Mental Multiplication (for Social Scientists) Made Easy
Transcript of Mental Multiplication (for Social Scientists) Made Easy
1
MENTAL MULTIPLICATION (FOR SOCIAL SCIENTISTS) MADE EASY
BEING A VERY BASIC INTRODUCTION FOR SOCIAL SCIENTISTS AND THE LIKE TO
THOSE AMAZING METHODS OF SEEMINGLY MIRACULOUS RAPID MENTAL
RECKONING WHICH ARE COLLECTIVELY CALLED BY THE TERRIFYING NAME OF
“SPEED MULTIPLICATION.”
J. T. Manhire*
ABSTRACT
This outline provides a basic introduction to techniques used to perform
mental multiplication—including squares—quickly and accurately. It is
intended for social scientists and other non-S.T.E.M. professionals obligated
to deal with numbers on a daily basis (e.g., tax attorneys, accountants, etc.).
After reviewing this brief paper and practicing the techniques, readers should
be able to perform complex multiplication calculations in a matter of seconds.
CONTENTS
I. INTRODUCTION ........................................................... xxx
II. A BRIEF NOTE ON RAPID ADDITION ........................... xxx
III. MULTIPLICATION ........................................................ xxx
A. Multiplying by 11 ................................................. xxx
B. Complementary Multiplication (Base Methods) .. xxx
C. Star Method.......................................................... xxx
D. Multiplying Any Pair of Two-Digit Numbers ...... xxx
E. Squares................................................................. xxx
IV. CONCLUSION .............................................................. xxx
* [email protected]. The subtitle is a paraphrase of Silvanus Thompson’s
classic book. See SILVANUS P. THOMPSON, CALCULUS MADE EASY: BEING A VERY-
SIMPLEST INTRODUCTION TO THOSE BEAUTIFUL METHODS OF RECKONING WHICH ARE
GENERALLY CALLED BY THE TERRIFYING NAMES OF THE DIFFERENTIAL CALCULUS
AND THE INTEGRAL CALCULUS (1910). “S.T.E.M.” is a common acronym standing for
Science, Technology, Engineering, and Mathematics. “Math” and “Maths” are
synonymous throughout. Nothing herein represents the positions or policies of the
government of the United States or any of its agencies. All analyses, conclusions, and
mistakes belong solely to the author.
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I. INTRODUCTION
Quickly, what is the 𝑟2 value (coefficient of determination)
when the correlation coefficient 𝑟 is 0.45? You have 103 survey
respondents and each survey has 107 questions. How many total survey
answers do you need to analyze? Your client can take an additional tax
deduction of $72 per dependent and she is claiming 11 dependents. How
much is her total additional deduction? Oh yeah, one more thing: make
sure you get each answer in less than five seconds without a calculator
or scratch pad. In other words, do the calculations in your head.
To those of you now belly-laughing in disbelief at the thought
that you could actually perform such operations in less than five
seconds, and completely in your head, I assure you that after reading
this brief monograph and practicing its very simple techniques, these
three calculations will be child’s play for you. You will not only be more
productive in your profession, but you will amaze your friends and
terrify your enemies with your seemingly supernatural powers of mental
calculation.
I know most of you are terrified of math. You’re not alone. We
live in a largely “mathphobic” world and most of us do whatever we
can to avoid dealing with calculations…especially mental calculations.1
If fact, I’ll bet that if you’re reading this you probably thought twice
about even downloading this paper because it had “multiplication” in
the title, and the only reason you got this far is because I’ve been able
to keep you mildly curious.
If you consider yourself bad at math, it’s not your fault. Like
most of the western world, you were taught from a very young age to
do calculations in the most difficult and arduous way possible. If I
believed in conspiracies, I’d say there must be a conspiracy of ignorance
in institutionalized education. How else can these terrible techniques be
1 The technical term for the fear of math is “arithmophobia,” which generally
means the fear of numbers, but also includes the fear of mathematical calculations.
See, e.g., Paula J. Williams, Kris Anne Tobin, Eric Franklin, and Robert J. Rhee,
Tackling “Arithmophobia”: Teaching How to Read, Understand, and Analyze
Financial Statements, 14 TENN. J. BUS. L. 341, 341 (2013).
