (1333)10(4) (3333)6 + (4444)6 = (12221)6

(2)6 + (55555)6 = (100001)6(5) The largest 6-digit number in base 6 is (555555)6. If we continue counting, thenext number is (1000000)6 = (66)10 (see (1)). Thus, (555555)6 = 66 − 1 = 46, 655in the decimal system.

Working with any other number system is analogous to what we have just done.Base 20, or vigesimal number system is based on twenty. (Why would anyone

pick twenty? One explanation is that it is the number of human fingers and toes.)This system was used in many countries, and by many cultures: Maya and Aztecin Central and South America, Inuit in Canada and Greenland, and in Africa (seeWikipedia, https://bit.ly/2LQdezA, for a complex system of Yoruba numerals).Base 20 names for numbers appear French, Welsh, Danish, Albanian, and a fewother languages.

If the base 10 system has 10 digits and the base 6 system has 6 digits, thenthe base 20 system must have 20 digits, representing decimal values 0, 1, 2, 3, . . .,18, and 19.

Inuit peoples have used Kaktovik Inupiaq numerals

1 15141312111098765432 19181716 0

Maya peoples used the digits below (left), with a stylized shell as a symbol for zero(for details, see Wikipedia under Vigesimal).


On the right is a stela (vertical stone pillar), on display at the Royal OntarioMuseum in Toronto, that was used to record a date (and hence the numbers comewith pictorial representations of units used in Maya calendars).

To work with base 20 numbers, we will use decimal numbers 0, 1, 2, 3, . . . , 18,19, and separate them by a semicolon, as in (9; 14; 0; 18)20. Its decimal equivalentis calculated analogously to what we’ve done before–write the powers above thedigits, from right to left:

and then keep in mind that the base is 20:

(9; 14; 0; 18)20 = 9 · 203 + 14 · 202 + 0 · 201 + 18 · 200

= 9 · 8000 + 14 · 400 + 0 · 20 + 18 · 1 = (77618)10

How did Maya people count in their system? Start with single-digit numbers:(0)20, (1)20, (2)20, . . . , (18)20, (19)20;

The next number is 20 in decimal notation; in base 20, it is (1; 0)20. We continue(1; 0)20, (1; 1)20, (1; 2)20, . . . , (1; 18)20, (1; 19)20,(2; 0)20, (2; 1)20, (2; 2)20, . . . , (2; 18)20, (2; 19)20;

The last set of two-digit numbers is(19; 0)20, (19; 1)20, (19; 2)20, . . . , (19; 18)20, (19; 19)20;

To write the next number we need to introduce a new place (value)(1; 0; 0)20;

By the way, (19; 19)20 = 19 · 201 + 19 = 399, and (1; 0; 0)20 = 1 · 202 = 400. Makessense! A new place value always starts with a new power of the base. Thus,(1; 0; 0; 0)20 = 1 · 203 = 8000, and (1; 0; 0; 0; 0)20 = 1 · 204 = 160000, and so on.

To resume counting:(1; 0; 0)20; (1; 0; 1)20, (1; 0; 2)20, . . . , (1; 0; 18)20, (1; 0; 19)20,(1; 2; 0)20, (1; 2; 1)20, (1; 2; 2)20, . . . , (1; 2; 18)20, (1; 2; 19)20,and so on ... the largest three-digit number is(19; 19; 19)20

whose decimal value is 7,999. We know that because it is 1 smaller than the nextnumber (1; 0; 0; 0)20 = 1 · 203 = 8000. Then, it is

(1; 0; 0; 0)20, (1; 0; 0; 1)20, (1; 0; 0; 2)20, and so on.

Here is an example of algebraic manipulation is base 20: using our usual algorithm,we add (17; 11; 0; 18)20 and (19; 9; 9; 15)20.


Exercises 4.2 Practice in Base 20

(1) What is the decimal value of (1;0; 0; . . . 0; 0︸︷︷︸

n

)20 if n is a positive number?

(2) Convert each number from base 20 into a decimal number:(3; 19)20(19; 19; 19; 19)20(14; 0; 0; 2)20(10; 10; 10)20

(3) Add the following numbers, and write their sum as a base 20 number:(4; 5; 6)20 + (13; 12; 11; 10)20(18; 0; 19)20 + (17; 0; 16)20(19; 19; 19)20 + (1; 1; 1)20(14; 15)20 + (18; 19; 9; 0)20

Partial answers:(1) As usual, above the digits of the given number, write the numbers 0, 1, 2,3, . . . , n from right to left. The only non-zero term is 1 · 20n = 20n. Thus, forinstance,

(1; 0; 0; 0)20 = (203)10 = (8, 000)10(1; 0; 0; 0; 0; 0)20 = (205)10 = (3, 200, 000)10.

(2) (14; 0; 0; 2)20 = 14 · 203 + 2 · 200 = 14 · 8000 + 21 = 112, 002

(3) (18; 0; 19)20 + (17; 0; 16)20 = (1; 15; 1; 15)20(14; 15)20 + (18; 19; 9; 0)20 = (19; 0; 3; 15)20

Sexagesimal number system is based on the number 60. Its roots can betraced to ancient Sumerians (3rd millennium BCE), so it is a bit over 5,000 yearsold.

Base 60 looks familiar? Ancient Egypteans divided a day into 24 hours. Baby-lonians, who inherited the system from Sumerans, took it from there and dividedone hour into 60 minutes, and one minute into 60 seconds.

As well, it is believed that Babylonians divided a full circle into 360 degrees,one reason being that 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, and by many morenumbers (10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360), andthus it was easy to mentally divide a circle into pieces, i.e., mentally calculate anangle.

Geographic coordinates (longitude and latitude) are also given in angle de-grees; for instance, McMaster University is located at the latitude of 43.2609 de-grees N (North of Equator) and the longitude of 79.9192 degrees W (West ofGreenwich, which is defined to be zero degrees).

In base 60, we need sixty digits, representing decimal numbers from 0 to 59.Babylonians used the following digits; they did not have a sybmol for zero, andjust used blank space (source: Wikipedia article Sexagesimal):


Exercises 4.3 Practice in Sexagesimal System

(1) Count forward from (45; 59; 57)60, by listing 5 numbers that come next.

(2) Count backward from (1; ; ; 2)60, by listing 5 numbers that come before it.(Recall that space represents zero.)

(3) Convert each number into its decimal equivalent. (Recall that space representszero.)

(1; )60(1; ; )60(10; 30; 50)60(1; 1; 1; 1)60

(4) What is the decimal equivalent of the largest two-digit base 60 number?

(5) Find (45; 51; 12)60 + (59; 38; 41; 36)60

Partial answers:(2) (1; ; ; 2)60, (1; ; ; 1)60, (1; ; ; )60, (59; 59; 59)60, (59; 59; 58)60, (59; 59; 57)60.

(4) The largest two digit number is (59; 59)60, which is one smaller than

(1; 0; 0)60 = (602)10 = (3600)10.

Thus, (59; 59)60 = (3599)10.

Hexadecimal and Binary Systems

The hexadecimal system is based on the number sixteen. The digits consist ofthe usual decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, to which we add letters a (=10), b(=11), c (=12), d (=13), e (=14), and f (=15); sometimes, uppercase letters A, B,C, D, E, and F are used instead. Because we have all digits as single symbols, wecan write numbers in the hexadecimal system as we write numbers in the decimalsystem, one digit next to another (with no separation symbols needed). Thus,

(12)16 = 1 · 16 + 2 = (18)10(209)16 = 2 · 162 + 9 = (265)10(5a)16 = 5 · 16 + 10 = (90)10(100)16 = 162 = (256)10(ef7)16 = 14 · 162 + 15 · 16 + 7 = (3831)10


and so on.Hexadecimal numbers are widely used in anything related to computers and

electronic communication. For instance, MAC (media access control) address is aunique code assigned to a device which is connected to the Internet. It consists of6 groups of two-digit hexadecimal numbers:

Internet protocol IPv6 is a communications protocol that locates and identifiesa device on the Internet, so that it can route traffic to and from it. In NetworkSystem Preferences of a device we can find IPv6 codes in hexadecimal form

To access certain sites (banking, utilities, Amazon, etc.) we need to enter apassword. How does it work? When we register and/or enter a new password, itis converted into a very long hexadecimal number (that number is called hash;it could have 64, or 128 digits, or more). That hash is then saved (our actualpassword is not stored anywhere). The Python code below shows how passwords‘a’ and ‘A’ are stored in the memory of the page we are trying to access. Wheneverwe enter a password, its hash is calculated and compared to the stored hexadecimalhash.

Note that the passwords thus created are case-sensitive: ‘a’ and ‘A’ have twocompletely different hashes.

A screen on any electronic device we use (phone, tablet, laptop, computer,etc.) is divided into pixels. To display a page, a device’s software knows whereeach pixel is located, and what colour to assign to it.

There are several conventions which are used to define colour for screens ofelectronic devices. One of the most commonly used system is the RGB (red, green,blue) system. In this system, a colour is viewed as an ordered triple of numbers(r, g, b) where the numbers (defined to be between 0 and 255) indicate the intensityof red, green (lime), and blue, respectively. The combination of these intensitiesgives the colour of each pixel.

For instance, (255, 0, 0) is maximum intensity red, no green, and no blue, so itrepresents pure red colour. Often, the triple is written as a string of six charactersRRGGBB, where RR, GG and BB are hexadecimal equivalents of intensity. Thus,instead of (255, 0, 0) we use ff0000 (recall that (ff)16 = 255 in the decimal system).Likewise, 00ff00, or (0, 255, 0) represent green (lime) colour, and 0000ff or (0, 0, 255)


is blue. The picture below shows the three basic colours, as well as three coloursderived from the basic red, green and blue colours.

Because colours are defied as numbers, we can do various algebraic operationsand/or apply algorithms (and this is exactly what is going on in Photoshop, whenwe modify a photo, for instance by making the sky more blue than it actuallywas).

The binary system is based on the number two – so it uses only two digits, 0and 1 (it does not get any simpler than that!). For instance,

(101011)2 = 1 · 25 + 0 · 24 + 1 · 23 + 0 · 22 + 1 · 21 + 1 · 20

= 32 + 8 + 2 + 1 = (43)10Let’s count (and keep in mind that every time we encounter a power of two, weintroduce a new place value):

(0)2, (1)2,(10)2 = (2)10, (11)2,(100)2 = (4)10, (101)2, (110)2, (111)2,(1000)2 = (8)10, (1001)2, (1010)2, (1011)2, (1100)2, (1101)2, (1110)2, (1111)2,(10000)2 = (16)10, (10001)2,

and so on. Algebraic operations with numbers in the binary system are simple,and there is very little we need to memorize. In particular, here is the entiremultiplication table: (0)2 · (0)2 = (0)2, (0)2 · (1)2 = (0)2, (1)2 · (0)2 = (0)2, and(1)2 · (1)2 = (1)2.

The binary system is so simple that it is believed that if there is life somewhereelse un Universe, very likely they will be able to figure out the binary system.


In 1972 and 1973, two spacecrafts (callen Pioneer 10 and 11) were launched intospace (the photograph and the drawing above show Pioneer 11; source: NASAArchive The Pioneer Missions https://go.nasa.gov/2mYEzDw).

By 1983, Pioneer 10 left our Solar system, and it sent its last signal to Earthin 1995. In January 2019, it was calculated to be about 16 billion kilometres fromEarth. Pioneer 11 was lost (i.e., stopped communicating) in 1995 as well.

Both spacecrafts carried identical plaques, in the case that they were found byextraterrestrial life. The plaques contained vital information about humans: whatwe look like (with the outline of the spacecraft in the background, to show howtall we are); where we live (the star-like diagram on the left shows the location ofour Solar system with respect to nearby pulsars, and the distances to them); onthe top is a representation of the hydrogen atom, and on the bottom is our Solarsystem.

All numeric information is given in the binary system! A vertical stroke (|) repre-sents 1, and a horizontal stroke (−) represents 0.

The distances between planets and the Sun are given with reference to the distancefrom Sun to Mercury, which is given as | − |− = (1010)2, whose decimal equiv-alent is 10. The distance from Earth to the Sun is given as || − |− = (11010)2,or 26 (reflecting the fact that Earth is 2.6 times farther from the Sun than Mer-cury). Jupyter is roughly 13.4 times farther from the Sun than Mercury, so theinformation is encoded as | − − −−||− = (10000110)2 = 28 + 22 + 2 = (134)10.Notes: (1) As quantum computing develops, the classical binary (positive charge-negative charge, that we denote by 0 and 1) is going to be replaced by a qubit,the smallest quantum information unit, which stil has two states–thus the binarysystem will remain highly relevant.(2) In the conference paper on computer architecture A DNA-Based Archival Stor-age System https://bit.ly/2LSJeTW, the authors from University of Washingtonand Microsoft discuss DNA to store (archive) information. We read that suchstorage “[...] is an attractive possibility because it is extremely dense, with a rawlimit of 1 exabyte/mm3 (109 GB/mm3), and long-lasting, with observed half-lifeof over 500 years.” What is interesting is that this technology requires that eitherbase 3 or base 4 numbers are used (these options are discussed in the paper.)


(3) For those interested in the history of math: Plimpton 322 is a Babylonianclay tablet, famous for its content: it shows ratios of sides in triangles of varioussizes (recall that that’s exactly what trigonometric functions are). The tablet is anevidence that trig functions are at least a thousand years older than what historiansbelieved. (Source: AncientPages.com Mysterious Clay Tablet Reveals BabyloniansUsed Trigonometry 1,000 Years Before Pythagoras https://bit.ly/2MmbIoy.)

Mayan Calendar

Maya civilisation, developed by the Maya peoples in the regions of Central Amer-ica, was a highly advanced civilisation, which made major advances in architecture,astronomy, mathematics, and art.

Maya people inherited, and further developed, a variety of methods of timekeeping. They simultaneously used three calendars, depending on the purpose:The Haab counted days and months, and was used for everyday events, and shortertime-scale events. The Tzolkin consisted of 13 months with 20 days each, and wasused to determine the time for various ceremonies and religious events. The LongCount calendar was used to keep track of historic and long-term events.

The Haab divided a year into 18 months of 20 days each, and then added 5days to the end, thus making a year 365 days long. As neither Haab nor Tzolkinkept track of the year, how did people figure out the year? They did it by aningenious combination of the two calendars.

Place each calendar on a wheel, as shown, and align them so that the wheelstouch at the day labelled 1 on both wheels.

Tzolkin Haab

3

2601

259

2

365

3

2

364

1

44

This position marks the date Tzolkin 1, Haab 1. The following day bothwheels advance by 1, and the date is

Tzolkin 2, Haab 2Then

Tzolkin 3, Haab 3Tzolkin 4, Haab 4Tzolkin 5, Haab 5


and so on, untilTzolkin 259, Haab 259Tzolkin 260, Haab 260

Tzolkin Haab

2

259260

258

1

259

262

261

258

260

2633

The Tzolkin wheel completed one full revolution, so the following day, it is reset;the Haab continues:

Tzolkin 1, Haab 261Tzolkin 2, Haab 262

and so on. Skipping 101 days,Tzolkin 103, Haab 363Tzolkin 104, Haab 364Tzolkin 105, Haab 365

Now the Haab wheel resets:Tzolkin 106, Haab 1Tzolkin 107, Haab 2Tzolkin 108, Haab 3

and so on. Obviously, the two calendars are out of phase, and that’s exactly whatis desired – that way, we keep getting unique combinations of the two numbers.However, it cannot go on forever - when will the two wheels realign again, i.e.,reach their initial position Tzolkin 1, Haab 1?

We use a small example to figure it out: on a piece of paper, draw two calendarwheels, one with 4 and the other with 6 days. How many unique combinationswill you get? I.e., if you start with 1,1 (as Mayan calendars) after how many dayswill the wheels get back to 1,1? (Answer: the least common multiple.)

The least common multiple of 260 and 365 is 18,980, which is exactly

18.980/365 = 52

Haab years. This is called a calendar round. In other words, there are 18,980unique combinations to label the days, and the two calendars synch again after 52(365-day long) years.

Thus, two days with identical Tzolkin and Haab numbers must 52 be yearsapart, or 104 years apart, or some other multiple of 52 years apart. If two peoplehave a birthday on the same day (say, Tzolkin 148, Haab 279), the are either ofthe same age, or one is 52 (or 104) years older.

The end of one calendar round (the 52-year cycle) and the start of anotherwas a huge event, accompanied by ceremonies, including human sacrifices.

The Long Count was an astronomical calendar, used to record historic events,and to track long time intervals. The reason for the word ‘count’ is because thecalendar actually counted days, starting with the day Maya astronomers believedthe Universe (in its present form) was created. Historians determined that, mostlikely, that date was Monday, 11 August 3114 BCE (according to the Gregoriancalendar, adjusted to reflect dates before it was introduced in 1582).


The counting system used in the Long Count calendar was a modified base 20system, with (of course) a day as a basic unit. The longer units are:

1 Uinal = 20 days1 Tun = 18 Uinals = 360 days1 Katun = 20 Tuns = 7,200 days1 Baktun = 20 Katuns = 144,000 days (about 394 (365-day long) years)

13 baktuns (1,872,000 days = 5,125.36 years) form a Great Cycle. Mayan peoplebelieved that at the end of each great cycle the existing Universe is destroyed, anda new one begins its life.

So how did the count work? A number was assigned to each of the five units:

The day when the existing Universe was created, and the start of the first baktun,was labelled as

and the following day was recorded as

It is common to write these as numbers separated by dots. Thus, the ‘CreationDate’ is the sequence of zeroes:

0.0.0.0.0 (also labelled as 13.0.0.0.0)From that day, then the long count advances:

0.0.0.0.10.0.0.0.20.0.0.0.3

... skipping 16 days ...0.0.0.0.190.0.0.1.0 (looks familiar? Addition in base 20!)0.0.0.1.10.0.0.1.2

... skipping many days ...0.0.0.17.19 (day 360)0.0.1.0.0 (keep in mind that the second place is like base 18, not base 20)0.0.1.0.1

... skipping many, many days ...0.0.19.17.19 (day 7,200)0.1.0.0.00.1.0.0.1

and so on, until:


0.19.19.17.19 (last day of the first baktun; day 144,000)1.0.0.0.0 (first day of the second baktun)

and it keeps going. This calendar does not care about the length of a year, sinceit counts days (so there is no need to make adjustments for leap years).

Smithsonian National Museum of the American Indian, on its page LivingMaya Time (https://s.si.edu/2n06D4N) keeps track of the long count calendar.Here is its report for 3 January 2020, showing Maya numerals as they would havebeen (associated with various gods)

Based on it, we compute that 3 January 2020 is the day number

13 · 144, 000 + 0 · 7, 200 + 7 · 360 + 2 · 20 + 9 · 1 = 1, 874, 569

Why was the world supposed to come to its end on 20 December 2012?

According to the Long Count calendar, 20 December 2012 was the last day ofthe 13th baktun (note the the first baktun is labelled by 0; the second bactun isthen labelled by 1, and the thirteenth battun labelled by 12).


In other words, that was the last day of the Great Cycle! According to Mayapeoples’ beliefs, the Universe as we know it should have ended. (By the way, Greatcycle resets itself once every 5,125.36 years, and we witnessed it, and survived!)

The Long Count recorded 20 December 2012 as

The following day was the beginning of a new Great Cycle and a new baktun, 21December 2012

which is also recorded as 0.0.0.0.0.

Notes: [1] Variations of the calendars we mentioned are still used in CentralAmerica, for instance in parts of Guatemala.

[2] If you’re interested: the baktun we are in at the moment will end on Sun-day, 25 March 2407; its long count will be 0.19.17.19.19 (sometimes written as13.19.19.17.19), and the long count for Monday, 26 March 2407 will be 1.0.0.0.0.0(or 14.0.0.0.0).

[3] Why did Maya astronomers define a month to be 20 days long? Historiansare not sure, one possible explanation is that they took 260 (=13 · 20) as it is theinterval (actually approximate interval) between conception and birth of a baby.Plus, Mayans had a thing for number 13.


5 Proportional Relationships

We have already seen some ways of talking about quantities: stating a number todescribe it (absolute), relating that number to something else (relative; as in perminute, per 100,000 people); using orders of magnitude; comparing to human size,using our intuition, or visualizing by comparing to something we know, or haveexperience with.

To describe how quantities change, we often look for patterns. Understandingpatterns and what they say about the change is an essential part not just ofmathematics, but of the way we think mathematically about almost anythingthat surrounds us.

How do we know that, unless we all decide to make a change and actually dosomething, the global temperature will rise by about 1.5 degrees Celsius?

First, scientists defined what the term global temperature means, and thenstarted measuring and recording its values (easier said than done). By looking atthese (historic) values, scientists have been trying to identify a pattern, and thenapply that pattern to look into the future–predicting what would happen 10, 20,or 50 years from now, and what the consequences would be.

In its report Global Warming of 1.5 degrees C the IPCC (= IntergovernmentalPanel on Climate Change, a United Nations body which studies climate change)states (http://www.ipcc.ch/report/sr15/) “Why is it necessary and even vital tomaintain the global temperature increase below 1.5C versus higher levels? Adap-tation will be less difficult. Our world will suffer less negative impacts on intensityand frequency of extreme events, on resources, ecosystems, biodiversity, food se-curity, cities, tourism, and carbon removal.”

Pattern: Proportional Quantities

We say that two quantities A and B are proportional if the ratio of their val-ues A/B (or B/A, does not matter) remains fixed as the quantities change. Forinstance, consider the measurements of A and B taken once a second for 6 seconds:

Initially, when the time is 1, the ratio is A/B = 9/6 = 1.5. Note that thesame ratio is kept throughout the changes in A and B: when the time is 3,A/B = 3/2 = 1.5, when it is 6, A/B = 12/8 = 1.5 (check the remaining values).Thus, we conclude that A and B are proportional.

However, the quantities A and C are not proportional: initially, when the timeis 1, A/C = 10/2 = 5, whereas when the time is 4, A/C = 5/6 ≈ 0.7833 and whenthe time is 5, A/C = 8/10 = 0.8.

Case Studies 5.1 Proportional Relationships

(1) The excerpt below discusses two major electoral systems: Proportional rep-resentation is an electoral system where the proportions of votes reflect in the

Chapter 5 Proportional Relationships 85

composition of the elected body (such as a parliament). Under the first-past-the-post system each voter indicates the candidate of their choice, and the candidatewith the most votes wins. The winning candidate (their party) then wins mostseats in elected body (thus forming, for instance, a majority government). Readmore about these - as they are important for Canada - in Wikipedia articles ‘Pro-portional representation’ and ‘First-past-the-post voting’.

In Why Trudeau’s broken electoral reform promise could rebound on him (CBCNews, 26 July 2019 https://bit.ly/2YqqDW3) we read: “In June 2015, Trudeauvowed that the federal election of that year would be the last conducted under thefirst-past-the-post system. In February 2017, as prime minister, he decided to walkaway from that commitment. Whatever the merits of that decision (Trudeau hadmisgivings about the ramifications of moving toward proportional representationand feared that a national referendum would be divisive), electoral reform is easilyclassified as a ‘broken’ promise.”

Related to a possible change in electoral reform, on 20 Decemcer 2018, GlobalNews reported that British Columbians reject proportional representation, vote tostay with first-past-the-post (https://bit.ly/2KlwFNG).

(2) In the report Chronic Anemia May Affect White Matter Volume in theBrain and Cognitive Performance published on 1 August 2019, Hematology Ad-visor (https://bit.ly/2Kf9Psl) reports that “According to results published in theAmerican Journal of Hematology, lower hemoglobin levels appear to be associatedwith reduced white matter volume throughout the brain, regardless of patients’sickle cell disease (SCD) status. Thus, the severity of anemia, not disease state,predicts white matter volume.” It then continues, by qualifying the relationshipas proportional: “ Patients with chronic anemia demonstrated a decrease in brainwhite matter volume proportional to anemia severity.”

(3) In some situations (could be quite often) the word proportional is usedin a more loose sense, where it just means related. For instance, in the article12 Things Homebuyers Should Look For In A Real Estate Agent published inForbes, at https://bit.ly/2ZvAsOV the author writes “I have found a proportionalrelationship between the amount of time it takes an agent to get back to you andtheir level of professionalism. If an agent doesn’t miss emails or calls, this is anindication that they are on the ball and detail-oriented. Conversely, agents thatare inattentive tend to be equally sloppy in details of the transaction.”

To emphasize: in a proportional relationship, one variable mimics the (multiplica-tive) changes in the other variable. (Another way to say this is to state that bothvariables react the same under scaling.)

Assume that A and B are proportional. This means that if A doubles, so doesB; if A decreases by 75%, then B decreases by 75% as well. If B quadruples, thenA quadruples.

In a more technical language, A and B are called proportional if they aremultiples of each other, i.e., there is a number m (which is not zero), such that

A = mB

Of course, then A/B = m, which shows that the ratio of A to B remains constant,and equal to m.

What does it look like visually? Consider the table below which establishesa proportional relationship between quantities A and B (recall that A/B = 1.5)and graph the points in a coordinate system:


B

A

(6,9)

(8,12)

(2.5,3.75)(2,3)

(4,6)(3,4.5)

Note that B is on the horizontal axis, so the first coordinate of each point is thevalue of B.

So, all points lie on the same line, and the slope of that line is 1.5, which isequal to the constant of proportionality m.

Example 5.2 Proportional Reasoning: Distance and Speed

It is good to remember that 1 mile is approximately 1.6 kilometres (as a matter offact, 1 mile = 1.60934 kilometres, but for all practical purposes we can say that 1mile = 1.6 km). Then 2 miles is 2 · 1.6 = 3.2 kilometres, 10 miles is 10 · 1.6 = 16kilometres, and so on. Thus, the relationship is proportional, and we can write(use the equals sign but keep in mind that it’s an approximation)

x miles = 1.6x kilometres

Thus, 390 miles (roughly the distance from Buffalo to New York City) is approxi-mately 390 · 1.6 = 624 km.

Multiplying 1 mile = 1.6 km by 60, we obtain that 60 miles = 1.6 · 60 = 96 ≈100 kilometres. Dividing both sides by hours (thus getting the speed) we obtainanother useful thing to remember:

100 km/h is roughly 60 mph

This relationship is also proportional, so we can halve it to obtain

50 km/h is roughly 30 mph

Computing 80% of it, we obtain

80 km/h is roughly 48 mph ≈ 50 mph

Multiplying by 1.2 gives

120 km/h is roughly 60 · 1.2 = 72 mph ≈ 75 mph

and so on.Using the conversion rate we just discussed, you can check that all entries in

this chart from tripsavvy (https://bit.ly/31fJiAD) are correct, or approximatelycorrect:


Example 5.3 Proportional Reasoning: Scaling

Scientists have calculated that Earth is about 4,540 million (i.e., about 4.5 billion)years old, with a margin of error of about 50 million years (thus, the range forEarth’s age can be written as 4,540 ± 50 million years). Dinosaurs lived throughthe Mesozoic Era, which started about 245 million years ago, and whose end ismarked with their extinction about 65 million years ago. It is hard to determinewhen humans emerged, because it is hard to define what ‘human’ actually means.For the sake of this example, we define it as the emergence of homo erectus about1.8 million years ago.

4.5 billion, 65 million, 1.8 million – how do we make sense, how do we visualizethese numbers? One way is to scale them to something that we are all familiarwith–for instance, the length of a day, 24 hours. So we take 24 hours to represent4,540 million years.

Dinosaurs went extinct 65 million years ago, whose ratio with regards to theage of the Earth is 65/4,540 = 0.014317. Now apply this same ratio to 24 hours, toget 24 · 0.014317 = 0.34361 hours. Thus, dinosaurs went extinct less than half anhour before the clock turned to 24. More precisely: 0.34361 ·60 = 20.6166 minutes,which is 20 minutes and and 0.6166 · 60 = 36.996 ≈ 37 seconds. We conclude thatthe dinosaurs went extinct at approximately 23:39:23.

Repeat the calculation for the time of 1.8 million years: 1.8/4,540 = 0.000396Apply this ratio to 24 hours, 24·0.000396 = 0.009504 hours. This is 0.009504·60 =0.57024 minutes, which is 0 minutes and 0.57024 · 60 = 34.214 ≈ 34 seconds. Weconclude that homo erectus appeared on Earth at approximately 23:59:26.

Now we can easily see that, compared to Earth’s geological age, even theextinction of dinosaurs is a recent event.

00:00:00

24:00:00(today)

23:39:2323:59:26

To practice this thinking about proportionality, we discuss a couple of exam-ples (we will do many more in the following sections).(1) A 30 kg bag of cement costs $ 7.80. How much does 5 kg of cement cost?Answer: 5 kg is one-sixth of 30 kg, so the price must be one-sxith of $ 7.80, whichis $ 7.80/6 = $ 1.30. How much does 100 kg of cement cost? Apply the ratio100/30 of weights to the prices (as their quantities are proportional, the sameratio applies), to get 100

30 · 7.80 = $ 26.How much cement can we buy for $ 3.90? Answer: Since $ 3.90 is one half of

$ 7.80, we can buy (the same ratio, i.e.,) one half of 30 kg, which is 15 kg.


(2) You are driving from Hamilton to Montreal (driving distance of about 610 km)and just passed Oshawa, which is about 128 km from Hamilton. If you were torepresent the distance from Hamilton to Montreal with a 10 cm piece of string,where would you have to place Oshawa?

Answer: At the moment when we are in Oshawa, we covered 128/610 of thedistance to Montreal. Applying the same ratio to the string, we get 128

610 ·10 = 2.098.Thus, we would have to place Oshawa just a bit over 2 cm (2.098 cm) from theend of the string which represents Hamilton.

Next, we study several important proportionality relations, and then we in-vestigate other common relationships. Just to contrast with proportional thingswe have studied–what is not proportional?

The area of a square with side 3 is 9. The area of the square with side 6(double the side) is 36, which is not the double of 9. Thus, the area of a squareand its side length are not in a proportional relationship.

Human bones do not grow in a proportional way, in the sense that the lengthof a bone and its diameter (thickness) are not proportional. Studying bones ofvarious animals, Galileo (16/17th c. CE) realized that that was the case. Hisillustration shows that a bone three times longer is not three times thicker, butmore than that. (Note: he exaggerated the thickness - the longer bone should beabout 5.2 times thicker than the shorter one).

Currency Exchange

When we asked Google (on 4 August 2019 around 1pm) what the exchange ratebetween Canadian Dollar and Euro is, we saw this:

Thus 1 Euro equals 1.47 Canadian dollars, which we think of as

A Euro = 1.47 · A Canadian dollars


Thus, 100 Euro is worth 1.47 · 100 = 147 Canadian dollars, and 5 Euro is worth1.47 · 5 = 7.35 Canadian dollars. So, this currency conversion is a proportionalrelationship, where the constant of proportionality is the exchange rate of 1.47.

Note that the diagram on the right shows that the exchange rate is not con-stant, but changes over time.

Dividing both sides of A Euro = 1.47 ·A Canadian dollars by 1.47, we obtainthe reverse exchange rate:

11.47

· A Euro = A Canadian dollars

i.e.,A Canadian dollars = 0.68 · A Euro

So, if we wish to buy 100 Euro, we need to pay 147 $ (Canadian), and to buy100 $ (Canadian), we need 68 Euro. Mathematically this is correct, but in reallife banks and other financial institutions involved in currency exchange need tomake money. Thus, when currency exchange is involved, there are two rates, calledbuy rate and sell rate. The buy rate refers to the rate at which a bank (financialinstitution) buys currency from customers, and the sell rate is the rate at whichthey sell it to their customers. Here is a photo of the buy and sell rates from thecurrency trader ice at Pearson airport in Toronto (taken in 2018):

Note that buy rates are smaller than sell rates.If we wish to buy 100 Euro, we look at the sell rate of 1.805410, and compute

that we will need 1.805410 · 100 = 180.54 $ (Canadian). But if we have 100 Euroand wish to sell it, ice will give us 1.316786 · 100 = 131.68 $ (Canadian). Thedifference is the profit that ice makes. Sucks!

Of course, the smaller the difference between buy and sell rates, the less moneycurrency exchange people/institutions make.

When travelling, keep in mind that different entities (banks, financial institu-tions, money exchange kiosks) offer different exchange rates, and that the ratesoffered at certain places (airports, train stations) are often a lot more extremethan the ones a bank in the centre of a city will offer. So don’t change your moneyat airports (unless you really have to); as well, avoid money exchange kiosks, assometimes they offer exchange fees on top of unfavourable buy and sell rates (someof these places are really shady and do not fully discolse their rates until it’s toolate).


Sometimes the currency exchange places will post the rates, without identify-ing buy and sell rates. How do you know which is which? Just figure out whichworks against you.

Assume that, in Vancouver, you see two rates for the Australian Dollar: 0.8547and 0.9216. Assume that 0.8547 is the sell rate (the rate at which the currencyexchange kiosk sells to us): we need 85.47 $ (Canadian) to buy 100 $ (Australian).Now we sell Australian dollars back: applying the buy rate (i.e., the rate at whichthe kiosk buys currency from us) of 0.9216 we get 92.16 $ (Canadian). So, in thisprocess we made 92.16 − 85.47 = 6.69 $ (Canadian). And then, if we repeat thishundred times, we would make over 669 dollars! Of course this cannot happen –it’s the other way around: 0.8547 is the buy rate and 0.9216 is the sell rate.

By the way, these days there is an efficient way to avoid changing actual(paper) money – use credit or cash cards (bit more about it soon), as they areaccepted in almost every country in the world. However, there might be placeswhich are still cash only (on the other extreme, there are places which no longeraccept cash).

One more thing to keep in mind when abroad: it is becoming more and morecommon that shops (especially duty free shops at airports), restaurants, ATMs,and varuous businesses offer conversions, so that you pay in your home currency,instead of the local currency.

Here is a screen from an ATM at Heathrow airport:

(Source: The Conversation article Save money when traveling abroad by thinkinglike an economist at https://bit.ly/2YKXfJj).

If you plan to travel, read the above article! It strongly discourages you frompaying in your home currency (Continue with conversion option), as the exchangerates are unfavourable, and there are extra charges involved (which again, mightor might not be clearly spelled out). In any case, when abroad, always select topay in local currency.

In the above article from The Conversation, we read: “Increasingly, however,retailers, restaurants and ATMs are offering travelers the option to pay or with-draw money in terms immediately converted into their home currency. Companiesoffering the service call it ‘dynamic currency conversion.’ For example, an Ameri-can tourist visiting Paris is able to use her credit card to pay for a fancy meal ata French bistro in U.S. dollars, instead of euros. This may seem innocuous – oreven convenient – but agreeing to use your home currency in a foreign land cansignificantly inflate the cost of every purchase.”

The article goes on explaining what the extra charges are, concluding thatthese can add a significant amount to our bill. We read: “[...] Even some bankerswarn against consumers doing this [i.e., paying in one’s home currency whenabroad] because the exchange rate used is much worse than the one your bankwould offer.”


The article Holiday money: how to find the best cards and currency ratespublished in the Guardian (23 June 2018, https://bit.ly/2lCNAzt) offers identicaladvice, and provides a “[...] guide on avoiding bank charges and making your cashgo further this summer.”

Read these articles, and consult an online currency converter to get a good feelabout the currency of the country you are travelling to. Think about this beforeyou go, as quite often, when things actually happen (say, rushing through dutyfree because you have to catch your flight, or standing in front of an ATM with10 people in line behind you) you will not have time to reflect and make a gooddecision. Keep in mind that exchange rates depend on many factors, includingthe time of day (the rates are never stable), who is involved (bank, financialinstitution), what we are paying with (cash, travellers cheques), and the type ofcard we use (credit, debit).

Conversion Between Units

To convert metres to inches, we multiply by 39.3701 (or, very often, by 39.37): sothree metres are equivalent to 39.37 ·3 = 118.11 inches, half a metre is 39.37 ·0.5 =19.685 inches, and so on. A general conversion between units has the form of aproportional relationship:

unit = conversion factor · another unit

One kilogram is equivalent to 2.20462 pounds (which is often rounded off to 2.2pounds). In general,

A kg = 2.2A lb

As any proportional relationship, we can visualize it as a line (through the origin,since 0 kg = 0 lb). The slope of the line is the conversion factor 2.2.

lb

kg

(1,2.2)

(6.4,14.08)

(10.7,23.54)

(3,6.6)

To compute the reverse conversion, we divide by 2.20462 :

A lb =1

2.20462A kg = 0.45359A kg

In other words, to figure out the mass of something in pounds, we multiply itsmass in kilograms by 0.45359.

Exercises 5.4 Conversion Between Units

(1) If x metres is converted to y kilometres, which number is larger, x or y?

(2) If x mph (miles per hour) is converted to y km/h, which number is larger, xor y?

(3) If x km/h is converted to y km/s (kilometres per second), which number islarger, x or y?

(4) How many litres are there in a cubic metre?


(5) Human hair grows at about 1.25 cm per month. Convert this speed into metresper second.

Partial answers: (1) Say we walk for 400 metres - we still have not walked for1 kilometre (to be precise, 400 metres is 400 · 1

1000 = 0.4 kilometres). Thus, x(the number of metres we walked) is larger than y (the number of kilometres wewalked).(3) Driving at 120 km/h, we cover the distance 120 kilometres in one hour. Defi-nitely we will cover a much smaller distance in one second, so y is smaller than x.If we’re not convinced (keep in mind that 1 hour = 3600 seconds):

120kmh

= 120 · 13600

kms

=130

kms

(so x = 120 and y = 1/30).(4) A good visual to remember: one litre of water fills a 10 cm × 10 cm × 10 cmcubic box. In other words, the volume of 1 litre is equivalent to the volume of(10 cm)3 = 1000 cm3.

Now multiply the relation 1 m = 100 cm by itself three times (i.e., cube it):

1 m3 = 1000000 cm3

and thus1 m3 = 1000 · 1000 cm3 = 1000 litres.

(5) Assume that a month has 30 days. Then

1.25cm

month= 1.25

0.0130 · 24 · 60 · 60

ms

Here we used the fact that 1 metre has 100 centimetres, and thus 1 centimetre =1/100 = 0.01 metres. In the denominator, we converted days into seconds. Using acalculator, we find that the number on the right is 4.8225 ·10−9 metres per second.

Although this number is not of much help, we learned something: one wayto make a small (or a large) number more manageable size is to change the unitsinvolved. 4.8225 ·10−9 metres per second is not what many of us can comprehend,however 1.25 cm per month we can visualize and work with.

Notes: (1) There are a number of online unit converters that we can use. When wedo, we realize how many different units there are, and it’s all quite chaotic. Thatis why countries around the world have been adopting the International System ofUnits (known as the SI system), which is based on metre (length), kilogram (mass),second (time), kelvin (temperature), ampere (electric current), mole (amount ofsubstance) and candela (luminous intensity).

However, for many practical purposes we use other units (imagine a weatherforecast for an August day given as 303.15 degrees Kelvin; so we stick to degreesCelsius or Fahrenheit). A pint is lot more common in some contexts than litres ormillilitres (Challenge: try to order 568 millilitres of beer. By the way, 568 ml iscalled a British pint. An American pint is smaller, and contains 473 ml).

(2) Why is it important to have a unique, clearly defined system of units? Buyingone pound of bananas instead of one kilogram is usually not a big deal. How-ever, sometimes the errors caused by messing up units could be (and are) deadly.Wikipedia article Korean Air Cargo Flight 6316 (https://bit.ly/2GC6yRu) recallsthe 1999 cargo plane crash which killed three people on board. We read: “[...]investigation carried out by CAAC (= Civil Aviation Administration of China)showed that the first officer had confused 1,500 metres, the required altitude, with1,500 feet, causing the pilot to make the wrong decision to descend.”

On 23 July 1983, an Air Canada flight from Montreal to Edmonton ran outof fuel mid-air. The plane involved was the first Air Canada plane that used


metric (SI) measurements; see Gimli Glider https://bit.ly/1T25uTu). We read:“The Aviation Safety Board of Canada (predecessor of the modern TransportationSafety Board of Canada) reported that Air Canada management was responsiblefor ‘corporate and equipment deficiencies.’ Their report praised the flight andcabin crews for their ‘professionalism and skill.’ It noted that Air Canada ‘ne-glected to assign clearly and specifically the responsibility for calculating the fuelload in an abnormal situation.’ [...]”

(3) Not all conversions are given by a proportional relationship. We will see soonthat conversion between degrees Celsius and degrees Fahrenheit involves a non-proportional relationship.

(4) Numbers that come with units behave in ‘weird’ ways: for instance, the math-ematically correct equation 1 = 1 might not be correct when units are involved:for instance, 1 metre is not equal to 1 centimetre. As well, the mathematicallyincorrect statement 1 = 2.54 can be made correct if we add units: 1 inch = 2.54cm.

(5) In Canadian grocery stores, prices for both pounds and kilograms are displayed,but in many cases the prices per pound are in larger print. Below are screenshotsfrom a Sobeys flyer:

One reason for this, as researchers tell us, is psychological: shoppers are moresensitive to prices than to quantities. So, as long as the price is lower (even if thequantity is smaller), they are more likley to buy. Read more about this in theBBC News article The food you buy really is shrinking https://bbc.in/2OBQzJJ.