2015] Mental Multiplication Made Easy 3
taught generation after generation? Of course, there is no conspiracy.
The truth is, most math teachers only know the old and difficult methods
because that’s how they were taught and it’s what the textbook tells
them to teach, so they teach those methods and the cycle perpetuates
itself.
Being afraid of math is not shameful, but it is curable. That’s
what I’m hoping to accomplish with this paper. I want to cure as many
mathphobes as possible by showing you just how easy math can be. You
do not need to be born with special abilities or be some sort of “rain
man” to excel at mental multiplication. These are simple techniques that
anyone can learn very quickly.
Let me prove to you how easy mathematics can be, beginning
with the 𝑟2 problem. When squaring any number ending in the digit 5,
the last two digits of the answer will always be 25. To get the first two
digits, simply add 1 to 4 (the first digit of the number to be squared) to
get 5, and then multiply the two numbers (4 × 5 = 20). Put the first and
last two digits together and you get your initial answer: 2025. Since
you’re squaring a number with two digits after the decimal place, move
the decimal of your initial answer four places to the left and you get
your final answer: 𝒓𝟐 = 0.2025.
If you have 103 respondents to a survey with 107 questions, you
want to find the answer to 103 × 107. To get the first part of your
answer, add the last digit of the second number (7) to the first number
(103). This gives you 103 + 7 = 110. Since 103 and 107 are close to
100 (which has two zeroes), you need two more digits to get your
answer (1 1 0 _ _). Take the difference of the first number and 100
(103 − 100 = 3) and the same for the second number (107 − 100 =
7). Multiply 7 and 3 to get 21 and those are the last two digits of your
answer (1 1 0 2 1). So your answer is 11,021 questions to analyze.
The client with 11 dependents and a $72 deduction per
dependent is even easier. For 72 × 11, simply split the numbers 7 and
2 and make room for a third number in between (7 _ 2). Add 7 and 2 to
get 9 and that becomes your third digit (7 9 2). So your answer is an
additional deduction of $792.
With just a little practice, you’ll easily be able to perform these
and much more (seemingly) difficult math operations in your head in a
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matter of seconds. This outline provides a very basic introduction to
techniques used to perform mental mathematics quickly and accurately.
It is intended for social scientists and other professionals obligated to
deal daily with numbers (e.g., tax attorneys, accountants, etc.).
Mental multiplication is based on just two foundational
concepts. The first is a base. This paper refers to standard bases, such
as base 10 and base 100, throughout (like the survey problem). Another
general concept is splitting. This means that you physically split the
answer into at least two separate answers and then combine them to get
the final answer (like the tax deduction and 𝑟2 problems). Familiarity
with these foundational concepts will help you succeed with these
techniques, but only practice can ensure success. Like anything else
worth doing, you must commit to practicing these new concepts
daily…even if only for five to ten minutes.
This paper only addresses mental multiplication, with a brief
mention of rapid addition. Other common operations, such as
subtraction, division, fractions, cubes, and roots, are not included. These
will be covered in future work.
Of course, this is only a brief introduction, and there are many
more techniques that lie below the surface of this proverbial iceberg.2
The concepts and techniques are fundamental, and any practitioner of
this craft will want to pursue this discipline deeper and broader than
what is exposited here. Armed with this short paper alone, anyone can
become a mental mathematics tyrannosaurus. I ask only that you use
your newly-found powers for good instead of evil.