Percent

Recall that to convert a number into percent we multiply by 100 (move the decimalpoint two places to the right), and to convert percent into number we divide by100 (move the decimal point two places to the left). Thus,

0.45 = 45 %1.28 = 128 %0.07 = 7 %0.0004 = 0.04 %

12.4 % = 0.1241 % = 0.010.8 % = 0.008231 % = 2.31

If A is a quantity, then a percent of A is calculated by multiplication (hence it’s aproportional relationship):

16 % of A is 0.16A, 1.6 % of A is 0.016A, and 160 % of A is 1.6A.

Recall that a proportional relationship means that a multiplicative input gen-erates identical response (output). Computing a percent is proportional in two


different ways: with respect to the percent, and with respect to the quantity in-volved.

Assume that 15 % of some quantity A is equal to 80. Then (triple the rela-tionship) 45 % of A is equal to 240 (triple the value), (halve the relationship) 7.5 %of A is equal to 40 (halve the value), (take one tenth of the relationship) 1.5 % ofA is equal to 8 (one tenth the value), and so on.

Again, assume that 15 % of some quantity A is equal to 80. Then (triple therelationship) 15 % of 3A is equal to 240 (triple the value), (halve the relationship)15 % of A/2 is equal to 40 (halve the value), (take one tenth of the relationship)15 % of A/10 is equal to 8 (one tenth the value), and so on.

The most important thing about percent is that it is relative – so wheneverwe work with information presented as percent, we must ask percent of what?Although this might seem obvious, there are many cases when we forget about it,or various sources (purposely or not) omit the reference information.

For instance, information contained in the phrase “The unemployment rate inManitoba rate fell by 0.2 %” is useless, until we make clear what the reference is,such as “compared to the last year,” or “when compared to the first quarter ofthis year.”

Does 10 % + 10 % equal to 20 %?It depends! The first thing we need to be clear about is the meaning – 10 %

of what? 20 % of what? If the above phrase means 10 % of A + 10 % of the samequantity A then this sum is 0.1A + 0.1A = 0.2A, which is 20 % of A. Otherwise,it could be many different things.

Is 10 % larger than 4 %? Again it depends!

If a quantity A grows by 34 %, then its value is A + 0.34A = 1.34A. We can thinkof this using the so-called 1 plus rule: 1.34 = 1 + 0.34, where 1 represents theoriginal quantity, and 0.34 is the 34 % increase. Thus, 1.5A can be interpreted as(1.5 = 1 + 0.5) a 50 % increase in the value of A, and a 3 % percent increase in Acan be written as 1.03A.

Likewise, if a quantity loses 18 % of its value, then its new value is A−0.18A =0.82A, i.e., 82 % of its original value. We can think of this using the so-called 1minus rule: 0.82 = 1 − 0.18, where 1 represents the original quantity, and 0.18 isthe 18 % loss. Thus, 0.59A can be interpreted as (1 − 0.41 = 0.59) a 41 % declineof the value of A. If A loses 78 % of its value, what is left is 0.22A.

The 1 plus rule and the 1 minus rule can help us figure our more complexsituations. Look at the following example.

Example 5.5 Using 1 Plus and 1 Minus Rules

Assume that A represents the price of something.

(1) If A is discounted by 7 %, and then the new price is further discounted by12 %, then the final price of A is 0.88(0.93A) = 0.8184A, i.e., the total discount is(1 − 0.8184 = 0.1816), i.e., 18.16 %.(2) If A is discounted by 7 %, and then the new price is increased by 12 %, thenthe final price of A is 1.12(0.93A) = 1.0416A, i.e., the final price is 4.16 % higherthan the original price.(3) Reverse the situation in (2): if A is first increased by 12 %, and then the newprice is discounted by 7 %, then the final price of A is 0.93(1.12A) = 1.0416A, i.e.,the final price is 4.16 % higher than the original price – the same as in (2).(4) If A is increased by 12 % and the new price is increased again by 12 %, is thetotal increase going to be 24 %?


No, it will be more than 24 %. The second increase is calculated not basedon A, but based on the increased value of A: 1.12(1.12A) = 1.2544A, which is a25.44 % increase.(5) If A is increased by 12 % and the new price is discounted by 12 %, is the finalprice going to be A?

No, it will be less than A, because the 12 % discount is calculated not basedon A, but based on the larger price (A increased by 12 %). To check, we compute:0.88(1.12A) = 0.9856, i.e., the final price is not A, but 98.56 % of it (equivalently,the final price is A discounted by 1.44 %).

Case Study 5.6 Caesarean Births are ‘Affecting Human Evolution’

In the report Caesarean births ‘affecting human evolution’ (BBC News, 7 Decem-ber 2016, https://bbc.in/2g4kpQv) we read: “The researchers estimated that theglobal rate of cases where the baby could not fit through the maternal birth canalwas 3 %, or 30 in 1,000 births. Over the past 50 or 60 years, this rate has increasedto about 3.3− 3.6 %, so up to 36 in 1,000 births. That is about a 10-20 % increaseof the original rate, due to the evolutionary effect.”

Let us check the math: is it really 10-20 % increase of the original rate, asclaimed?

First, we look at the lower end of the range: 3.3 % is 33 per 1,000. Thus, theincrease is from 30 to 33. The absolute change is 3 births, and the relative changeis 3/1, 000 = 0.003, or 0.3 %. However, if the relative change is computed basedon the initial count of 30, then it is 3/30 = 0.1, or 10 %.

Similarly, the upper end of the range is 3.6 %, or 36 per 1,000. The absolutechange is 6 births, and the relative change could be 6/1, 000 = 0.006, or 0.6 %, ifcompared to the 1,000 births, or 6/30 = 0.2, or 20 %, if compared to the initialrate of 30 (in 1,000 births).

Thus, mathematically both ranges 0.3-0.6 % and 10-20 % are correct. Verylikely, the author picked the 20 % angle to make a stronger point.

Case Study 5.7 (Mis)interpreting Risk: ‘Pill Scare’

This case study refers to the risk of developing blood clots (venous thromboem-bolism) when taking third- and fourth- generation contraception pills (called CHC= combined hormonal contraception), which have been introduced to the generalpublic in 1990s and 2000s, respectively, and which are in widespread use.

On its web page, Hormonal Birth Control and Blood Clot Risk, U.S. organiza-tion National Women’s Health Network (https://bit.ly/2yDvAeL) states that “[...]CHC methods are birth control methods containing the hormones estrogen andprogestin. Tens of millions of people safely use CHCs – including birth controlpills, patches, and vaginal rings – to help space births and prevent unintendedpregnancy.”

Later, on the same page, we find risk statistics: “Blood clots are generally rarebut sometimes occur in otherwise healthy people, even those not taking CHCs.Between 1 and 5 of every 10,000 women (who are not pregnant and not usingCHCs) will experience a blood clot in any given year. This number increasesslightly if the person uses CHCs. Between 3 and 9 of every 10,000 CHC users willexperience a blood clot in any given year.”

Note that the article states that the risk increases slightly for a person whouses CHCs.

Let us analyze this change from 1-5 per 10,000 (no CHC, no pregnancy) to3-9 per 10,000 (CHC) in a different way.


The average of the interval 1-5 is (1+5)/2 = 3, and the average of the interval3-9 is (3+9)/2 = 6. Thus, we can say that the average risk doubles. Alternatively,the change from 1-5 per 10,000 to 3-9 per 10,000 could be interpreted as doublethe rate, as the doubles of numbers from 1 to 5 are 2 to 10, which is close to the3-9 range.

Unfortunately, the news that the risk of developing blood clots doubles forCHC users found its way into media on both sides of the Atlantic. For instance,on 14 November 2012, ScienceNordic (an Independednt news organization whichfocuses on writing about research in Scandinavian countries) published the follow-ing article:

(Source: https://bit.ly/31b8tV8.) ScienceNordic states that “Now a Danish lit-erature review confirms what has been suspected for years: that the so-calledfourth-generation OCP (= oral contraceptive pill) doubles the risk of blood clotscompared to second-generation OCPs.”

Later in the same article, we find the following statistics:

The narrative states that there is no cause for panic. However, in the last line,it says that the risk (fourth generation OCPs) increases six-fold, as the authorscompared it not to the women who take OCPs but to those women who do nottake contraception.

In the piece Some contraceptive pills said to have DOUBLE the risk of bloodclots published in Cosmopolitan (27 May 2015, https://bit.ly/31jj7t2) we read: “Anew study has revealed that some third-generation combined contraceptive pillsare twice as likely to cause blood clots in the arm or leg than older brands – butit’s not quite time to panic yet.”

The messages “no cause for panic” and “it’s not quite time to panic yet” didnot catch people’s attention nearly as much as the claims about doubling the riskor six-fold increases in the risk.

The key observation is that (even when the risk is doubled, or increased six-fold) the numbers are still very small, i.e., the actual risk of developing blood clotswhen taking new generation oral contraceptives is still very small.

However, it was too late. Many women decided to stop taking the pill, andsome switched to other methods of contraception. The consequences were serious.


In the report Oral Contraceptives and the Risk of Venous Thromboembolism:An Update published in 2010 by The Council of the Society of Obstetricians andGynaecologists of Canada (https://bit.ly/2yFTWV2) we read:

“Recent contradictory evidence and the ensuing media coverage of the ve-nous thromboembolism (= blood clots) risk attributed to [...] certain newer oralcontraceptive products have led to fear and confusion about the safety of oralcontraceptives. ‘Pill scares’ of this nature have occurred in the past, with panicstopping of the pill, increased rates of unplanned pregnancy, and no subsequentdecrease in venous thromboembolism rates.”

The abstract of the paper Pill scares, an avoidable side effect published inThe European Journal of General Practice (https://bit.ly/33hiaTY) echoes theCanadian report: “Whilst their [oral contraceptives] benefits have been obvious towomen, the majority of medical publications have focused on the risks that may beassociated with their use. ‘Pill scares’ are becoming a part of contraceptive practice– they worry women and their families, cause considerable morbidity (throughtermination of unwanted pregnancies) and complicate the work of doctors andother health professionals.”

Epilogue: the 1995 ‘pill scare’ in England and Wales resulted in additional (i.e.,over the yearly average) 26,000 unplanned pregnancies, with a bit over one half(13,600) resulting in abortions (see the diagram on the left; source: [1], page 54).

Subsequent ‘pill scares’ have had similar consequences elsewhere. The diagramon the right shows the large upward trends in both the number of abortions andthe abortion rates due to the two pill scares in France in 2000 and 2012 (source:Institut National d’Etudes Demographiques (INED), https://bit.ly/2GMP8Si).

Notes: [1] The two case studies discussed here are really important: the first one(Caesarean Births) shows how the same information can be presented in differentways (this is called frame or framing, and we will talk more about it), and,consequently, can generate different reactions from people.

The second case study (‘Pill Scare’) is based on a conflict between two frames:small increase in risk, vs. doubling the risk. It shows how a wrong choice offraming the given statistics can lead to very serious negative consequences.

Here is a classical example which underlines the importance of framing. Aphysician is describing the risk of death from a surgery to their patient, who isconsidering undergoing the surgery. The physician can say “You have a 10 percent


chance of dying from the surgery” (this is called a negative frame) or “You havea 90 percent chance of surviving the surgery” (positive frame). Research showsthat a patient who is presented with a positive frame is more likely to give consentto undergo the surgery.

[2] To repeat the message of the ‘Pill Scare’ story: a small number, doubled, is stilla small number. But again, it’s psychology – we tend to exaggerate small chance,rarely occuring events. For instance, a story about people dying from smoking isnot a front page news, unlike a report on a plane crash (but it should be: in termsof morbidity, smoking deaths = three standard size passenger plane crashes daily).

Chapter rerefences[1] Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin,S. (2007). Helping Doctors and Patients Make Sense of Health Statistics. Psycho-logical Science in the Public Interest, 8(2), 5396. https://doi.org/10.1111/j.1539-6053.2008.00033.x

Chapter 6 Linear and Non-linear Relationships 99

6 Linear and Non-linear Relationships

When the UN Arctic Chief Jan Dusik said “Climate change isn’t linear – it’saccelerating” (source: https://bit.ly/2FKIXQe) what did he mean?

First – what is linear, what causes linear behaviour?Consider an example: the population of Ontario was 14.32 million in 2018,

and the current rate of increase is 0.258 million (258,000) people per year. Tomake a prediction about the future population count, we assume that the rate ofincrease will remain unchanged. Thus: 2018: 14.32 million 2019: increase of 0.258 million; the population is 14.32 + 0.258 = 14.578 million 2020: another increase of 0.258 million; the population will be 14.32+2 ·0.258 =14.836 million 2021: another increase of 0.258 million (the third since 2018); the populationwill be 14.32 + 3 · 0.258 = 15.094 million 2022: further increase of 0.258 million; the population will be 14.32+4 · 0.258 =15.352 million 2023: the population will be 14.32 + 5 · 0.258 = 15.610 millionAs every year the same number of people is added, we see a pattern emerging:

Population in year x

= population in 2018 + (number of years since 2018) · (yearly increase)= 14.32 + (number of years since 2018) · 0.258

Using P to denote the population of Ontario and t the number of years since 2018,

P = 14.32 + t · 0.258

The rate of increase of 0.258 million people per year is called a marginalchange. In general, a marginal change is the change in the value of a quantityfrom one moment to the next, i.e., over a small interval (in mathematics, we insiston really very small intervals, called infinitesimally small).

A quantity is called linear, or is said to change linearly if all marginalchanges, measured over intervals of identical length, are equal. The graph of aquantity which changes linearly is a line.

In the picture below, the population is shown in a blue line. All intervals areof length one (so ‘from one moment to the next’ in this case is one year), and allcorresponding changes in the population (red vertical segments) are equal.

0 41 5322018 2022202120202019 2023

14.32

popu

latio

n (m

illio

n)

time (year)

0.258 = marginal change


A linear quantity is represented using a linear function, as in P = 14.32+0.258t.

Recall that a linear function is of the form y = b + mx, where b representsthe vertical intersect (the value of the quantity y corresponding to the value 0 ofx), and m is the marginal change, also called the slope. The picture below (left)shows a quantity which increases over time, as its marginal changes (slope) arepositive. On the right, the marginal changes (slope) are negative, and the quantityloses its value.

Thus, a quantity that changes by accumulation, where, at identical intervals,a fixed amount (marginal change) is added, is a linearly increasing quantity.If a fixed amount is removed at identical intervals, the quantity is decreasinglinearly.

If a quantity experiences non-equal marginal changes, then it is called non-linear. For instance, the graph below shows a non-linear quantity: its marginalchanges are of different sizes; as well, some are positive (see the change from 1 to2 or from 10 to 11), and some are negative (see the change from 4 to 5 or from 6to 7).

0 1110987641 532 12

Back to the start: now we know that ‘Climate change isn’t linear – it’s ac-celerating’ means that the way climate changes cannot be represented by a line.The part ‘it’s accelerating’ suggests that the marginal changes are growing, as thequantity in the above graph between 9 and 12.

Example 6.1 Measuring Temperature

To most commonly used units for measuring temperature are defined based onthe Celsius and the Fahrenheit scales. The Celsius scale divides the temperaturebetween the freezing point of water (labelled 0◦C) and the boiling point of water(labelled 100◦C) into 100 equal parts. The Fahrenheit scale divides the tempera-ture between the freezing point of water (labelled 32◦F) and the boiling point ofwater (labelled 212◦F) into 180 equal parts.

By definition, both scales are linear.Note that the degrees Fahrenheit change ‘more quickly’ than the degrees Cel-

sius, as it takes 180 of them to reach the boiling point of water from its freezingpoint, whereas it takes only 100 degrees for the Celsius to do the same.


Based on the definition, conversion formulas have been calculated. In partic-ular,

F =95C + 32

converts degrees Celsius into degrees Fahrenheit. Note that the slope is 9/5 = 1.8,meaning that for each one degree change in the temperature measured in Celsius,the temperature measured in Fahrenheit increases by 1.8 degrees (this quantifiesthe ‘more quick’ change mentioned above). Thus a change of 5 degrees Celsiuscorresponds to a change of 5 · 1.8 = 9 degrees Fahrenheit, a change of 10 degreesCelsius corresponds to a change of 10 · 1.8 = 18 degrees Fahrenheit, and so on.

For instance, 12◦C is equivalent to 95 (12)+32 = 53.6◦F, and 25◦C is equivalent

to 95 (25) + 32 = 77◦F.

0

−40

212

100

32

degr

ees F

degrees C−40

Thus, we can write 25◦C = 77◦F. Subtract (keep in mind the above commentabout the slope) 5◦C = 9◦F from it, to get 20◦C = 68◦F, subtract again, to get15◦C = 59◦F. Add 10◦C = 10◦F to 25◦C = 77◦F to obtain 35◦C = 95◦F, and so on.

The reverse conversion is given by

C =59(F − 32)

Thus, 104◦F is equivalent to 59 (104 − 32) = 40◦C and −13◦F is equivalent to

59 (−13 − 32) = −25◦C

Note that −40◦F is equivalent to 59 (−40 − 32) = −40◦C, so the two scales

meet there: −40◦C =−40◦F.

Linear vs Proportional

Recall that two quantities are proportional if they are multiples of each other.Naming the quantities x and y, we say that they are proportional if there is anon-zero number m such that y = mx.

Note that this is the line y = mx + b, but without b (i.e., b = 0). Thus aproportional relationship is linear, and is represented by a line going thorugh theorigin (because b = 0).

A linear relationship, however, is not proportional (except when b = 0). Con-sider the temperature measured in degrees Celsius and degrees Fahrenheit in thetable on the left.

The scaling (a tell-tale sign of proportionality) does not work: for instance,doubling the temperature in degrees Celsius does not double degrees Fahrenheit:10◦C = 50◦F, but 20◦C is not equal to 100◦F.

As well, look at the graph above - it does not go through the origin.


To summarize: proportional is linear, but linear is not proportional (unless b = 0in Ey = mx + b).

Linear Regression

It is easy to figure things out when we are told what to expect, such as that arelationship is linear (as in the conversion between Celsius and Fahrenheit). Ingeneral, we do not have such information – but still, somehow, need to describethe relationship between the variables from the data which is available to us.

In this course, we focus on phenomena (data) which consist of one indepen-dent variable and one dependent variable. Depending on the context, thetwo variables are also referred to as input (variable) and output (variable),or cause (variable) and effect (variable), or explanatory variable and re-sponse variable, respectively.

The data (also called data points) we have to work with could look like this:

input (explanatory variable)

outp

ut (r

espo

nse

varia

ble)

It is (nearly) impossible to figure out a formula for a curve that would go throughall these points (or even if we could do it, very likely such a formula would not behelpful).

Why do we need to find a pattern, or, as is often called, model the rela-tionship between the variables?

The known data points determine the range of the independent variable (ex-planatory variable, input). We might need to know more about the data within therange (this is called interpolation), or about the data outside the range (calledextrapolation.

Assume that we computed a model for the population of Canada based onthe census data for 1996, 2001, 2006, 2011 and 2016 (thus, the range of data is1996-2016). Figuring out an estimate for the Canadian population in 2008 is amatter of interpolation, whereas predicting the population in 2025, or figuring outwhat it was in 1972 is a matter of extrapolation.

explanatory

res

pons

e

range

interpolation

extrapolation extrapolation

One of the simplest models are curves – and this is what we are going to do.In other words, we will try to ‘explain’ a relationship between the variables hiddenin the given data by using mathematically well understood curves (such as linesor exponential graphs).


To ‘explain’ means to approximate (in the sense that we will explain soon)the given data with a curve. If that curve is a line, the modelling process is calledlinear regression. In the case of one explanatory variable, it is also referred toas simple linear regression. For instance, the line that approximates the datagiven at the start of this section could look like this:

input (explanatory variable)

outp

ut (r

espo

nse

varia

ble)

Here is how we will do linear regression: using software, we will model thedata by a line (i.e., we will identify m and b in y = mx+ b) and then (again, usingsoftware) figure out a number that will tell us how well the relationship given bythe data can be explained by a linear function.

Example 6.2 Linear Regression Problem, Part 1

By the end of this section, we will figure out the question related to this data set:

Math 1LS3 course instructors are worried about the success of students in theircourse. Hence they are asking the following question: when we are half-way intothe course i.e., looking at the grades on the first two term tests, how well can wepredict students’ final grades?


The horizontal axis (explanatory variable) represents grades on the first twotests, and the vertical axis (response variable) are the final course grades, givenas percent. For instance, students who had between 20 and 40 percent on tests 1and 2 combined either failed the course, or received a grade below 60 % (exceptfor one student, whose final grade was a bit over 70 %).

We see that there is some kind of a trend - the better the grades on the firsttwo tests, the better the final course grade. But how confident can we in claimingthat, can we quantify our confidence?

We start small, with a data set consisting of two points: (0, 32) and (100, 212)(looks familiar?). We assume (have reasons to believe) that this data comes froma linear relationship, so we are looking for a linear model.

We use the online linear regression calculator https://bit.ly/31qZf7s for allregression-related calculations. Of course, in this case we can just compute theequation of the line which goes through the two points – but we are learninghow to use the calculator. (An alternative to using one of many online regressioncalculators is Excel.)

Note that the calculator writes the line in the form y = A + Bx (and not theone we are used to: y = mx + b).

Enter the data

and after you press Execute you will obtain the following output

From the table on the left we read the values of A and B, to identify the liney = A + Bx = 32 + 1.8x. Note that this is the Celsius to Fahrenheit conversionformula! Compare the graph on the right with the one we drew earlier. (Soonwe will explain what the correlation coefficient is; we will not use the informationabout the means of the data points).

In our next example, we consider a data set which consists of six data points:(2, 5), (3, 4), (1, 2.5), (6, 6), (5, 4.5), and (1, 5). First we plot them in a coordinatesystem:


x = explanatory variable

y =

resp

onse

var

iabl

e

1

1

5432 6

5432

6

0

Clearly, there is no line that goes through all six points, so instead we will tryto identify the line that ‘best fits’ the data points. The ‘best fit’ will be foundby considering all lines, and picking the one for which the sum of the distances(‘errors’) from the points to the line are the smallest (i.e., the sum of the lengthsof the blue segments in the picture below).


y =

resp

onse

var

iabl

e

1

1

5432 6

5432

6

0

In other words, we assume that the data points are random deviations from alinear relationship between the explanatory and response variables. The line thatwe pick is the one (among all lines) for which the sum of these deviations is thesmallest. That line is called a line of regression or a trendline. In our case,the line of regression is


y =

resp

onse

var

iabl

e

1

1

5432 6

5432

6

0

How did we find the line of regression? We entered the six data points into theonline regression calculator


and obtained the following output:

From it, we read the equation of the line of regression (with the coefficients roundedoff to two decimal places): y = A + Bx = 3.54 + 0.32x. Now we can interpolate:when x = 4, a likely value for y is y = 3.54+ 0.32 · 4 = 4.82 as well as extrapolate:when x = 8, a likely value for y is y = 3.54 + 0.32 · 8 = 6.10.

How well a line of regression describes the data that generated it, or equiv-alently, how well a linear model explains the relationship between the variablesis given by the correlation coefficient, which is usually denoted by r. In otherwords, the correlation coefficient measures the strength of a linear relationshipbetween the variables.

In the above case, r = 0.56 (if you are interested: scroll the results page in theonline calculator to find the formulas used to calculate the line of regression andthe correlation coefficient r). What is important for us to know is that r and theslope of the line of regression (B) are of the same sign (i.e., if one is positive thenthe other is positive; if one is negative, the other is negative; and if one is zero theother is zero as well).

The range of the values of the correlation coefficient is between −1 and 1,where 1 and −1 indicate strongest possible agreement with the linear model (goback to the example of two points – in that case we found the line that goesstraight trough them, so the errors (deviations) are zero).

To judge the strength of a relationship, we use the following convention; thereis no general agreement on this, and what we state is only one possible interpreta-tion. No need to memorize all this, just memorize the first and the last lines (i.e.,the cases of strong correlation).∗ When r > 0.7, the correlation is called strong positive, and the line of regres-sion is deemed to well represent the relationship between the variables. The lineof regression has a positive slope and describes a positive correlation.

♦ If 0.4 < r ≤ 0.7, the correlation is considered to be moderate♦ If 0.2 < r ≤ 0.4, the correlation is considered to be weak♦ If −0.2 ≤ r ≤ 0.2 there is no correlation♦ If −0.4 ≤ r < −0.2, the correlation is considered to be weak♦ If 0.7 ≤ r < −0.4, the correlation is considered to be moderate∗ When r < −0.7, the correlation is called strong negative, and the line of regres-sion well represents the relationship between the variables. The line of regressionhas a negative slope and describes a negative correlation.

Example 6.3 Illicit Drug Overdose Deaths in B.C.

In the report A year of overdoses: 7 charts that show the scope of B.C.’s drugcrisis (CBC News, 26 December 2016, https://bit.ly/2M43PER) we read that“despite declaring a public health emergency in the spring, despite more and more


funding announcements and despite naloxone becoming more and more available,the number of illicit drug overdoses in B.C. grew to record heights this year.”

Let us look at the numbers – on the same we page, we find a diagram, part ofwhich we reproduce here:

Assume that it is 2014, as part of identifying strategies to improve the situation,we need to estimate (predict) the number of deaths in 2016. (In other words, weare asked to extrapolate from the given data).

To simplify, instead of taking all data points, we take the data from 2008 to2014 (table on the left). As the data points seem to be close to a line (look atthe graph above), we decide to model the relationship between the time and thenumber of deaths using a linear model; thus, we will do linear regression.

Before we enter the data into our online calculator, we make one simplification(not necessary, but useful): to make the numbers smaller, we re-label the years,so that the earliest data point (2008) is assigned the label 0. Then 2009 is 1, 2010is 2, and so on, 2014 is 6; we are interested what happens in the year 8, which isa label for 2016.

The input, as well as the results of calculatios are shown below.

Look at the input data to see how it was entered.The correlation coefficient r = 0.96 is very close to 1, so the two variables, the

time (years) and the number of deaths, are strongly positively correlated. Positivecorrelation means that as one variable increases (i.e., the time increases from 2008on), the other variable increases as well. Strong means that their relationship isvery close to a linear – look at the graph (above right) to see that indeed the datapoints lie close to the line.

The line of regression is given by y = A + Bx = 171.89 + 30.89x, which weinterpret as

number of deaths = 171.89 + 30.89 · year

where (recall out labels) year = 0 represents 2008, and so on. Thus, the predictionfor the number of deaths in 2016 is

number of deaths in 2016 = 171.89 + 30.89 · 8 = 419.01.


We have data for 2016, so we can compare:

In 2016, there were over 900 deaths (914); so our model, predicting 419 deaths,severely underestimated the actual figures.

What does this mean? Our extrapolation was based on the assumption thatthe phenomenon we studied would continue its (linear) trend. However, this didnot happen, as the number of deaths increased in 2015, and then shot upward in2016.

In other words, this means that a linear model was not adequate as a predictorof what would happen in 2016. Of course, it’s easy to criticize once we know whathappened. In reality, in 2014, people would try (and did) all kinds of models (wewill explore exponential regression), and then look critically at all different answersto try to make a good prediction.

Recall that we picked only some of the available data. Of course, to have abetter chance of obtaining a more suitable model, we should use as much data asavailable (however in this case, that would not have helped, as nothing in historicdata suggests a large jump that occurred between 2015 and 2016).

Example 6.4 Linear Regression Problem, Part 2

Now that we know about linear regression, let’s quantify the relationship betweenthe grades on the first two tests (explanatory variable) and the final course grades(response variable).

Importing the Excel file into a regression calculator (or doing it in Excel), wefind the line of regression y = A + Bx = 18.95 + 0.7565x i.e.,

final course grade = 18.95 + 0.7565 · grade on the first two tests

The correlation coefficient r = 0.89588 tells us that the two variables are stronglypositively correlated (thus, the better the first two tests, the better the final grade,and vise versa: the lower the first two tests, the lower the final grade). This, ofcourse, is a general trend, and does not apply to every student.

The above formula tells us that a student who has 70 % on the first two testscan expect that their

final course grade = 18.95 + 0.7565 · 70 = 71.9

Someone who has 45 % on the first two tests can expect that their

final course grade = 18.95 + 0.7565 · 45 = 52.99


Notes: (1) Regression is one possible (but not necessarily reliable) way of makingsense of a given data set – it gives a model based on a function that we choose (inour case, it was a linear function; later, we will use exponential regression).

Keep in mind that regression is based on approximations and can in no waygenerate true values of the variables.

Another way of modelling a data set is to use curve fitting, where we computea formula for a function whose graph contains all (or most) data points. Althoughit is of some use, curve fitting cannot explain underlying mechanisms, so is notreliable when it comes to extrapolation.

(2) Linear regression is one of the most commonly used tools (we will use it againwhen we discuss climate change), but its limitations are often not recognized, orworse yet, known but ignored. In particular, blindly applying extrapolation (say,based on a line of regression) could lead to wrong conclusions. To avoid this, weneed to know something about the situation we are modelling.

Here is an illustration. Consider the following data set which shows how somevariable called Q changed over time, from 1965 to 2015. The second column, isthe usual re-labeling of time (time = 0 is assigned to the earliest measurement),so that we work with smaller numbers.

Using our linear regression calculator https://bit.ly/31qZf7s, we obtained the lineof regression y = A + Bx = 38.624 − 0.248x, i.e., we modelled the quantity Q bythe trendline

Q = 38.624− 0.248 · (time since 1965)

The correlation coefficient r = −0.992 signals a strong negative correlation. Byextrapolating, we obtain predictions for future values of Q:

in 2025, it will be Q = 38.624− 0.248 · 60 = 23.744,

in 2055, it will be Q = 38.624− 0.248 · 90 = 16.301,

in 2085, it will be Q = 38.624− 0.248 · 120 = 8.864,

in 2115, it will be Q = 38.624− 0.248 · 150 = 1.424,

and in 2145, it will be Q = 38.624− 0.248 · 180 = −6.016.

Now we reveal that Q is the percent (ratio) of children years 14 or younger aspart of the total world population. The data has been taken from the World Bankreport Population ages 0-14 (% of total population) https://bit.ly/2scII7k.

The extrapolated predictions do not make sense! Actually they might, forsome time, but it is hard to believe that in 2115, less than 1.5 percent of the entireworld population will be children age 14 or younger. Of course, the prediction for2145 makes no sense.

The problem here is that linear regression does not care what the data pointsactually are. These points could represent the ratio of children in the world popu-lation, or the average number of times dogs in Westdale bark in a single night, orthe maximum grades on an Oxford University math test – the line of regressionwill be the same.

As well, a correlation can be strong (as is the case here), however, extrapolationstill turns out to be useless, inappropriate or unrealistic.


(3) Here is another example that illustrates the point we made in (2). Looking atthe data points on the left, we can visualize the line of regression (centre graph),and extrapolate that whatever quantity is modelled, it will keep growing.

Now assume that the data on the left represents the altitude of a plane as it tookoff from an airport. Of course, the altitude cannot be increasing forever – at onepoint it will stabilize (cruising altitude), and then start decreasing as the planewill be landing.

(4) In the paper Chocolate Consumption, Cognitive Function, and Nobel Laureatespublished in New England Journal of Medicine (see [1] https://bit.ly/2ZJ9ymx,available from within McMaster) we read a report based on linking data about theconsumption of chocolate in certain countries and the Nobel prize winners fromthe same countries (the data is real, this is not a hoax).

We read: “There was a close, significant linear correlation (r = 0.791) betweenchocolate consumption per capita and the number of Nobel laureates per 10 millionpersons in a total of 23 countries.”

The article continues: “When recalculated with the exclusion of Sweden, the corre-lation coefficient increased to 0.862. Switzerland was the top performer in terms ofboth the number of Nobel laureates and chocolate consumption. The slope of theregression line allows us to estimate that it would take about 0.4 kg of chocolateper capita per year to increase the number of Nobel laureates in a given countryby 1. For the United States, that would amount to 125 million kg per year.”

Makes sense?


First of all, just because two quantities are correlated does not mean that onecauses the other. So people in the U.S. might eat all the chocolate in the worldwithout seeing any increases in the Nobel laureates from their country.

As well, a correlation could be due to a coincidence, and so there might be norelation between the variables at all.

(5) Keep in mind that uncorrelated does not mean that the two variables are notrelated – it means that they are not related in the particular way (such as beingin a linear relationship) that we tried to detect.

Exercise 6.5 Judging Correlations

In each case, state whether the correlation is strong or weak and positive or neg-ative, or alternatively, there is no correlation.

Partial answers: top left: strong positive correlation (the slope of the regressionline is small but that’s not relevant for the strength of the correlation). Top right:no correlation (horizontal line represents a constant, i.e., a variable whose valuesdo not change; as such it cannot represent the given data, where the variablesinvolved do change). Bottom centre: weak positive correlation. Bottom right:weak negative correlation.

Example 6.6 Recognizing an (Approximately) Linear Quantity

This left column in the table shows the actual values of a quantity, and in themiddle column these values have been rounded off.

Judging by the actual values, the quantity is linear – all marginal changes (notshown in the table) are equal to 1.1. But when we look at the rounded off values,


and compute marginal changes (rightmost column), they are not all equal. Notethat this happened because the numbers were rounded off.

In reality, numeric data is often rounded off, and it is hard (or impossible)to know the true values. So we work with approximations, and in that sense wecan say, based on looking at the centre and left columns in the table, that thisquantity is approximately (almost, nearly) linear.

Quadratic and Cubic Relationships

Consider the infographics which accompanies the page Stopping distances: speedand braking (at https://bit.ly/2IZi9tN, produced by Queensland Government inAustralia). The diagram shows how the reaction distance and the braking distanceon dry and wet roads depend on the speed of a car. Reaction distance is thedistance the car travels from the moment the driver realizes that they must stopthe car until they hit the brakes. Braking distance is the distance the car travelsfrom the moment the brakes are applied until it comes to a full stop.

Look at the reaction distance first – is there a pattern?

It is approximately linear, as in Example 6.6. As a matter of fact, the relation-ship between the speed of a car and the driver’s reaction distance is linear, so


this discrepancy is due to approximations coming from rounding off. The abovepage states that “In an emergency, the average driver takes about 1.5 seconds toreact.” Thus, the reaction distance is (time times speed) = 1.5· speed, which is aproportional relationship, and thus linear! (Note: to check the values in the tablewe would have to convert units, as we cannot multiply seconds and km/h.)

Now look at the braking distances on dry road, and compute the marginalchanges in this case:

This time, the marginal changes are not equal (nor approximately equal), so theway the braking distance depends on the speed of the car is not linear. But we dosee a pattern in the marginal changes – it’s approximately linear! There is a namefor such quantities.

A quantity changes quadratically if its marginal changes change linearly.(Keep in mind that for the marginal changes to be meaningful – so that we cancompare them – the measurements for the variable have to be taken at equallyspaced time intervals.)

Here is the model of a quadratic relationship – squaring a quantity; in mathterms, we write y = x2 or A = B2, whatever the names for the variables are.

Check that the calculations for the marginal changes in the table are correct.Thus, a quick way to check if a quantity is changing quadratically is to calculate

the second marginal changes and show that they are all equal (or approximatelyequal).

Exercise 6.7 Recognizing a Quantity Changing Quadratically

(1) Calculate the marginal changes and the second marginal changes for the quan-tity given below, to convince yourself that it is a quadratically decreasing quantity.

(Note that the times when measurements have been taken are given – to make


explicit the fact that they are equally spaced.)(2) Analyze the breaking distance on a wet road to show that it is approximatelyquadratic with respect to the speed of a car.

Note: the stopping distances infographics states that “Stopping distances increaseexponentially the faster you go.” As we just saw, this is not the case – the increaseis quadratic! There are many different patterns that can be used to model howa quantity changes, and some increase slower (or faster) than others. In theabove sentence, the word ‘exponential’ has likely been used in to suggest a ‘rapidincrease’, rather than to describe a specific, precisely defined pattern (exponential).

Example 6.8 Quantities Which Change Quadratically

(1) The area of a square of side x is x2. For instance, the area of a square of side3 is 9. If we double the side, the area is 62 = 36, which is a 4-fold increase. If wetriple the side, the area is 92 = 81, which is a 9-fold increase. If we take one-thirdof the side, the area is 12 = 1, which is one-ninth of the original area (note that(1/3)2 = 1/9). Thus, the area increases (or decreases) as the square of the scalingthat has been applied.

In general, if we scale the dimensions of a two-dimensional object by the sameconstant, its area changes by the square of the scale. Take a 2 by 5 metres rectangle(thus, its area is 10 metres squared). If we scale its width and height by a factorof 3, the dimensions are 6 by 15, and the area 6 · 15 = 90 is a scale squared (i.e.,32 = 9-fold) increase of the original area.

(2) In the case of no air resistance, the altitude of an object acting on by gravityonly changes quadratically (could be an object falling, or thrown upward from,say, Earth’s surface).

In reality, due to air resistance, a free falling object reaches its terminal velocityafter some time, i.e., its velocity cannot increase beyond a certain bound (unlikewhat is suggested by the quadratic growth pattern).

(3) A quantity which changes quadratically can be represented as a parabola,which is sometimes referred to as a U shaped curve. The picture below shows astandard parabola where the output values (i.e., the values on the y-axis) are thesquares of the input values (i.e., the values on the x-axis).

(4) Psychologists have determined that life satisfaction, or equivalently, perceptionof well-being viewed as depending on age assumes a roughly quadratic shape:


(Source: World Economic Forum report At what age does happiness peak? of 12November 2015, at https://bit.ly/2TR5miR Try to explain the left and the rightends of the curve, as well as find reasons for its lowest points.

Here is a graph (see [2]) showing the WB (= well being) ladder, i.e., well-beingscale with similar features. Think how you could define ‘mid-life crisis’ based onthe two diagrams.

(5) Another way to characterize a parabolic shape is based on the properties ofreflection by a mirror shaped as a parabola: rays (such as Sun rays or light rays)travelling parallel to each other and hitting a parabolic mirror reflect through asingle point, called the focus of a parabola.(Figure source: OpenStax CNX Project: University Physics Volume 3, section 2.2Spherical Mirrors, at https://bit.ly/30daq37.)

Solar furnaces, automobile headlights, satellite dishes, etc. operate based onthis principle.


(6) The article The U-shaped Life Cycle of Happiness (The Conglomerate, 20 Au-gust 2018, https://bit.ly/2Z0hoaq) discusses the inverted U curve approximatingthe probability of depression based on the age of the UK labour force.

(7) In Money can buy happiness, but only to a point CNBC News of 14 December2015 (https://cnb.cx/2GMhrxg) shows the following diagram:

Look at the red curve in the diagram. What does an increasing piece of the curverepresent? A decreasing piece? A level piece (i.e, the one that appears to behorizontal)? Interpret the piece of the curve that corresponds to the income of$ 80 and higher.

The meaning of “95 % confidence interval” is the following: the red curve wasobtained by surveying a certain number of families and analyzing their replies.If the identical survey is repeated (with a different choice of families) and thecorresponding curves drawn, it is expected that in 95 % of the cases these curveswould fall within the shaded range. I.e., if we repeat the survey 20 times and draw


curves, we expect 19 of them to be within the range (since 19 out of 20 is 95%).That’s all we know – we cannot tell which of the 20 curves is not in the range, orif there is such a curve).(8) Imagine that you kick a ball upward with exactly the same force on four differ-ent celestial bodies, and plot its height (in metres) against the time (in seconds)the ball takes to fall back to the surface from where it was kicked. The plot isa reversed parabola, showing that the ball would reach different heights beforestarting to fall back.

From physics we know that the bigger the mass, the stronger the gravity,and thus the stronger the force that prevents the ball from moving higher. Thefour diagrams represent the ball’s movements on Jupiter, Earth, Mars and Earth’sMoon. Note how big the differences are: the same force that you applied to kickthe ball 10 metres high on Jupiter would send the ball a bit higher than 50 metreson Mars and about 120 metres on Earth’s Moon.

(9) Consider a relationship between the revenue a government receives from taxesand the tax rate. (In this example we consider a simple case where everyone paysthe same percent of their income for taxes.) What curve could possibly representsuch a relationship?

Of course, collecting zero percent taxes yields no revenue. As the tax rateincreases, so does the revenue (below, left).

tax rate (percent)

reve

nue

100tax rate (percent)

reve

nue

0 2010 0 2010

But this trend cannot continue forever. Consider the other extreme: when thetax rate is 100%, we pay all of our income for taxes, and make nothing. In sucha scenario, very likely no one would work (at least not for a job where they haveto pay taxes), and the government revenue would be zero. Thus, there must be apoint where increasing taxes starts to diminish the revenue, and so the relationshipcould look like the one above, right.


(10) The strength of a bone or a muscle is proportional to their cross-sectionalarea, and not to their length or mass. Weight-lifters, bodybuilders and ArnoldSchwarzenegger know that to increase their strength, they have to work out tomake their muscles wider, and not longer or just arbitrarily larger.

(Photograph source: CNN.com article Before Arnold Schwarzenegger was the ’Ter-minator’ at https://cnn.it/31ZttPb)

In the table below we recorded several values of quantities which are linear,quadratic, and – a new one – cubic. To obtain a value in the last column, lookat the corresponding value in the fist column and multiply it by itself three times(for instance, 6 · 6 · 6 = 216).

All three quantities increase, and as we can see, at different rates. A cubic quantitychanges much more quickly than others: for instance, when a linear quantityreaches 6, the quadratic is at 36, and the cubic is 6 times larger, at 216.