2 Some of the first popular books on “speed math” include ANN CUTLER &
RUDOLPH MCSHANE, THE TRACHTENBERG SPEED SYSTEM OF BASIC MATHEMATICS
(1960); HENRY STICKER, HOW TO CALCULATE QUICKLY: FULL COURSE IN SPEED
ARITHMETIC (1945); EDWARD STODDARD, SPEED MATHEMATICS SIMPLIFIED (1965);
GERARD W. KELLY, SHORT-CUT MATH (1969). Other popular books that explore
mental mathematics in greater detail include EDWARD H. JULIUS, RAPID MATH TRICKS
& TIPS (1992); EDWARD H. JULIUS, ARITHMETRICKS: 50 EASY WAYS TO ADD,
SUBTRACT, MULTIPLY, AND DIVIDE WITHOUT A CALCULATOR (1995); BILL HANDLEY,
SPEED MATHEMATICS: SECRET SKILLS FOR QUICK CALCULATION (2003); SCOTT
FLANSBURG, MATH MAGIC: HOW TO MASTER EVERYDAY MATH PROBLEMS; DHAVAL
BATHIA, VEDIC MATHEMATICS MADE EASY (2005); ARTHUR BENJAMIN & MICHAEL
SHERMER, SECRETS OF MENTAL MATH: THE MATHEMAGICIAN’S GUIDE TO
LIGHTNING CALCULATION AND AMAZING MATH TRICKS (2006);
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II. A BRIEF NOTE ON RAPID ADDITION
Although I am purposely restricting the scope of this paper to
multiplication, many of the multiplication techniques require some
rapid addition. Therefore, I will spend just a few paragraphs discussing
how to do “speed addition.”
Most of us know how to add from right to left. Start with the
ones column, then the tens column, hundreds column, etc. Most mental
math techniques involve adding in the opposite direction; from left to
right. The trick is to remember which column you are in at any one time
(hundreds, tens, ones, etc.).
For example, take a look at the equation:
35
67
+ 56
Most of us can do this in our head if we really try, but we usually go
about it by adding the ones column (5 + 7 + 6 = 18; bring down the 8;
carry the one), and then we add the tens column (3 + 6 + 5 = 14 plus
the carried 1 = 15). Bring down the 15 and we have the answer: 158.
This method works, but it very slow and difficult to do mentally.
Instead, try adding from left to right starting with the tens
column. Recall that it’s called the tens column because all of the digits
in the column have the value of the digit times 10. So the digit 3 in 35
has the value 30 (3 × 10), the digit 6 in 67 has the value 60 (6 × 10),
and so on. So when you see the 3 in the tens column, think “30;” a 6,
think “60;” a 5, think “50.” You get the picture.
So for the equation 35 + 67 + 56, first think “30 + 60 + 50.”
Most of us can do this pretty quickly and get to 140. Then just keep
counting in the ones column: 140 + 5 = 145; 145 + 7 = 152; and
152 + 6 = 𝟏𝟓𝟖. We get to the same answer, just faster.
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Try another one:
26
11
+ 79
Go to the tens column and start adding up the tens; “20 + 10 + 70 =
100.” Then continue with the ones column; “100 + 6 = 106; 106 +
1 = 107; 107 + 9 = 116. And you’re done. You can even start looking
for groupings that add to 5 or 10 to make it easier. For example, once
you add up the tens column and get 100, you can quickly scan the ones
column and see that there’s a grouping of 9 and 1 to quickly get to 110.
After that, you just have to add the 6 to get to 116. See, this stuff is
actually pretty easy.
Now let’s try adding three digit numbers. Take
326
678
245
+ 567
326 + 678 + 245 + 567. Use the same technique only this time start
with the hundreds column, which means every digit is multiplied by
100. Start from the left and go to the right: 300 + 600 + 200 + 500 =
1600. Then keep adding with the tens columns: 1600 + 20 + 70 +
40 + 60 = 1790. Then go to the ones column: 1790 + 6 + 8 + 5 +
7 = 1816. And there’s your answer.
Although I’m using equations for this paper, I do not think with
plus signs, and maybe you don’t either. What works for me is simply
counting off the sums as I go. So instead of 300 + 600 + 200 + 500 =
1600, my brain goes “300, 900, 1100, 1600.” Same with the tens and
ones. Everyone is a little different, so practice, practice, practice and
find what works best for you.
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III. MENTAL MULTIPLICATION
Rapid addition can be mildly amusing, but the truly amazing
mental math starts with multiplication. It’s amazing because most of us
think multiplication is so difficult, so when someone calculates the
product of 98 and 96 in less than two seconds, our jaws drop because
inside we are thinking, “there’s no way I can do that.” But you can. In
fact, some of the multiplication techniques are easier than the addition
technique.