Let’s look a bit closer at a cubic quantity by computing its marginal changes:

We might not recognize the pattern in marginal changes, but we certainly recognizethe linear patern of second marginal changes. Recall that linear marginal changesindicate a quadratic quantity! Thus, a quantity is cubic if its marginal changeschange quadratically.

Quadratic and cubic are examples of power relationships, which we formallywrite as y = xn: n = 1 (linear), n = 2 (quadratic), n = 3 (cubic), n = 4 (fourthpower), and so on. We use different powers as one possible model for (i.e., todistinguish between) quantities which change at different rates.


Example 6.9 Cubic Relationships

(1) The volume of a cube of side x is x3. For instance, the volume of a cube ofside 1/2 metres is (1/2)3 = 1/8 metres cubed, and the volume of a cube of side1.4 mm is 1.43 = 2.744 mm3.

How does the volume behave under scaling? From

(mx)3 = m3x3

we see that the volume of a cube whose side has been scaled by a factor of m isthe scale cubed times the original volume. This is true in general: if we doublethe radius of a sphere, its volume increases 8-fold (since 23 = 8). If we scale eachof the length, the width and the height of a rectangular box by 0.1 (i.e., makethem 10 times smaller), then the volume of the scaled box (because 0.1 = 0.001)is thousand times smaller than the original box.

In practice, we usually state this result in the following way: if a linear di-mension is scaled by a factor m, then the volume changes as m3 (recall that thearea changes as m2).(2) If an object is homogeneous (i.e., built of the same material), then its massis proportional to its volume. More technically, since density = mass/volume itfollows that

mass = density · volume

Assume that dogs are homogeneous (which, in reality, is a very reasonable as-sumption). Pineapples (a real dog name) is twice as long as Fido (i.e., Pineapples’linear dimension is twice as large as Fido’s linear dimension). Thus, Pineapples’volume is 8 times the volume od Fido, and consequently, Pineapples’ mass is 8times larger than Fido’s mass.

Example 6.10 Could King Kong (as Envisioned by Hollywood) Exist?

An averge gorilla is about 6 ft tall, and has a mass of about 150 kg. Consultingmultiple King Kong movies, we assume that King Kong is about 160 ft tall, i.e., itslinear dimension is 10 times larger than the liner dimension of an average gorilla.

average gorilla

King Kong

(Monkeys shown to scale. Picture source: BoredPanda https://bit.ly/2Z54cpu)


An animal 10 times taller than an average gorilla would have the volume, andthus the mass, which are 103 = 1000 times larger. Thus, a real King Kong wouldweigh about 150, 000 kg, or 150 tonnes; for comparison, the heaviest land animalis elephant, whose mass can reach 7-10 tonnes.

The heaviest known animal is blue whale, known to weigh at most 190 tonnes.Land animals (with calcium-based skeletons) cannot weigh nearly as much, astheir bones would crumble under the weight. (The buoyancy of water reduces theweight in water.)

Assume, for now, that weight is not a problem. Recall the fact that thestrength of bones and muscles is proportional to their cross sectional area. So, a10-fold increase in linear dimension means a 102 = 100-fold increase in area, andthus in strength – which is no match for a 1000-fold increase in mass.

Thus, to match the increase in mass, the linear dimension of the cross-sectionalarea (of a bone or a muscle) needs to be such that its square is 1000. Since thesquare root of 1000 is approximately 31.6 (i.e., 31.62 = 1000), to match the massincrease, the width of the bones and muscles has to scale by a factor of 31.6.

In conclusion, to properly scale an average gorilla to become King Kong, wehave to make it 10 times taller and 31.6 times wider! But as Hollywood likes thin,they disregarded the calculus of growth we just did. So Hollywood’s King Kongsare fiction and cannot be real.

Example 6.11 How Bones Grow

Human bones and muscles must grow (and they do) in a way to support theincrease in body mass (otherwise we would not be living on this planet).

Example: as a baby girl grows to twice her size (so linear dimension doubles),her volume, and thus her mass, increase eightfold. Recall that the strength of abone is proportional to its cross-sectional area. So if a bone in the baby’s bodygrows so that its radius doubles, the cross-sectional area would quadruple – whichis not enough to match the eight-fold oncrease in her mass.

What is the proper scaling? That scale must be the number whose square is 8,which is (approximately) 2.83. In other words, to support an eightfold increase inmass, brought on by the babys dimensions all doubling, her bones must increasetheir radius by a factor of about 2.83.

(Source: see [3] in References, page 69). Repeat this argument to show that if abody grows to three times its original size (so the length of a bone triples), thewidth of the bone will have to grow by a factor of about 5.20 to maintain itsstrength.


Note: this is a simplified, ‘back of an envelope’ calculation, which illustratesthe principles of growth and development. The actual scaling factors could bedifferent (but not much dfferent).

Example 6.12 Hot Paws: Hard to Find a Pattern

The paper Thermal Contact Burns From Streets and Highways (see [4] in Ref-erences) discusses the relationship between high air temperatures and the corre-sponding temperatures of paved surfaces, such as walkways, streets, and highways.Part of the data is presented in this infographics, warning us to be aware of hightemperatures when we walk our pets.

(Source: CrazyRebels.com https://bit.ly/30qSuT0)It is not possible to determine a pattern, as we do not have enough data (nor

is air temperature equally spaced – so we cannot meaningfully compare marginalchanges). The paper lists data only, and does not underline principles based onwhich we could figure out a pattern.

As exercise, convert the temperatures in the picture into degrees Celsius.

Chapter references[1] Messerli, F. H. (2012). Chocolate Consumption, Cognitive Function, and NobelLaureates. New England Journal of Medicine 367:1562-1564 DOI: 10.1056/NEJ-Mon1211064

[2] Stone, A. A., Schwartz, J. E., Broderick, J. E., and Deaton, A. (2010). Asnapshot of the age distribution of psychological well-being in the United StatesProceedings of the National Academy of Sciences U.S.A., Jun 1;107(22):9985-90.doi: 10.1073/pnas.1003744107. Epub 2010 May 17.

[3] Adler, F. R., Lovric, M. (2015). Calculus for the Life Sciences: Modelling theDynamics of Life. Toronto: Nelson Education.

[4] Berens, J. J. (1970). Thermal Contact Burns From Streets and HighwaysJAMA, 214(11), pp. 20252027. doi:10.1001/jama.1970.03180110035007


7 Quantities Changing Exponentially

If we start with some quantity Q and keep adding the same (fixed) amount a toit (thus creating identical marginal changes), we obtain linear growth:

Q, Q + a, Q + 2a, Q + 3a, Q + 4a, . . .

(if we keep subtracting a, we obtain linear decrease).What happens if we start with Q and keep multiplying by the same (fixed)

amount r?For example, assume that a quantity Q grows by 9 percent per year. At the

end of the first year, it will grow to 1.09Q. At the end of the second year, it willgrow by 9% of the first year’s value: 1.09 · (1.09Q) = 1.092Q. At the end of thethird year, it will grow by 9% of the second year’s value: 1.09 · (1.092Q) = 1.093Q.Continuing, we obtain exponential growth or exponential increase:

Q, 1.091Q, 1.092Q, 1.093Q, 1.094Q, 1.095Q, . . .

Repeated multiplication by a positive number smaller than 1 generates exponen-tial decrease or exponential decay:

Q,12Q,

14Q,

18Q,

116

Q,132

Q,164

Q, . . .

Remember:

(1) Accumulation by addition (or subtraction) creates a linear pattern, and accu-mulation by multiplication (or division) creates an exponential pattern.(2) If, for some quantity,

next value = present value · rthen the quantity grows exponentially if r > 1, and decays exponentially if 0 <r < 1.

Exercise 7.1 What Works Better?

You are negotiating a salary for a job scheduled to start on 1 January 2020 andend on 31 January 2020, and are given two options:(a) Accept $ 10 million as a salary for the job(b) Accept the following option: on 1 January, you are paid 1 cent; on 2 January,you are paid 2 cents, on 3 January 4 cents, on 4 January 8 cents, and so on; i.e.,each day you are paid double of what you were paid the previous day.Which option would you select, hoping to maximize your salary?

Answer: Let’s figure out (b): start with 1 cent on 1 January, and double on eachfollowing day:

Does not look promising: on 15 January, your salary is a bit less than $ 164, a farcry from the $ 10 million in option (a). However, there is something that suggestsbetter news – look at the marginal changes: on 2 January you were paid only 1

Chapter 7 Quantities Changing Exponentially 123

cent more than on 1 January, but on 15 January we were paid $ 163.84 − $ 81.92= $ 81.82 more than the day before.Continue the calculation:

Note how your salary starts to increase more quickly, and the marginal changeon 24 January (compared to 23 January) is $ 41,943.04. Still far from 10 million,however:

The dollar amounts are skyrocketing, and on 31 January you take home about$ 10.7 million! By the way, that’s just the amount you are paid on the last day.Your total salary is much higher, as you need to add up all the amounts you werepaid from 1 January to 31 January.

In the diagrams we show the salary for the first 20 days, and then for the entiremonth.

As the accumulation is multiplicative (each entry is equal to the fixed number (2)times the previous entry), the growth is exponential. Note the distinct feature: a


slow initial growth is followed by a more and more rapid increase (the units of thevertical axis in the diagram on the right are ten millions; 1e7 means times 107).

Similarly, a tell-tale sign of an exponential decay is a rapid initial decrease,followed by a quick slow-down in the rate of decrease. The diagram shows thevalues of a quantity which loses 7 percent of its value per day.

Going back to the salary exercise - we introduce a formula which tells us howto add the powers of a number, so that we can calculate our total salary for themonth:

1 + r + r2 + r3 + · · · + rn =rn+1 − 1

r − 1

For instance, when r = 2 (multiplication factor) and n = 30 (number of days forwhich the value (salary) is doubled)

1 + 2 + 22 + 23 + · · · + 230 =231 − 12 − 1

= 231 − 1 = 2, 147, 483, 647

i.e., about 2.1 billion. Note that this is exactly the total amount (in cents) thatwe would receive under option (b) in Exercise 7.1. Converting, it gives a bit over21.47 million dollars.

Note: it’s good to remember the values of three powers of 2:210 = 1, 024 is about a thousand (kilo)220 = 1, 048, 576 is about a million (mega)230 = 1, 073, 741, 824 is about a billion (giga).

Example 7.2 Fun With Exponential Growth and Decay

(1) Read about the Wheat and chessboard problem, related to the powers of 2, onWikipedia https://bit.ly/1OaaCXe.(2) Take a sheet of paper and fold it in half 20 times (in theory, because in practiceyou will not be able to do it). Assuming that a stack of 100 sheets is about 0.5cm in height, how tall would the folded paper be? How tall would it be if you foldit 30 times?(3) Take an apple (of, say, diameter equal to 7 cm) cut it in half, then take oneof the halves and cut it in half, and so on. What happens after you repeat thisroutine 10 times? 20 times? 30 times? Compare to the approximate diameter of a


water molecule (which is very small in relation to other molecules), which is about0.275 nanometers.

How do we recognize a quantity which changes exponentially?One way is to remember that it’s a multiplicative accumulation: so the next

value must be equal to a fixed number times the previous value (or equal to theprevious value divided by a fixed number, whichever is more convenient). Keep inmind that this works, as in the case of marginal changes, under the assumptionthat the values we are looking at correspond to equally spaced values of the variablewith respect to which the quantity is changing.

Another way is to look at the marginal changes. The quantity in the tablebelow is an exponentially increasing quantity, where the fixed number we keepmultiplying by is 2.

Note that marginal changes show the pattern identical to the pattern of changeof the quantity we study: to advance to the next marginal change, multiply theprevious one by 2. In the following example the quantity grows exponentially(this time, the next value = 3 times the previous value), and the marginal changesfollow the identical pattern (the next value = 3 times the previous value):

Finally, the table below shows an exponentially decreasing quantity (note that thenext value is equal to 0.2 times the previous value, or the previous value dividedby 5). Again, the marginal changes follow the same pattern.

The quantities that we study in this course, in most cases, change with time (hencethe references to time as the input or independent variable).

An exponentially growing quantity is characterized by its doubling time: thetime needed for the quantity to grow to twice its original size. An exponentially


decaying quantity is characterized by its half-life: the time needed for the quantityto decay to one half of its original amount.

Assume that the doubling time of some quantity is T (in some time units,say hours). If its initial amount was q, then after T hours it will grow to 2q;after another T hours expire this new amount will double to 4q. After three timeintervals T have expired (i.e., three doubling times), the quantity will reach 8q. Ingeneral, after n doubling times have expired, the quantity will reach 2nq.

Likewise, assume that the half-life of some quantity is T. It will decay accordingto the following schedule:

Number of half lives Amount left

zero q

one 12q = 0.5q

two 122 q = 0.25q

three 123 q = 0.125q

four 124 q = 0.0625q

five 125 q = 0.03125q

six 126 q = 0.015625q

. . .

n 12n q

Thus, after four half-lives, a bit over 6% of the original amount is left, and afterfive half-lives, the quantity is down to about 3% of its initial amount.

Most common drugs (such as pain killers or caffeine) become ineffective insmall amounts, and it’s generally viewed that after 4-5 half-lives they become inef-fective. However, not all drugs decay according to exponential decay (for instance,the way our body absorbs alcohol follows a different non-linear regime). Below arehalf-lives of several drugs:

Comment on advantages and disadvantages of taking Zalkeplon vs. Doxylamine,or Aleve vs Advil.

Radioactive decay is an example of exponential decay. The half-life sched-ule above shows why certain areas contaminated by radioactive fallout, such asFukushima in Japan or Chernobil in Ukraine, (following major disasters in nu-clear power plants), are still uninhabitable: some radioactive substances have along half-life (years, or decades); as well, some which decay more quickly are dan-gerous even in very small quantities.

Case Study 7.3 Breast Cancer: Clinical Exam vs. Mammogram

In the abstract of the article Is clinical breast examination an acceptable alternativeto mammographic screening? published in British Medical Journal (see [2] in


References at the end of this section) we read: “Breast cancer screening andmammography have almost become synonymous in the public perception, yet thisshould not necessarily be the case. Ideally, a screening tool for breast cancerwould reduce mortality from breast cancer while having a low false alarm rate andbeing relatively cheap. Screening should not be at the expense of the symptomaticservices nor inappropriately divert scarce resources away from equally deservingareas of the NHS [National Health Services, UK equivalent of OHIP] that are lesspolitically sensitive.

An ideal screening test would be simple, inexpensive, and effective. Of thethree modalities of breast cancer screening - breast self examination, clinical breastexamination, and mammography - breast self examination fulfils the first two cri-teria, but early results of two randomised trials conducted in Russia and Chinasuggest that it would not be effective in reducing mortality from breast cancer.Clinical breast examination is also relatively simple and inexpensive, but its ef-fectiveness in reducing mortality from breast cancer has not been directly testedin a randomised trial. Mammography is complex, expensive, and only partiallyeffective. We believe that there is sufficient circumstantial evidence to suggest thatclinical breast examination is as effective as mammography in reducing mortalityfrom breast cancer and that the time has come to compare these two screeningmethods directly in a randomised trial.”

Like other cancers, breast cancer begins with certain changes within one cell,which then propagate as the cell grows and multiplies (by doubling, i.e., by pro-ducing two daughter cells). The doubling times depend on the patient’s age andpossibly other factors. Here is information from [3]:

Age Median doubling time and interval (days)

less than 50 80 (44-147)

50 − 70 157 (121-204)

over 70 188 (120-295)

The doubling times vary quite a bit: for instance, for women aged 50-70, theshortest doubling time recorded was 121 days (about 4 months), and the longestwas 204 days (almost 7 months). For one half of the women, the doubling timeis below 157 days, and for the other half it is above 157 days (that’s what themedian tell us).

Going back to the article, we read: “[...] if you consider the exponential growthrate and doubling time of breast cancer you find that a single breast cancer cell hasto undergo 30 doublings to reach a size of 1 cm, when it will contain 109 cells and beclinically palpable. Since the average size of a non-palpable, mammographicallydetected cancer can be assumed to be about 0.5 cm, the lead time gained bymammography over clinical breast examination would be of the order of only onedoubling. Whether this lead time equivalent of one doubling in the natural courseof 30 doublings would lead to a significantly greater reduction in mortality isquestionable.”

Let’s try to understand what the authors are saying. After 30 doublings,the cancer grows (ideally, assuming that no cells die) to 230 cells, which is aboutone billion (i.e., 109). So we verified the first statement. The term ’size’ in thiscontext means a linear dimension (i.e., the diameter of the cancer as measured ina mammogram with a measuring tape/ruler).

The doubling time is based on the cell count, so during one doubling time,the number of cells doubles. Thus, during one doubling the volume of the cancerdoubles.


How long does it take for a cancer to grow from a diamemeter of 0.5 cm to adiameter of 1 cm?

It cannot be one doubling as claimed in the paper! If the diameter (lineardimension) doubles, then the volume would increase 8-fold, not double. So, some-thing is not right.

We know that volume depends on the third power of a linear dimension, sowe write V = k ·a3, where k is some constant and a is a linear dimension (k = 1 ifa is the length of a side of a cube; k = 4π/3 if r is the radius of a ball). If we wantthe volume to double, i.e., V = 2(k · a3) = 2k · a3, then the new linear dimensionA must satisfy the equation

2k · a3 = k · A3

i.e., A3 = 2a3, and A ≈ 1.26a. Thus, an increase of approximately 26% in thelinear dimension guarantees that the volume doubles. With this in mind, we workbackward:♦ 30th doubling: 230 ≈ 1.07 billion cells, linear dimension = 1 cm♦ 29th doubling: 229 ≈ 536 million cells, linear dimension = 1/1.26 = 0.79 cm♦ 28th doubling: 228 ≈ 268 million cells, linear dimension = 0.79/1.26 = 0.63 cm♦ 27th doubling: 227 ≈ 134 million cells, linear dimension = 0.63/1.26 = 0.50 cm!

Thus, it takes three doublings for the cancer to grow from the size (lineardimension) of 0.5 cm (when it is detectable on a mammogram) to 1 cm (which islarge enough to be detected in a clinical breast examination).

We conclude that the claim made in the paper: “the lead time gained bymammography over clinical breast examination would be of the order of only onedoubling” is incorrect! It should be three doublings, which is a lot more significantas a lead time. The source of error is the ambiguous term ‘size,’ which the authorsof the paper interpreted incorrectly as both linear dimension and volume.

Epilogue: although the journal and the authors have been warned about theerror, no one has decided to correct the paper, nor to publish a warning that oneof its major statements is wrong.

If a quantity grows by R percent in a given time interval, when will it double?If it decays by R percent in a given time interval, what is its half-life? Note thathere R represents a percent, so 17% means that R = 17 (and not 0.17).

People working in finance very often do mental estimates of doubling time andhalf-life, using either the rule of 70 (divide 70 by the rate R) or the rule of 72(divide 72 by the rate R). Which rule is better?

The rule of 70 gives a closer approximation for the half-life (for all decay rates),and for the doubling time with the growth rate up to about 5%. The rule of 72


gives a better approximation of the doubling time when the growth rate is 5% ormore.

As the differences in estimates are not large, in practice either rule is used(often depending on convenience). The rule of 72 is sometimes easier for mentalcalculations, as 72 is divisible by many numbers: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,and 72. Likewise, we can use the rule of 70 when it’s easier to divide 70 by thegiven rate.

Here is a sample of mental calculations one can do: if we invest a sum of moneyat a rate of 8% per year, our investment will double in approximately (use the ruleof 72) 72/8 = 9 years. If we invest at a rate of 3.5% per year, our investment willdouble in approximately (use the rule of 70) 70/3.5 = 20 years.

Inflation decreases money’s buying power (i.e., money is worth less over time).For instance, at a constant rate of inflation of 4% per year, it would take about72/4 = 18 years for any amount of money (not invested, just kept, say in a sock)to halve. Under a 14% per year inflation, it would take about 70/14 = 5 years forany amount of money (not invested) to halve.

In the table below we compare the exact values for the doubling time and thehalf-life with the approximations obtained from the rule of 70 and the rule of 72.

The formulas for the exact values are derived in the following example.

Example 7.4 Calculus: Computing the Doubling Time and the Half-Life

A quantity which grows or decays exponentially can be described using the formula

q = q0(1 + r)t

where q0 is the initial quantity. When r > 0, then 1 + r > 1, and q is increasing(as it’s multiplied by a number larger than 1); when r < 0, then 1+r < 1, and q isdecreasing (as it’s multiplied by a number smaller than 1). Here r is the decimalinterpretation of the rate, so 17% is expressed as r = 0.17.

To calculate the doubling time, we need to find t so that 2q0 = q0(1 + r)t, i.e.,(1 + r)t = 2. Using the natural logarithm,

t =ln 2

ln(1 + r)

For the half-life, we need to find t so that q0/2 = q0(1+r)t, i.e., (1+r)t = 1/2.Using the natural logarithm,

t =ln(1/2)

ln(1 + r)

As an example, assume the annual growth of 7%. The doubling time is

t =ln 2

ln(1 + r)=

ln 2ln 1.07

≈ 10.24 years

Compare this exact value with the two approximations: the rule of 70 gives 70/7 =10 years, and the rule of 72 gives 72/7 ≈ 10.29 years.


Assume the yearly decay of 7%. The half-life is

t =ln(1/2)

ln(1 + r)=

ln(1/2)ln 0.93

≈ 9.55 years

Compare this exact value with the two approximations: the rule of 70 gives 70/7 =10 years, and the rule of 72 gives 72/7 ≈ 10.29 years.

Example 7.5 Quantities Changing Exponentially

(1) The global consumption of plastic has been increasing at alarming rates, creat-ing huge amounts of plastic waste. The diagram (taken from [4], see References),shows the trend in the total production of plastic since 1950s, when it emerged asa major material (the first fully synthetic plastic was invented in 1907).

The units on the vertical axis are megatonnes (Mtones); recall that 1 megatonneis one million tonnes. Graph source: https://bit.ly/2koqm2Q. The authors ofthe paper [4] encourage recycling as “the best treatment for plastic waste since itcannot only reduce the waste but also reduce the consumption of oil for producingnew virgin plastic.”

The European Commission in-depth report Plastic Waste: Ecological and Hu-man Health Impacts (see [5]) shows an exponential increase in plastic productionthat ends around 2007, and is replaced by a downward trend:

The units are the same in both diagrams. Reading the reports, we realize that thedifferences in the two diagrams are due to their authors using different definitionsof what plastic is. The European Commission report states an alarming statisticthat we should all be aware of: “About 50 per cent of plastic is used for single-use disposable applications, such as packaging, agricultural films and disposableconsumer items.”


(2) The Economist on 15 May 2019 published a piece America’s avocado supply isset to tighten (https://econ.st/2JnJ3P2) in which they claim that “the market foravocados has undergone a transformation unlike that of any other fruit (yes, fruit)over the past few decades – from chic canape of the 1970s to millennial stapletoday. In 2018 Americans consumed 3.5kg (7.7lbs) of avocado per person – nearlyfour times the level in 2000. Even McDonalds is serving up guacamole. Yet theoutlook for American avocado lovers is a concern.”

The graph below shows the trend in avocado consumption. As the last sen-tence in the quoted text suggests, the (almost) exponential growth pattern cannotcontinue for much longer.

(3) The richer the people are, the fewer children they have. This statement isgiven evidence in the following graph (taken from The Economist, August 302019, at https://econ.st/2jRfVoe) which relates fertility (the number of childrenper woman) with the GDP (gross domestic product), which is an indicator of acountry’s wealth.

The graph suggests an exponential decay (but it does not have to be exponential;it could be some other pattern).


(4) Writer and blogger Darrin Qualman describes himself as “a civilizationalcritic, a researcher and data analyst, and an avid observer of the big picture.”In his blog post Another trillion tonnes: 250 years of global material use data(https://bit.ly/2kr0l2S) he writes “Our cars, homes, phones, foods, fuels, clothes,and all the other products we consume or aspire to are made out of stuff – out ofmaterials, out of wood, iron, cotton, etc. And our economies consume enormousquantities of those materials – tens-of-billions of tonnes per year.”

The legend points to different sources of information for the graph. Qualmancontinues: “The graph above shows 250 years of actual and projected materialflows through our global economy. The graph may initially appear complicated,because it brings together seven different sources and datasets and includes aprojection to the year 2100. But the details of the graph aren’t important. Whatis important is the overall shape: the ever-steepening upward trendline – theexponential growth.”

(5) On its web page https://bit.ly/32ftWwA, Monex, an Australian securities com-pany, tracks worldwide profits of Netflix, a media service provider and productioncompany:

In the commentary, we read: “Netflix increased annual revenue 35% to $ 16 billionin 2018 and nearly doubled operating income to $ 1.6 billion with an operatingprofit margin of 10%.”


Example 7.6 7 Percent per Year Growth; 10 years to Double

By the rule of 70, a quantity which grows at 7% per year will double in about70/7 = 10 years. It is amazing how many different quantities show this trend;here we explore a few of them. Keep in mind that just because something doublesin some time interval does not make it an exponentially growing quantity.

(1) The UK Newspaper telegraph report Cost of university accommodation doublesin 10 years (at https://bit.ly/2Lj6Ka3) states that “[...] the price of a room inuniversity halls of residence has doubled in just 10 years amid rising fears overstudent debt levels, a major report has found.”

(2) Statistics New Zealand (https://bit.ly/2zDX58t) presents data which provesthe claim made in the title of their report Kaumatua population [in New Zealand]doubles in 10 years (Kamatua is a respected elder of either gender in the Maoricommunity. In this context, it refers to people aged 80 or older in general.)

Compare 2002 and 2012 data for both males and females to check the claim.

(3) On 7 September 2016, BBC News published a report titled Number of childrenwho are refugees doubles in 10 years (report and photo: https://bbc.in/2PyD123).

We read: “The number of children who are refugees has doubled in the last 10years says charity UNICEF. Many of the children have escaped from danger and


wars in countries like Syria, Afghanistan, Eritrea or Iraq. The report by UNICEFsays that more than half of all refugees in the world are children.”

(4) In Saskatchewan diabetes more than doubles in 10 years, part of worldwidetrend, CBC News (6 April 2016, https://bit.ly/1T0q4G6) reports that “over aquarter of province’s population has been diagnosed with diabetes or pre-diabetes;An estimated 97,000 people in Saskatchewan have diabetes, according to the Cana-dian Diabetes Association. That’s a 59 per cent increase from 10 years ago, andpart of an international trend.”

Note: The 59 percent increase does not make it double. Read the article tosee why the author claims that the rates doubled.

(5) In the video report Digital: NY Student Loan Debt Doubles in 10 Years NCCNews (21 September 2016, https://bit.ly/2MKaFQp) we hear that “the averageNew Yorker with college loads owed $ 22,200 in 2015 [...] Student load debt hasmore than doubled in NY in the last decade.”

(6) In the article World heritage tourists destroying the sites they love, in NewZealand Herald (11 July 2018, https://bit.ly/2AJYptI) is critical of the way how,even with best intentions, tourists are bad news (so much so that local authoritiesworldwide started limiting the number of people visiting certain locations).

As well, we find this statement: “The number of people travelling by airinternationally has increased by an average of around 7 per cent a year since 2009.This growth is expected to continue at a similar rate.”

Example 7.7 Exponential Model is Sensitive With Respect to its Exponent

After some initial period of seemingly slow increase, a quantity growing exponen-tially picks up the pace and increases at larger and larger rates. By comparingtwo exponential growth patterns, in our case 1.1x and 1.11x, we discover that, ontop of rapid increases, the two patterns grow apart from each other as well (eventhough there is a difference of one hundredth between the two numbers 1.1 and1.11); when we divide them, 1.11x/1.1x, that pattern is also exponential.

When x = 100, the two quantities are of the same order of magnitude. However,when x = 1, 000, 1.11x is 4 orders of magnitude larger than 1.1x, and when x =10, 000, it is 40 orders of magnitude larger than 1.1x.

Note how large the numbers in the table have become. There is absolutely nothingin universe that we could relate these numbers to. For instance, the total numberof elementary particles in universe is believed to be somewhere between 1078 and1082. Even if we wish to measure the diameter of the known universe by using astick whose length is the diameter of an electron we would not need numbers solarge.

Hence an important conclusion: an exponential model, no matter the quantityinvolved, becomes unrealistic after some time. In other words, a quantity growingexponentially, sooner or later, will change its growth pattern. (This is not trueonly for exponential growth, but for any model which assumes unlimited growth,such as a line with positive slope.)


Exponential Regression

Of all exponential growth patterns (functions), there is one that we view as special,and call it a natural exponential function: y = ex, where e = 2.71828 . . . is adecimal number which keeps going (i.e., has infinitely many decimals), in such away that there is no periodicity in its decimals (such numbers are called irrational).Why someone would choose e to call “natural” (as opposed to, say, 2, or 3, or 10)is clarified in calculus courses. In real life situations, of course, we use the onethat’s most convenient.

Many online exponential regression calculators work with natural exponentialfunctions.

Recall that regression is an attempt at describing a data set by an objectwhich is easier to work with, such as a curve (in the case of linear regression, it’sa line). Knowing linear regression only does not suffice, as there are phenomena(data sets) which cannot be described well by a line.

For instance, the data set below represents cranial capacity (brain volume) ofhumans dating about 3 million years (the diagram has been drawn based on datafrom [1]):

In this case, as suggested, an exponential function could represent (we can alsosay ‘summarize’) the data.

To figure our exponential regression we use an online calculator, as we did for thelinear regression. The link is https://bit.ly/31dfxAj.

As illustration, we consider Stats Canada information (or Wikipedia, under Pop-ulation of Canada, Census data) on the population of Canada:

Based on the 1966-2006 data, we will obtain an exponential regression model,which we will use to predict the population in 2016, and compare with the actual


2016 Census data.As in the case of linear regression, we enter the data into the calculator,

relabelling the years so that the earliest data (1966) is represented by zero. (Thus,1976 is 10, 1986 is 20, and so on.)

The top line tells us that we are looking for the model y = AeBx. Hitting ‘Execute,’we obtain the following output:

The correlation coefficient r ≈ 0.998 indicates a strong correlation; thus, thepopulation data we have can be well described/ explained by an exponential model(note how the data points lie very close to the exponential curve). This exponentialmodel is given by

y = 20, 236.058e0.01141x

where y represents the population (in thousands) and x is the time, measured from1966.

We rounded off A to three decimal places, and B to 5 decimal places (recallthat exponential functions are sensitive to their exponents, as illustrated with the1.1x vs. 1.11x example).

This model extrapolates that, in 2016, the population of Canada is

y = 20, 236.058e0.01141(50) = 35, 800.65

i.e., about 35.8 million. Compared to actual data, we see that the model givesan overestimate; in other words, the population grew slower than the exponentialpattern based on the given data.

We will look at further examples of exponential regression in our study of climatechange and human population.

Limited Growth

Resources (food, water, energy, etc.) are essential for human (and animal, plant)survival on our planet. Nature, excluding humans, is able to self-regulate and con-


trol the balance between the demand for resources and their availability. However,humans are the problem. As human population grows, so does the demand forresources.

A simple, but accurate model of this dynamic is shown in the diagram below.The resources we need to live are growing, but in a linear fashion (for instance, webuild another power plant, convert 10,000 hectares of desert into farmland, etc.).However, the demand, based on the exponential growth of human population, isexponential.

time

reso

urc

es

resources

when?

demand for resources

Assuming that these trends continue, there will be a time when the demand forresources will meet and then overcome the resources available – which is going tobe a huge problem. When, or if, this will happen is not clear.

As our planet is a limited environment, all resources are limited. So neither linearnor exponential growth models can predict what will happen in a more distantfuture, since both imply unlimited growth. To understand things, we need todevelop limited growth models.

One of the most commonly used models of limited growth is called a logisticmodel, visualized by the logistic or S-shaped curve.

inflection

input / explanatory

outp

ut / re

sponse

carrying capacity

rapid initial growth rate

growth rate slows down

growth rate approaches zero

Logistic model is characterized by an initial period of rapid growth that mimicsexponential growth. For instance, when an infectious disease emerges, its initialspread within a population is exponential (as there is a large “pool” of people whocould contract it).

After some time, the growth slows down since it becomes more difficult fora disease to find people who are not infected; so although the quantity we study(say, the number of infected people) still increases, it does so at a decreasing rate.The value where this happens is called an inflection (see the figure above).

Finally, in the long run, the growth rate approaches zero; i.e., the quantitystabilizes. We also say that it reaches its carrying capacity, or that it plateaus.

This is one possible way to model limited growth. Note that although thismodel is more appropriate in many circumstances, it assumes that the quantityremains at its plateau. Sometimes, that could be the case (for instance, an animalpopulation achieving ecological balance, where its size matches the resources


available), but sometimes not (an infectious disease will eventually die out, andthe number of infected people will be decreasing).

We can explain logistic growth pattern in terms of its marginal changes.

As they describe a growing pattern, all marginal changes are positive. Initially,they increase (drawn in red), until the growth reaches the point where the marginalchanges start decreasing (green). The moment when the pattern changes (red-green rectangle) is the inflection.

For a quick estimate related to any quantity which increases according to thelogistic model – it is good to remember the mathematical fact that the value ofthat quantity at inflection is one half of its carrying capacity (plateau).

In reality, all growth is limited, but its pattern, of course, is not necessarilylogistic.

Example 7.8 Limited Growth, Logistic Curve, Plateau

(1) The diagram below shows airspeed records (in km/h) in the last century (inother words, how fast was the fastest airplane at the time). We recognize thetell-tale signs of an (initial) exponential pattern: slow initial increase followed bya stronger and stronger growth. However, after some time, the growth shows thesigns of slowing down, thus suggesting logistic growth. (Source: Wikipedia Flightairspeed record https://bit.ly/32AJc70.)

(2) The Rockefeller University web publication Bi-Logistic Growth discusses logis-tic growth model and suggests an improvement, named bi-logistic growth, whichcombines two logistic curves (https://bit.ly/34PzQGe). We read “Many processesin biology and other fields exhibit S-shaped growth. Often the curves are wellmodeled by the simple logistic growth function, first introduced by Verhulst in1845. Although the logistic curve has often been criticized for being applied to


systems where it is not appropriate, it has proved useful in modeling a wide rangeof phenomena.”

The following diagram, taken from the publication, shows the growth of asunflower (height vs age):

(3) The number of mobile phone subscriptions shown in the following diagram hasbeen taken from Our World in Data (https://bit.ly/2Q33KCx)

Note that the vertical axis represents the number of cellular subscriptions per 100people.

ITU (International Telecommunications Union) estimates that by the end of2018 there were 8.2 billion mobile subscribers worldwide, which averages 107 mo-bile phones for 100 people.

(4) In his blog bit-player, the science writer Brian Hayes discusses (among otherthings) trends in world economics (https://bit.ly/2pOsHai). He says: “Writing in


The New York Times, the business columnist Eduardo Porter quotes Paul Ehrlichquoting Kenneth Boulding: ‘Anyone who believes exponential growth can go onforever in a finite world is either a madman or an economist.’ Then Porter, sidingwith the economists if not the madmen, proclaims that economic growth mustcontinue if ‘civilization as we know it’ is to survive. (He doesnt explicitly say thegrowth needs to be exponential.)”

Then Brian Hayes presents his vision of historic and future economic growth:

He explains: “For hundreds or thousands of years before the modern era, averagewealth and economic output were low, and they grew only very slowly. Life wassolitary, poor, nasty, brutish, and short. Today we have vigorous economic growth,and the world is full of wonders. Life is sweet, for now. If growth comes to an end,however, civilization collapses and we are at the mercy of new barbarian hordes(equipped with a different kind of horsepower).”

(5) Godwin’s law, when originally formulated, referred to newsgroup discussions.These days it is more broadly used, but more or less in the same context. Accord-ing to Wikipedia https://bit.ly/2q1kn75 “As an online discussion grows longer,the probability of a comparison involving Nazis or Hitler approaches 1. If an on-line discussion (regardless of topic or scope) goes on long enough, sooner or latersomeone will compare someone or something to Adolf Hitler or his deeds, the pointat which effectively the discussion or thread often ends.”

length of online discussion

pro

ba

bili

ty o

f re

fere

nce

to H

itle

r o

f N

azis

1

Note: In linguistics, the term “inflection” represents the change of form thatwords undergo to accommodate for number (egg vs. eggs), tense (go vs. went),gender, person, case, etc.

Inflection is also used to signify change, or a particular moment in the processof change. For instance, in the report Hong Kong airport shuts down amid pro-democracy protest by CP24 News (12 August 2019, at https://bit.ly/2CvsSd6) weread that “One of the world’s busiest airports cancelled all flights after thousandsof pro-democracy demonstrators crowded into Hong Kong’s main terminal [...]”Characterizing thousands of people fighting for freedom as terrorists, the regime inBeijing stated that “One must take resolute action toward this violent criminality,showing no leniency or mercy [...]” and then “Hong Kong has reached an inflectionpoint where all those who are concerned about Hong Kong’s future must say ‘no,’to law breakers and ‘no’ to those engaged in violence.”


The title of the article A Historic Inflection Point In Capitalism’s BattleAgainst Climate Change published in the Forbes Magazine on 26 April 2019(https://bit.ly/2WYg1to) announces that the authors are reporting on some majorchange (in this case, an advancement in direct air capture technology).

Logarithms

The paper Hopes for the Future: Restoration Ecology and Conservation Biology(see [6] at the end of this chapter) discusses restoration ecology and its impor-tance for the preservation of our nature and natural resources. In the abstract, weread: “Conversion of natural habitats into agricultural and industrial landscapes,and ultimately into degraded land, is the major impact of humans on the naturalenvironment, posing a great threat to biodiversity. The emerging discipline ofrestoration ecology provides a powerful suite of tools for speeding the recovery ofdegraded lands. In doing so, restoration ecology provides a crucial complement tothe establishment of nature reserves as a way of increasing land for the preserva-tion of biodiversity. An integrated understanding of how human population growthand changes in agricultural practice interact with natural recovery processes andrestoration ecology provides some hope for the future of the environment.”

Later in the article we find the following diagram depicting the relation be-tween the spatial scale of natural (ovals) and human caused disasters (rectangles)and their recovery time.

Note that the scales (tickmarks) on the axes are not linear: the same amountof space is used for the numbers between 1 and 10 as between 10 and 100 orbetween 100 and 1000. (Recall that on a linear scale, the numbers (labels) 1, 2, 3,4, etc. are equally spaced).

What is this scale, why was it used, and how was it done?

Some disasters affect small areas: for instance a lightning strike affects aboutone hundredth (10−2) of a square kilometre; a land slide could destroy about onesquare kilometre area (hundred times larger than a lightning strike). Industrialpollution can destroy an area of more than 100 km2, and a meteor strike could


affect anywhere between 1,000 and 10,000 km2. As well, the range of the recoverytimes is large: from about 2 years for a lightning strike, to 100 years for modernagriculture, to 1000 years for a meteor strike.

Imagine drawing a linear scale on which we have to show the range of numbersfrom 0.01 (10−2) to 10 thousand. If we use 1 cm for the unit distance (i.e., thelabels 0, 1, 2, 3, 4, etc. on the number line are 1 cm apart from each other), theline would have to be 10,000 cm (100 metres) long. Sometimes we draw a linearscale in the following way

627021 1080.047 10610

where the zigzag pieces indicate where the number line has been collapsed. (Truesizes cannot be shown: the distance from 1 to 21 is 20 units; the distance between21 and 6270 is 6249, which is supposed to be roughly 297 times larger than thedistance between 1 and 21.) Thus, at best, a linear scale can show the order ofthe numbers correctly (i.e., show which is larger, which is smaller), but cannotfaithfully represent their relative sizes.

There is a way to make things look better, and it’s called a logarithmic scale.This is what it looks like, compared to a linear scale (i.e., the usual number line):

0

10010 10000

0.001 0.01

51

0.1

1000

4

1

32

0

−1−4 −3 −2logarithmic

linear

A logarithmic scale is defined for positive numbers only (this is not a big issue).Note its important feature: at the same time, it reduces large quantities (moreprecisely, quantities larger than 1) and expands small quantities (those between 0and 1). In this scale, 1 corresponds to 10; 2 to 100; 4 to 10,000; −1 to 0.1; and−3 to 0.001 = 1/1000. How is this done?

Logarithms were invented in the 17th century to simplify calculations withnumbers, based on the following idea. Suppose we need to multiply two large num-bers A and B, for which the usual pencil-and-paper algorithm takes a long time.Using specially designed tables, we find the logarithms of these two numbers, log Aand log B, and then add them (addition is much easier to do than multiplication)to form log A + log B. These logarithms have a very special property:

log A + log B = log(AB)

Now we know the logarithm of the product AB that we need to compute. Lookingat the “inverse” tables, we find the desired product AB.

It took mathematicians quite some time to generate (by hand, of course) thetables to read the values of the logarithms, and the inverse tables to recover anumber from its logarithm. Such tables were published around 1620s, and havebeen in used, in one form or another (slide rule), until the 1970s, when we startedusing computers and then pocket calculators.

How are logarithms defined?The logarithm (more precisely the logarithm base 10) of a number is

defined as the power that we need to raise 10 to in order to get the given number.For instance, the logarithm of 100 is 2, because 102 = 100; the logarithm of 100, 000is 5, because 105 = 100, 000; the logarithm of 0.001 is −3, because 10−3 = 0.001.