A. Multiplying by 11
I want to start with something simple so you can quickly amaze
yourself. Why? Because success breeds success, and not only hitting the
ball but hitting it hard the very first time you step up to the plate will
give you the confidence you need to continue reading, which means you
will continue learning. So let’s start with a quick and easy way to
multiply any number (even a big one) by 11. I will start with some
special techniques and end with a general technique that applies to any
number you want to multiply by 11 quickly without ever swiping for
that calculator app.
1. Single Digits: Clone Them
The special technique for single digits is one you probably
already know. All you do is clone the single digit by which you are
multiplying 11. For example, if you multiply 11 by 1, you simply take
the number 1 and create a clone (another 1) and put them next to each
other. So the problem would read 11 × 1 = 11. Notice how the answer
is simply a clone of the original number (1 in this case) placed next to
the original.
The same is true for 11 × 2. Since 2 is the original number,
clone it (make another 2), and place it next to the original. So the
equation reads 11 × 2 = 22. You can repeat this pattern for all single
digit numbers, including zero.
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Here are the rest:
11 × 3 = 33
11 × 4 = 44
11 × 5 = 55
11 × 6 = 66
11 × 7 = 77
11 × 8 = 88
11 × 9 = 99
and
11 × 0 = 00.
2. Double Digits: Split and Add
I realize most of you already knew the cloning technique for
single digits, but I’ll bet this next technique will be new to most of you.
For any two-digit number (10 through 99), simply split the first and
second numbers apart, and then put the sum of the original two digits in
the middle. For example, if you are multiplying 27 by 11, take the
number 27 and split the number into two “bookends” so you have the 2
on the left and the 7 on the right with room for one more number in
between them, like this:
27 → 2 ___ 7.
After you split, add the original two digits together and put their sum in
the middle of the split numbers. Since the sum of 2 and 7 is 9, put the 9
in the middle of 2 and 7 like this:
27 → 2 __ 7
2 + 7 = 9 → 2 𝟗 7
and you have your answer: 11 × 27 = 297. Go ahead…double check
it on your calculator app.
Let’s do another one. Try 11 × 78. Step 1, split 78 into two
bookends so you have 7 on the left and 8 on the right, with room for a
third number in the middle, like this:
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78 → 7 __ 8.
Step 2, add 7 and 8 and place the sum in between the bookends, like
this:
78 → 7 __ 8
7 + 8 = 15 → 7 𝟏𝟓 8.
Notice that the sum gives you the two-digit number 15, but when
you split 7 and 8 you only have room for one digit. When this happens
you need to carry the number in the tens place (1 is the number in the
tens place for the number 15) and add it to your left bookend number (7
in this case) and keep only the ones digit of the sum (5 in this case). So
with the result 7 15 8, you carry the 1 and add it to the 7 and leave the
5 in the middle. Since 7 + 1 = 8, your new left bookend number is 8,
so 7 15 8 becomes 8 5 8, and that’s your answer. 11 × 78 = 858.
What does your calculator app say? It’s right, isn’t it?
3. Three-Digits and More: Add to the Neighbor
When you multiply digits greater than 99 by 11, you need to use
a general technique. This general technique is sometimes called adding
to the neighbor. It simply means that you add a digit to the digit
immediately to the right. Let’s look at some examples.
Say you want to mentally multiply the number 12,345 by 11.
First, put lines to the left and right of the number and add zeros on either
side (when first trying this, use a paper and pencil, but pretty quickly
you’ll be able to set it up in your head). Your new number should look
like this:
0 | 12,345 | 0.
Next, add to the neighbor. Start with the last digit of the original
number (5 in this case) and add it to the right-most zero.
5 + 0 = 𝟓
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This will be the right-most digit of your answer. Then add the neighbor
to the left of the last digit of the original number and the last digit of the
original number. So in this case the neighbors would be 4 and 5.
4 + 5 = 𝟗
Keep adding neighbors, moving all the while to the left, until you add
the two left-most digits (0 and 1 in this case), like this:
3 + 4 = 𝟕
2 + 3 = 𝟓
1 + 2 = 𝟑
0 + 1 = 𝟏
If you line up all of your sums from right to left, you get your
answer. So line the boldfaced digits up from right to left and you get: 1
3 5 7 9 5. And that’s your answer. 11 × 12,345 = 𝟏𝟑𝟓, 𝟕𝟗𝟓.