In general, to compute logarithms we use calculators. In the tables below werecord numbers and their logarithms, as well as their orders of magnitude.

Number Logarithm Why Order magn.

1 0 100 = 1 0

10 1 101 = 10 1

100 2 102 = 100 2

1,000 3 103 = 1, 000 3

1,000,000 6 106 = 1, 000, 000 6

Thus, for the powers of ten, the logarithm is equal to the order of magnitude. Thisworks for the powers smaller than 1 as well:


1 0 100 = 1 0

0.1 −1 10−1 = 0.1 -1

0.01 −2 10−2 = 0.01 -2

0.001 −3 10−3 = 0.001 -3

0.000001 −6 10−6 = 0.000001 -6

In general, the logarithm of a number is a decimal number:


3.456 = 3.456 · 100 0.53857 100.53857 = 3.456 0

34.56 = 3.456 · 101 1.53857 101.53857 = 34.56 1

345.6 = 3.456 · 102 2.53857 102.53857 = 345.6 2

3456 = 3.456 · 103 3.53857 103.53857 = 3456 3

Note that for the numbers in the above table (all larger than 1), the integer partof the logarithm is equal to the order of magnitude. It’s different for the numberssmaller than 1:


0.3456 = 3.456 · 10−1 −0.46143 10−0.46143 = 0.3456 −1

0.03456 = 3.456 · 10−2 −1.46143 10−1.46143 = 0.03456 −2

0.003456 = 3.456 · 10−3 −2.46143 10−2.46143 = 0.003456 −3

0.0003456 = 3.456 · 10−4 −3.46143 10−3.46143 = 0.0003456 −4

Remember that:If log B = x, then 10x = B

So, if we know that the logarithm of some number is 2.7, then we can recover thenumber by computing the power of 10, i.e., 102.7 = 501.187. The number whoselogarithm is −1.95 is 10−1.95 = 0.01122.

In calculus, we often work with the natural exponential function ex (wheree = 2.71828 is a famous math constant), and its inverse, the natural logarithmln x. (We used ex in working with the exponential regression.)


We will stick to the logarithm base 10, as is common in many applications.To work with logarithms, we need to use a calculator. For instance, to compute

the logarithm of 0.0352, we use the log key and enter

log 0.0352 =

to obtain −1.45346. Remember that this means that 10−1.45346 = 0.035.

Inverse calculation: to find the number whose logarithm is 4.16, we use inverselogarithm (i.e., exponential function); so press shift and log keys and enter

shift log 4.16 =

to obtain 14, 454.39771.

The actual calculation performed was 104.16 = 14, 454.39771.

Example 7.9 Logarithmic graph (logarithmic scale)

In the graph below, the output variable is shown on a logarithmic scale, meaningthat the labels are the logarithms of its values.

A

B

C

D

log

x

0.6

3

−1−1.4

E2.6

For instance, the data point B represents the value 10−1.4 = 0.0398, and the datapoint E represents the value 102.6 = 398.107. (Keep in mind that the negativevalues of the logarithm mean that the actual (positive) number is smaller than 1.)Going down vertically means than a value is getting closer to zero.

Note that the difference on the log scale between the data points A and Dis 3 − (−1) = 4. This means that A is 104 = ten thousand times larger than D.Indeed, A represents the value 103 = 1, 000, and B represents the value 10−1 = 0.1,and A/B = 1, 000/0.1 = 10, 000.

Likewise, the difference on the log scale between the data points E and C is2.6 − 0.6 = 2. This means that E is 102 = 100 times larger than C.

Often, instead of the values of the logarithm, we label the actual values:

A

B

C

Dx

0.01

E

0.1

100

10

1

1000

10000

This is how both axes on the human and natural disasters graph were done.


In the picture below both variables are shown on a logarithmic scale.

600

3020

40

500

40003000

The gridlines in a coordinate system with linear scales on its axes are equallyspaced. That’s not the case for logarithmic scales: the gridlines are packed tighterand tighter before the the next power of ten starts. We can verify this with acalculation (below): as we move from 10 to 100, the next value of the logarithmis getting closer to the previous value (i.e., the marginal changes are decreasing).

Example 7.10 Richter Scale: Measuring the Power of an Earthquake

Determining the magnitude of an earthquake involves measuring (using a seis-mograph) the amplitudes of seismic waves that travel through the interior of ourEarth at a specific distance from the source of an earthquake.

Below is a sample recording of a seismic wave. The amplitude of a wave is thelargest displacement from its equilibrium; in the picture below the equilibrium isthe second horizontal line from the bottom.

Routinely, a number of seismographs record the amplitudes of the waves. This data


is collected and adjusted to account for their distance from the epicentre, resultingin a number (Richter scale) which represents the magnitude (or the power) ofthe earthquake. By the way it is constructed the Richter scale is logarithmic.

Every year, our Earth experiences several million earthquakes of magnitude be-tween 1 and 1.9 on the Richter scale. Called microearthquakes, these earthquakesare sensed by seismographs, but we can only rarely feel them (some animals arequite better than humans in feeling earthquakes).

Earthquakes of magnitude 5 to 5.9 are still fairly common (occur about 2-3per day). Depending on the area they hit and the quality of buildings, they cancause significant damage and loss of life. An earthquake of magnitude 6 or higheris considered major.

The earthquake that occurred on 16 April 2016 in Ecuador was measured tobe of magnitude of 7.8 and caused significant damage and loss of life.

(Source: Reuters.)Because the scale is logarithmic, each integer increase in magnitude represents a10-fold increase in the amplitude of the incoming seismic wave. Thus, the differencein amplitudes between a Richter magnitudes 5 and 6 earthquakes is tenfold. Usingphysics, we can compute that the amount of energy released is 31.6 times largerfor a scale 6 earthquake than it is for a scale 5 earthquake. The graphic belowshows the effects of earthquakes of different magnitudes (source: US GeologicalSurvey, https://bit.ly/32BuE7h).

The strongest earthquakes in human history (recorded directly, or calculated fromgeological data) have been of magnitude 9. The nuclear bomb dropped on Hi-roshima in 1945 caused an earthquake of magnitude 6. The super-eruption of theToba volcano (Sumatra, Indonesia) about 75,000 years ago caused an estimatedmagnitude 9.15 earthquake.

The strongest earthquake ever that we know of (magnitude 13) was causedby a large meteor that collided with Earth (in the area of Yucatan Peninsula inMexico) about 65 million years ago, and is believed to have caused the demise ofsignificant fraction of life on Earth, including the dinosaurs.


Example 7.11 Logarithmic Scales

Whenever the range of values of a variable is large, it’s convenient to use a loga-rithmic scale.[1] In Wikipedia (https://bit.ly/2CA6aAs) we read: “The Internet of Things(IoT) is a system of interrelated computing devices, mechanical and digital ma-chines, objects, animals or people that are provided with unique identifiers (UIDs)and the ability to transfer data over a network without requiring human-to-humanor human-to-computer interaction. There are a number of serious concerns aboutdangers in the growth of IoT, especially in the areas of privacy and security; andconsequently industry and governmental moves to begin to address these.”

The graph below shows the number of devices (on a logarithmic scale) con-nected to the Internet of Things. Note that 100B (100 billion) is 100 thousandtimes (6 orders of magnitude) larger than 1M (1 million).

A quantity represented by a straight line on a logarithmic scale is growing ordecaying exponentially. Why?

[2] Often used in chemistry, pH is a measure of acidity or alkalinity of a solu-tion. It is defined as the negative of the logarithm base 10 of the hydrogen ionconcentration, in moles per litre solution. (Note: the negative sign in front of thelogarithm is for practical reasons - it makes pH a positive number.)

[3] A decibel (dB) is used to measure the relative loudness of sounds. Given twosounds S1 and S2, a decibel is defined as 10 times the logarithm base 10 of theratio S1/S2. When S2 is taken to be the intensity of the sound barely detectableto the human ear (such as a faint whisper), we obtain the usual measurement ofloudness. Normal human communication is measured to be around 60 dB, and aloud motorcycle engine produces a 95 dB sound. A prolonged exposure to soundsabove 70 dB could cause permanent damage to our hearing.

[4] The Hick’s law, or the HickHyman law, relates the time it takes for a personto make a choice, given several options. Increasing the number of options has beendetermined to increase the decision time logarithmically. As a matter of fact, theaverage reaction time (i.e., the time it takes to make a decision) is approximatelyproportional to the logarithm of the number of options.


Chapter references[1] McHenry, H. M. (1994). Tempo and mode in human evolution.Proceedings ofthe National Academy of Sciences, USA. 91:6780-6.[2] Mittra, I., Baum, M., Thornton, H., and Houghton, J. Is clinical breast exami-nation an acceptable alternative to mammographic screening? BMJ. 2000 Oct 28;321(7268): 10711073. https://bit.ly/2W2rCtb[3] Peer, P. G., Dijck, J. A., Hendriks, J. H., Holland, R., and Verbeek, A. L.(1993).Age-dependent growth rate of primary breast cancer. Cancer. 1993 Jun 1; 71(11):3547-51. https://bit.ly/39BOpj9[4] Wu, G., Li, J., and Xu, Z. (2013). Triboelectrostatic separation for granu-lar plastic waste recycling: a review. Waste management, 33(3): 585-97. doi:10.1016/j.wasman.2012.10.014. Epub 2012 Nov 28.[5] Science for Environment Policy (2011). Plastic Waste: Ecological and HumanHealth Impacts. European Commission[6] Dobson, A., Bradshaw, A. and Baker, A. (1997). Hopes for the Future: Restora-tion Ecology and Conservation Biology. Science. 277. 515-522. 10.1126/sci-ence.277.5325.515.

Chapter 8 Covid-19 149

8 Covid-19

The outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)was identified in Wuhan, China, as early as November 2019. On 11 March 2020,the World Health Organization declared this outbreak (identified as COVID-19) apandemic. As of 5 May 2021, there have been more than 154 million cases world-wide (very likely many more, as due to the low number of tests many asymptomaticand mild cases have not been included). About 3.2 million people died. COVID-19has affected and disrupted every aspect of human life and has had a devastatingeffects on local and global economies.

In this chapter we focus on understanding numeric information and principlesrelated to the pandemic (which are more general, and apply to any infectiousdisease).

Reproduction RatesTo quantify how infectious a disease is, researchers use two numbers. The basicreproduction rate or the basic reproduction number R0 is the mean (aver-age) number of people who are infected, for the duration of the disease, by a singleinfected person. This number is based on the assumption that there is no immu-nity in the population from past exposure or from vaccination, and no interventionmeasures have been implemented. Thus, the basic reproduction rate is suitablefor an initial outbreak of a disease, when all, or most members of a populationare susceptible (i.e., not infected, but can be infected). This rate varies within apopulation, as it depends (among other factors) on the density (different contactrates) and cultural differences.

This example shows how R0 works. If R0 = 2.6, this means that a singleinfected individual will, on average, pass on the infection to 2.6 additional people.(Easier to think this way: ten infected people will spread the infection to 26additional people.) Assume that initially, five people are infected (shown in red):

Each of the five people will infect 2.6 additional people, so after one cycle, therewill be 5 (infected from before; shown in orange) + 5 · 2.6 (new infections, shownin red) = 5 + 13 = 18 infected people:


(A cycle is the average time needed for a susceptible individual to become infected,and identified as infected.) In the next cycle, there will be 18 (infected from before;orange) + 13 · 2.6 (new infections; red) = 51.8 ≈ 52 infected people (note that 13,and not 18 people infect additional people in this cycle; the initial five infectedpeople already infected additional people in the first cycle):

One more cycle: of the 52 infected people, 18 have already caused additionalinfections. Thus, the total number of infected people in the next cycle is 52(infected from before; orange) + (52−18) ·2.6 (new infections; red) = 140.4 ≈ 140infected people:

As we can see, the growth is exponential. Note the pattern: start (cycle zero): 5 cycle 1: 5 + 5 · 2.6 cycle 2: 5 + 5 · 2.6 + (5 · 2.6) · 2.6 = 5 + 5 · 2.6 + 5 · 2.62

cycle 3: 5+5·2.6+(5·2.6)·2.6+((5·2.6)·2.6)·2.6 = 5+5·2.6+5·2.62+5·2.63

and, in general, in cycle n the number of infected people is

5 + 5 · 2.6 + 5 · 2.62 + · · · + 5 · 2.6n = 5 · 2.6n+1 − 12.6 − 1

So, in cycle 4 there will be

5 · 2.65 − 12.6 − 1

≈ 368

infections, and in cycle 10

5 · 2.611 − 12.6 − 1

≈ 114, 695

infections.When R0 > 1, the infection will spread exponentially. If R0 < 1, the infection

will spread slowly, and will eventually disappear. So R0 < 1 is what we want!Note: R0 = 1 means that every infection causes another new infection. In theory,this means things stay at the same level. However, if people get infected fasterthan they recover then the number of infected people will increase.


Recall that R0 is determined when there is (near) zero immunity in the pop-ulation, and under the assumption that everyone is susceptible. As an infectiontakes its course, these assumptions no longer hold. What is more useful is to havea number like R0, but measured at any time in the life of an infection.

The effective reproduction rate or the effective reproduction numberRe is the mean (average) number of people who can be infected by a single infectedperson, at any given time. Unlike, R0, the effective rate Re changes over time.

Re is based on the number of people who are infected, and on the number ofsusceptibles who might get in contact with infected people. As people recover ordie from the infection, Re decreases.

To read more about reproduction rates, read the CBC News article “COVID-19 modelling explained: The calculations used to make public health decisions inSask.” (4 May 2020, at https://bit.ly/3fsFUdN). The following graphic has beentaken from it:

See [1] at the end of this chapter for further information about R0 and Re. Asof early May 2020, Re for COVID-19 has been estimated to be about 1.1 (forCanada) and 2.3 (for Ontario); see [2] and [3]. Of course, the estimates for theserates change over time, as more (and beter) data becomes available.

The relation between the basic and essential reproduction rates R0 and Re is givenby

Re = R0 · (1 − Pimmune)

where Pimmune is the proportion (expressed as percent) of immune people withinthe population. Because 1−Pimmune ≤ 1, we conclude that Re ≤ R0, which makessense: as an infection takes its course, it’s harder to find non-infected people whowould be passed on the infection by an infected individual.

Herd Immunity

Herd immunity, (also known as herd effect, social immunity, community immu-nity, or population immunity) is a state when a large proportion of a populationhas become immune to an infectious disease. In such populations, assuming that


immune members are highly unlikely to spread the infection, the chains of trans-mission are either broken or the transmission is slowed down.

In [1], The Centre for Evidence-Based Medicine states that “Initial reportssuggested that one of the UK Government’s strategies in tackling the pandemicwas to allow the virus to spread within the community, in a controlled way, so thatimmunity, so-called herd immunity, could develop across the population. [...] Theproblem with leaving people to catch the infection spontaneously, leading to herdimmunity, is that the death rate would increase as a result. For example, on 10April, the number of confirmed cases in Sweden [which has been trying to achieveherd immunity] was 9685 with 870 deaths (9.0%), compared with Norway with6219 confirmed cases and 108 deaths (1.7%) and Denmark with 5830 confirmedcases and 237 deaths (4.4%).”

The percent of immunized or immune individuals within a population neededto achieve herd immunity (i.e., to prevent a sustained spread of the infection) isgiven by

Phi = 1 − 1R0

Thus, if R0 = 2, then 50% of the people need to be immunized (or to become im-mune by contracting the infection and recovering). If R0 = 4, then the proportionrises to Phi = 1 − 1/4 = 3/4, i.e., 75%.

If R0 = 10, then the proportion to achieve herd immunity is 90%. Thus, as R0

increases, so does the proportion of the population that needs to be immunized.In [1], we read: “Thus, if R0 is 10, a child with measles will infect 10 others ifthey are susceptible. When other children become immune the infected child whoencounters 10 children will not be able to infect them all; the number infected willdepend on Re. When immunity is 90% or more the chances that the child willmeet enough unimmunized children to pass on the disease falls to near zero, andthe population is protected.”

This diagram (from [1]) visualizes the herd immunity requirements.

How to interpret this graph? If 60% of a population is immune (purple line),the reproduction number Re will fall below 1 (that’s what we need to eradicatethe infection) only if R0 is not greater than about 2.5. Herd immunity of 80% (redline) will work (i.e., will push Re below 1) if R0 is less than 5. On the other hard,herd immunity of 90%will push Re below 1 even if R0 = 10.

The way we use this diagram is to go backwards: if R0 = 4 then Re will fallbelow 1 when herd immunity is larger than 70% but can be smaller than 80% (thegreen line (70%) is above 1, but the red line (80%) is below 1).


In [2], we find R0 = 2.3 for Ontario (as of April 2020). In Saskatchewan,it is about 3.12 (source: CBC News, 4 May 2020, at https://bit.ly/3fsFUdN).According to [1], the mean (average) value of R0 for Covid-19 in the U.K. hasbeen estimated to be between around 2.6, with a maximum estimate placing it at4.6.

Using the above formula: Ontario R0 = 2.3 herd immunity is achieved at Phi ≈ 57 % Saskatchewan R0 = 3.12 herd immunity is achieved at Phi ≈ 68 % U.K. mean R0 = 2.6 herd immunity is achieved at Phi ≈ 62 % U.K. high R0 = 4.6 herd immunity is achieved at Phi ≈ 78 %

Case Fatality Rate and Recovery Rate

The case fatality rate is the proportion (percent) of infected people who diedfrom the infection. The recovery rate is proportion (percent) of infected peoplewho recovered from the infection (and might, or might not, become immune).

Both of these numbers are important parameters that quantify a disease. Theyare notoriously hard to calculate, as the denominator (the number of infectedpeople) is hard to determine, for instance, because of the unknown number ofasymptomatic cases. Of course, when the case fatality rate goes up, the recoveryrate declines, and the other way around.

The diagram below (from https://www.coronatracker.com/country/ca) showsthe two rates for Canada, estimated on 4 May 2021:

Thus, as of 4 May 2021, of all people in Canada who have been diagnosedwith Covid-19, 2% have died, 91.4% have recovered, and the remaining 6.6% areunder treatment (hospitals, ICUs, quarantines, self-isolation).

“Flattening the Curve”

What curve, and what does it mean to flatten it?The “curve” represents a projected number of people who will contract Covid-

19 over some period of time (in case of Covid-19, it is based on new cases reporteddaily). Starting with known data, this curve is obtained by using mathematicalmodels, and is only as good as the models and the data that are used. Nevertheless,it is better than nothing - we need to predict how the pandemic will spread, sothat we can plan accordingly.

The following curve for Canada has been generated as part of the Predic-tive Monitoring of COVID-19 project at Singapore University of Technology andDesign (https://ddi.sutd.edu.sg/) based on data up to 7 May 2020.


The purpose of any model is to identify a trend (pattern) and then apply it to makepredictions for future. (Unfortunately, the site is no longer publishing updates onthe model.)

The red curve is the “curve,” and the blue bars are reported new daily cases. Themodel (red curve) predicts that by July 2020 the daily number of new infectionswill be very close to zero. (Note: it did not happen as we know all too well.)

On https://bit.ly/3fJ4VRI we read: “The curve takes on different shapes,depending on the virus’s infection rate. It could be a steep curve, in which thevirus spreads exponentially (that is, case counts keep doubling at a consistent rate),and the total number of cases skyrockets to its peak within a few weeks. Infectioncurves with a steep rise also have a steep fall; after the virus infects pretty mucheveryone who can be infected, case numbers begin to drop exponentially, too.”


“The faster the infection curve rises, the quicker the local health care system getsoverloaded beyond its capacity to treat people. A flatter curve, on the other hand,assumes the same number of people ultimately get infected, but over a longerperiod of time. A slower infection rate means a less stressed health care system,fewer hospital visits on any given day and fewer sick people being turned away.”

Dynamics of the Spread of Covid-19We look at how data about the spread of Covid-19 has been represented in media.

Under Tracking the coronavirus: Confirmed COVID-19 cases in Canada byprovince and territory (https://newsinteractives.cbc.ca/coronavirustracker/) CBCNews publishes daily updates (graphs below are from 3 May 2020). This first graphrepresents the cumulative number of cases in Canada, Ontario and Quebec:

We can clearly see that the numbers increase. But to understand them better, welook at marginal changes, i.e., at the new daily cases:


The numbers of daily cases fluctuate, and for now do not show a tendency todecrease (for neither of the three populations depicted). In the case of Ontario(blue line), they hover around 500 newly diagnosed patients per day. This suggestsa linear growth, which is reflected in the shape of the blue curve in the top diagram.

On the above web page, we read: “Daily numbers give a sense of whether thenumber of new infections is growing and how quickly. When experts talk aboutusing physical distancing to ‘flatten the curve’ and keep the strain on the health-care system manageable, this is the curve they’re talking about. However, thereare a few days between each of the following: infection, the onset of symptoms,testing, and test results, meaning that the numbers typically reflect new infectionsa couple of weeks earlier.”

On the same site we find more diagrams, including the one that shows thecumulative number of people who dies from Covid-19. We read: “Because thenumber of reported cases depends on how much testing is done and how targetedor widespread it is, epidemiologists consider deaths to be a better gauge of theactual number of infections and the progress of the epidemic. While it takes acouple of weeks for an infected person to be reported as a positive case, deathoccurs, on average, more than three weeks after a person has been infected. Thatmeans while deaths represent information that may be more accurate and precise,the figures are also more out of date than reported cases.”

On its site Coronavirus in Canada: These charts show how our fight to ’flattenthe curve’ is going Maclean’s (https://bit.ly/2zp4iMM published the followingdiagram:

Note the logarithmic scale on the vertical axis. Since a line with a positive slopein a logarithmic plot represents exponential growth, we see here that the increasesin the number of cumulative cases are slower than exponential.


This conclusion is confirmed by the following graph with a linear scale on thevertical axis (cumulative cases):

Each bar is coloured according to the contribution from our provinces and territo-ries to the total number of infected people. Clearly, Quebec and Ontario dominate.We read: “Regional differences have always been a reality within Canada, and theCOVID-19 pandemic is proving to be a crisis defined by regionalism.”

Further analysis, as well as additional diagrams are available at Maclean’s sitehttps://bit.ly/2zp4iMM.

Below is a diagram from Johns Hopkins University of Medicine CoronavirusResource Center (7 May 2020, at https://coronavirus.jhu.edu/map.html), showingcumulative cases in Canada on a linear scale (left) and logarithmic scale (right):

The next two diagrams (from the same Johns Hopkins site) contrast daily cases(marginal changes) with the cumulative situation. In the first diagram, marginal


changes suggest exponential increase in the daily cases, which is reflected in theexponential growth of cumulative cases:

The second diagram is what we hope to achieve everywhere, soon: after reachinga peak, the number of daily cases decreases to very small values, and hopefully tozero. The cumulative number shows a logistic curve, approaching a plateau.

Moving averagesIf we look at any graph depicting the number of new cases, we see that the

numbers (represented by bars in the above two graphs, on the left) are ‘all overthe place,’ possibly obscuring the underlying pattern. Here is another example,from the same site:


This diagram represents the number of new daily cases in Spain, but there is aproblem. The gaps indicate that there is no data published for some days (often,these are weekends), and then the data for these days is aggregated and reportedas the daily number of new cases for the following day (that’s why the bars rightafter a gap are higher). As well, it could happen that the daily new cases for agiven day include some previously unreported cases.

So, how do we make sense out of this? One way to do this is to calculatemoving averages (also called rolling averages, running averages, or movingmeans). Thus, on top of reporting the number of new cases each day, we reportthe average number of cases for the previous seven days (does not have to be seven- could be three five, ten, etc.).

Here is an example. Imagine that the numbers of new daily cases are:

3, 4, 6, 0, 0, 21, 11, 14, 2, 3, 5, 8, 0, 24, 9, 3, 4, 5, 19, 3

(represented by the light blue bars in the diagram below). As we have no dataprior to day 1, the first seven-day average we can report is on day 8 - it is theaverage of the number of cases for days 1 to 7:

average of days 1 to 7 =3 + 4 + 6 + 0 + 0 + 21 + 11

7=

457

= 6.43

This is where the red curve (moving averages curve) starts in the diagram. Themoving average on day 9 is


7=

567

= 8

The moving average on day 10 is


7=

547

= 7.71

and so on.

Moving averages are shown in red, with individual values (points) connected withline segments (which helps to identify a pattern).

We see how moving averages ‘control’ large fluctuations in the numbers, andthe resulting pattern (red graph) suggests no major changes in the numbers.

The diagram below, published as part of the Globe and Mail article Coron-avirus tracker: How many COVID-19 cases are there in Canada and worldwide?


The latest maps and charts (3 May 2021, https://tgam.ca/3vDEuo0) shows thenumber of daily cases, and the seven-day moving averages.

Note how the moving averages clearly suggest three waves of the pandemic inCanada (as of writing this, in May 2021, we’re in the midst of the third wave).

Math aspects of testing for a diseaseThe usual way to test a population for the presence of, say, a virus is to administerone test for each member of the population. If the test turns positive, the personis declared to be infected, and must be treated. Otherwise, the person is declaredto be virus-free.

This approach has potential weaknesses: it is slow (one person’s sample istested at a time), and requires a large number of test kits (one for each memberof the population). During a pandemic, it is important to have rapid testing, sothat a large number of people can be tested in a short period of time (assumingthat there is a sufficient number of test kits).

How is it possible to test n people by using fewer than n test kits? Ini-tially in any situation where a disease spreads, the ratio of infected people isfairly small, could be 1 in 50, or 1 in 20. Even later, the ratio could remainpretty small; for instance, in early May 2021 in Alberta, at the height of the thirdwave, about 11% of tests for Covid-19 have been positive (Source: Maclean’s, 4May 2021, https://bit.ly/3ehuTxe). In its COVID-19 daily epidemiology updateat https://bit.ly/33fvdGx, on 4 May 2021, the Government of Canada states thatof the total of just over 32 million tests performed since the start of the Covid-19pandemic, 4.1% turned out positive.

Assume that 10% of people who are tested for some disease will test positive.This means that in a group of 10 people, we expect to see one positive test. Wedivide this group into two subgroups of 5 people each, and mix the samples of eachsubgroup together, and test the two mixed samples. One of these two samples willtest negative, and all we have to do is to individually test the 5 members of theother subgroup. Thus, we used 6 tests to test 10 people.

In the Conversation article The maths logic that could help test more peoplefor coronavirus (9 April 2020, https://bit.ly/3eGvI0q) we find further suggestions,all involving math reasoning about combinations.


We read: “A sample taken from each of our theoretical patients is distributedto half of the test tubes that we have, in different combinations. If we have tentest tubes, for example, we would distribute the samples from each patient into adifferent combination of five of them. Any tube that tests negative tells us thatall the patients that share that test tube must be negative. Meanwhile, test tubesthat test positive could contain samples from a number of positive patients - andan individual patient will test positive only if all their associated test tubes arepositive.”

The scenario we presented earlier - positive test means infection, negative testmeans no infection - is, unfortunately, not how things work. For many reasons(including the design of the test, or the moment when an individual is tested),someone could test negative, even though they are infected. Such cases are calledfalse negatives, and are the least desired outcome of any testing. The falsenegative rate is the proportion of the people who have an infection, but testnegative. Thus, if the false negative rate is 12%, in a group of 100 infected people,12 will test negative.

A false positive, or a false alarm is a situation where a person who is notinfected nevertheless tests positive.

(Source: M Health Lab https://bit.ly/2RsZejj)

The CNN News article Don’t get a false sense of security with Covid-19 testing.Here’s why you can test negative but still be infected and contagious (3 Novem-ber 2020, https://cnn.it/33bpYHL) is clear about it; their answer to “If I gotinfected yesterday, would a test today pick that up?” is “Probably not. A studyin the medical journal Annals of Internal Medicine examined false-negative testresults of people who actually had Covid-19. The study estimated that duringfour days of infection before symptoms typically started, the probability of gettingan incorrect/negative test result on Day 1 was 100%.”

Reading further, we find an explanation: “On the day people started showingsymptoms, the average false-negative rate had dropped to 38%, according to thestudy. Three days after symptoms started, the false-negative rate dropped to 20%.‘The virus just takes time to replicate in the body to detectable levels,’ said JustinLessler, a senior author of the study and associate professor of epidemiology atthe Johns Hopkins Bloomberg School of Public Health. ‘You can get infected byjust a few viral particles, but these will not be detectable until they have time toreplicate to adequate levels to be detected,’ he told CNN by email.”

In conclusion, a possible answer to “How many days should a person waitafter possible exposure to get tested?” is “‘There is no hard and fast rule, but theevidence suggests getting a test before the third day after exposure is not of muchuse,’ Lessler said.”

In the chapter on probability we will examine medical (and other) testing indetail. Here, for introduction, we look at the article Coronavirus: surprisingly bigproblems caused by small errors in testing published in The Conversation (5 May2020 https://bit.ly/31pMInY). We read: “Perhaps the least well understood ofthe concerns is the accuracy of the tests. The US Food and Drug Administration(FDA) has granted an emergency use authorisation to seven manufacturers to


bring antibody tests for COVID-19 to market. One of the first tests to gainauthorisation was developed by Cellex. If you have antibodies against COVID-19their test will tell you this correctly 93.8% of the time (this is the test’s sensitivity).If you dont, it will get this correct 95.6% of the time (this is the test’s specificity).Getting the correct result more than 90% of the time sounds pretty encouraging.”

Sounds good, however, “let’s consider what would happen if the test were givento 10,000 people as in the diagram below. Although estimates vary significantly,the WHO suggested recently that as few as 3% of the global population may havehad COVID-19 and recovered. This means that 9,700 of the 10,000 tested will nothave had the disease and only 300 will have. Of the 300 recovered patients, 93.8%– or 281 – will be correctly told they have antibodies against the disease. Of thevast majority (9,700) of people who haven’t had the disease, 4.4% – or 427 – willbe incorrectly told that they have had the disease and recovered.”

(Source: The Conversation, https://bit.ly/31pMInY)Thus, of the total of 281+427 = 708 people who tested positive, 281 have Covid-19, and we conclude that if one tests positive for Covid-19, the chance that theyactually have Covid-19 is 281/708, or 39.7%.

More Math

To book an appointment for Covid-19 vaccination, residents of Ontario are re-ferred to the page How to book a COVID-19 vaccine appointment at https://covid-19.ontario.ca/book-vaccine/. A user, trying to determine whether or not they arethey are eligible, sees this:

Note the use of the double negative in “If you are NOT the member of one of these


groups, you may NOT be currently eligible”. Rewrite this sentence in a more clearway, avoiding the double negative.

In the article Alberta to announce new public-health measures amid highestCOVID-19 rates in North America published in The Globe and Mail (3 May 2021,https://tgam.ca/3he0W3h), we find this diagram comparing the trends in Covid-19 infections in five Canadian provinces. .

Note that the vertical axis represents relative values - the number of daily infectionsper 100,000 people. This is why it is possible to compare the five provinces (sincetheir populations (numbers) are not the same). By the way, the graphs representmoving seven-day averages.

The risk is often represented as percent, and interpreted as probability orchance. For instance, if the risk of something happening is 13%, this means thatwe can expect that in 100 cases, we will observe that something happening in 13cases. [We talk lot more about risk and probability later in this book.]

In The Conversation article Blood clot risks: comparing the AstraZeneca vac-cine and the contraceptive pill (9 April 2021, https://bit.ly/3aUqezq) we read: “Inthe UK, blood clots have occurred in people taking the AstraZeneca vaccine at arate of roughly one in every 250,000, whereas blood clots caused by the pill areestimated to affect one in every 1,000 women each year.”

The article continues: “If blood clots occur when they shouldn’t, this can befatal. In some people who have had the first dose of the AstraZeneca vaccine,unwanted clotting is being reported in the brain, known as cerebral venous sinusthromboembolism (CVST). With this in mind, it’s worth remembering that whilethere is a small risk of clotting in some individuals who take the AstraZenecavaccine, this clotting risk is much less than with many other things, includingcontraceptive pills and significantly less than the risk of clotting after a COVID-19 infection.”


Note that the risk is represented numerically (“one in every 250,000” and “onein every 1,000”), and verbally (“a small risk” and “significantly less than the risk”).How risk is presented (presenting information in a certain way is called framing)is extremely important, especially in communicating with people who might notbe familiar or comfortable with numbers, or with verbal descriptions. And thereis more to it: although the phrases “90% chance of survival” (positive frame) and“10% chance of death” (negative frame) and mathematically equivalent, researchshows that the way people perceive them, and react to them, differ significantly.For instance, people presented with a positive frame are more likely to undergosurgery than those who were communicated a negative frame.

In The Conversation article Pfizer vaccine: what an ‘efficacy rate above 90%’really means (10 November 2020, https://bit.ly/2Sh9D21) we read: “There wasrightfully a lot of excitement when Pfizer and BioNTech announced interim resultsfrom their COVID vaccine trial. The vaccine, called BNT162b2, was reported tohave an ‘efficacy rate above 90%.’ This was soon translated in the press to be 90%‘effective’ at preventing COVID-19.”

Math teaches us that we need to know exactly what we’re talking about (howis it defined?); so are efficacy and effectiveness the same thing, or if not, how dothey differ?

The efficacy is defined as the performance of a drug or a treatment in ideal andcontrolled circumstances, which, for instance, occur while they are being developedor lab tested, or clinically trialled. On the other hand, the effectiveness of a drugor a treatment is the measure of their performance in real-world conditions. Boththe efficacy and effectiveness are expressed as percent, with ideal, full success being100%.

The article continues: “The Pfizer/BioNTech vaccine reports 90% efficacy,which means that - of the 94 confirmed cases of COVID-19 - their vaccine preventedCOVID-19 symptoms for 90% of those who received the vaccine compared withplacebo. This is very high and will probably change by the end of the study. Thepress release reported the results for 94 participants they need 164 to completethe trial, which shouldn’t take long. Safe vaccines with efficacy above 50% areexpected to be approved for COVID-19.”

Once a vaccine has been approved for general use, one can measure its effec-tiveness. “Monitoring of vaccines does not stop after they are approved for use.When the vaccine is deployed, data will continue to be collected to study howwell it works over the years for all vaccinated people. [...] We dont know whatthe overall effectiveness of the vaccine will be in preventing COVID-19 symptoms,severe disease or deaths, and it may be several years before studies report on theeffectiveness of BNT162b2 for different groups. However, it is unlikely that it willbe 90%. But then very few vaccines aside from measles and chickenpox are 90%effective. The flu vaccine is around 40%- 60% effective, but it still saves millionsof lives. And thats something to celebrate.”


[1] Quartz, 16 April 2020 (at https://bit.ly/31AgiY3) published the article AngelaMerkel gave one of the clearest explanations of how coronavirus transmission works(Angela Merkel is the first female Chancellor of Germany.) In the article, we read:“We dont often expect politicians to grasp the nitty gritty of the mathematicalmodelling of infectious diseases, let alone explain it in a concise and accuratemanner. German chancellor Angela Merkel – who has a doctorate in quantumchemistry – just did both those things, and it’s a clear demonstration of howimportant it is for policy makers to understand the nuts and bolts of science whendealing with a public health emergency like the coronavirus pandemic.”


“At the core of the ‘suppress and lift’ strategy is one key number: Rt, or thereal-time effective reproductive number. Rt tells us a viruss actual transmissionrate at a given time, t. That is, in a particular population at a particular time, howmany other people will catch the disease from a single infected person?” (Note:Rt is another notation for the effective reproduction rate Re.)

“This is the concept that Merkel explained so well yesterday. She noted thatthe Rt in Germany was currently around one, meaning that on average a personwith the virus infects one other person. One is the critical threshold: below one,the epidemic gradually fades out. Above one, it will grow, possibly exponentially.”

“Merkel then sketched out what it would mean if Germany’s Rt edged up to1.1. ‘If we get to the point where everybody infects 1.1 people, then by Octoberwe will reach the capacity level of our health system, with the assumed level ofintensive care beds,’ she said. And if it edges up further still, to 1.2, ‘everyoneis infecting 20% more.’ But 20%is arguably an abstract number, and hard forthe average citizen to grasp. Merkel seemed to recognize this, and explained thepercentage more concretely: ‘Out of five people, one infects two and the rest one.’At this rate, Germanys health care system will reach its limit in July. At an Rt

of 1.3, the health care system maxes out in June. ‘So you see what little leewaywe have,’ she said.”

Watch her at https://www.youtube.com/watch?v=22SQVZ4CeXA

[2] In The Guardian article Global report: Wuhan to test all as Germany pinpointsnew Covid-19 outbreaks (12 May 2020, https://bit.ly/3g9FdFP) we read:

R-number is, of course, a reference to the effective reproduction rate.

[3] In The Conversation piece Coronavirus: is the R number still useful? (19 March2020, at https://bit.ly/305HY6b) we read: “R is widely used as a key metric fordetermining UK public policy. The rationale is that R gives a quick assessment ofour control of an outbreak. If R is greater than one, the outbreak is growing. If itis less than one, the outbreak is under control and will eventually die out. Hence,the government’s focus is on keeping R below one as lockdown is slowly eased.”

Of course, there are many parameters that go into a mathematical model forthe spread of an infection, However, for the purpose of informing general popula-tion, a simplified – yet accurate – information needs to be presented (think abouthow many people are familiar with systems on nonlinear differential equations).

The article continues: “However, a group of scientists who call themselvesIndependent SAGE (a parallel group to the governments own SAGE – Scientific


Advisory Group for Emergencies) is now drawing the value of this metric intoquestion. In a recently published report, the group argues that R can be misleadingand the government shouldnt rely so heavily on this one metric for determiningpolicy.”

[4] BBC News (https://bbc.in/3xwYDha) reports about a study that investigatedthe new variant of COVID-19 which is lot more transmissible than the previousversion of the virus in the article Covid-19: New variant ’raises R number by up to0.7.’ The study concludes that “the new variant increases the Reproduction or Rnumber by between 0.4 and 0.7.” and continues “The UK’s latest R number hasbeen estimated at between 1.1 and 1.3. It needs to be below 1.0 for the numberof cases to start falling.”

Prof Axel Gandy of London’s Imperial College told BBC News that “There isa huge difference in how easily the variant virus spreads. This is the most seriouschange in the virus since the epidemic began.”

[5] Here is an article which suggests a different parameter as a major metric inquantifying the spread of Covid-19. In The Conversation article Is the K numberthe new R number? What you need to know (https://bit.ly/386qHMd) we read:“Just a few months ago, no one, aside from epidemiologists and their ilk, hadheard of the R number. Now, thanks to the coronavirus, everyone has heard of itand most people can tell you that it’s the reproduction number, an indicator ofwhether the number of infected people is increasing or decreasing.”

Next, we learn about dispersion. “Different pathogens will have different waysin which they spread and statisticians use K, the so-called dispersion parameter,to describe how variable the infection can be. For some diseases, the variation willnot be large, as shown below.”

“Simply put, a low K value suggests that a small number of infected peopleare responsible for large amounts of disease transmission. For the 1918 influenza,the number K is thought to be around 1, and perhaps 40% of infected peoplemight not pass on the virus to anybody else. But for diseases like Sars, Mers andCOVID-19 with K as low as 0.1, this proportion rises to 70%. In contrast, largeoutbreaks will be initiated by only few super-spreaders, as shown below.”


[6] Further support for studying the K number (from [5]) appears in The Conver-sation article A few superspreaders transmit the majority of coronavirus cases (5June 2020, https://bit.ly/2Vo7rET).

We read: “Early in the outbreak, researchers estimated that a person carryingSARS-CoV-2 would, on average, infect another two to three people. More recentstudies have argued, however, that this number may actually be higher. As early asJanuary, though, there were reports out of Wuhan, China, of a single patient whoinfected 14 health care workers. That qualifies him as a super spreader: someonewho is responsible for infecting an especially large number of other people.”

[7] Have we reached herd immunity? In The Conversation article 3 May 2021 Herdimmunity: can the UK get there? https://bit.ly/3toXAwB we read: “The mostcommon estimates for the herd-immunity threshold for the original coronavirusstrain the so-called wild type are at 70% or more immunity. For the new vari-ants, the threshold is probably higher. Also, we know that vaccines have limitedeffectiveness, which further increases the proportion of the population that needsto be treated.”

“It is important to note, too, that there are pockets of the population wherevaccination level is lower than the national average. These pockets are potentiallya breeding ground for super-spreading events. These sorts of outbreaks, althoughinitially small, can lead to the establishment and subsequent spread of highlyinfectious variants. The danger is well illustrated by the spread of the Brazilvariant in Canada or the UK variant in Europe.”

Chapter references

[1] University of Oxford, CEMB (The Centre for Evidence-Based Medicine) “Whenwill it be over?” An introduction to viral reproduction numbers, R0 and Re. Avail-able at https://www.cebm.net/covid-19/when-will-it-be-over-an-introduction-to-viral-reproduction-numbers-r0-and-re/

[2] A. R. Tuite, D. N. Fisman, A. L. Greer. Mathematical modelling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada.CMAJ 2020. doi: 10.1503/cmaj.200476; early-released April 8, 2020.

[3] https://epiforecasts.io/covid/posts/national/canada/ Centre for MathematicalModelling of Infectious Diseases at the London School of Hygiene and TropicalMedicine, University of London, UK.

Mathematics of Everyday Life - Math @ McMaster University

Documents

Transcript of Mathematics of Everyday Life - Math @ McMaster University