Let’s do another one, this time where we have to carry a digit.
Multiply 5,179 by 11. Again, set up the number with zero bookends like
this:
0 | 51790 | 0
Next, add the neighbors from the right starting with 9 + 0 = 9, which
gives you the last digit of your answer. When you add the next
neighbors, 7 and 9, notice that 7 + 9 = 16. Here, the 6 becomes the
second digit of your answer and you need to carry the 1 over to the next
round of adding. This means when you add the next neighbors, 1 and 7,
you need to also add the 1 you carried over from the 16. So the third
digit of your answer will be 1 + 7 + (1) = 9. Next, add the 5 and 1 to
get 5 + 1 = 6, and finally add the 0 and 5 to get the last digit of your
answer, 0 + 5 = 5.
Here’s a recap of all the “add to the neighbors” from the
problem. The bold digits are the ones that become your answer. Notice
that you must carry the 1 in the second operation:
2015] Mental Multiplication Made Easy 11
9 + 0 = 𝟗
7 + 9 = 1 𝟔
1 + 7 + (1) = 𝟗
5 + 1 = 𝟔
0 + 5 = 𝟓
So your answer is 5,179 × 11 = 𝟓𝟔, 𝟗𝟔𝟗.
4. An Alternate Method
There is an alternate method for multiplying any number by 11.
Its general form might seem scary because it looks a lot like algebra,
but it’s actually quite simple. The formula is 11 × 𝑛 = 𝑛 + 10𝑛. Let’s
walk through it together.
All you do is take the number by which you’re multiplying 11
and call that 𝑛. So in the last two-digit example 11 × 78, 𝑛 = 78. Using
our alternate formula and plugging 78 in for 𝑛 we get 11 × 78 = 78 +(10 × 78). Since 10 × 78 = 780, we find that 11 × 78 = 78 + 780.
Adding from left to right we get 700, 850, 858 (see the addition section
for a review if necessary). And that’s our answer: 858.
This method works for any value of 𝑛, so it’s pretty versatile and
some might find it easier than the “add to your neighbor” method.
Again, practice, practice, practice and play around with the different
techniques until you find what works for you.
B. Complementary Multiplication (Base Methods)
As mentioned in the introduction, base methods—also known
generally as complementary multiplication—are some of the most
popular in mental mathematics, and for good reason. Not only are they
efficient methods for mental calculation, but they also cause the greatest
amazement in those observing someone using the technique. We are
trained from a young age that multiplying small numbers is easier than
multiplying large numbers. The base methods introduced here will show
you that this is not necessarily true. Large numbers can be multiplied
just as easily as small numbers.
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Although the next section covers the vertical and crosswise
method of multiplying any two numbers, the techniques themselves will
be introduced with the base methods. A base is just a “home number;”
one you can remember easily and one that is easy to use in calculations.
The most popular bases with this method are 10 and 100, but other bases
can and should be used, such as 20, 50, 200, and even 1,000. There are
really only two rules for the base methods that you need to remember:
(1) add across diagonally, and (2) multiply vertically (up and down).
Let’s see this simple rule in action.
Let’s start with a pair of numbers that are each less than 10. This
is the best introduction since you probably know your multiplication
tables from 1 to 10 from rote memory. Notice I didn’t say you know
how to calculate these numbers. You only know them through
memorization. Let’s look at how to actually calculate them, starting
with 7 × 6. Of course you know the answer is 42 from your
multiplication tables. That’s why this is a good introduction to the
method…you already know the answer.
First, figure out the difference between each number and the
base 10. Since 10 − 3 = 7, we know that 7 is −3 from the base 10.
Likewise, since 10 − 4 = 6, we know that 6 is −4 from the base 10.
With this information we can set up the problem with the differences
off to the right side (do this on paper for now, but soon you will be able
to set it up in your head).
7 −3
× 6 −4
First, add crosswise, so either add 7 and −4 or 6 and −3. As
you can see it doesn’t matter which one you choose because you always
get the same answer. In this case the answer is 3. Place the 3 in the left
bottom box under the original equation like this:
7 −3
× 6 −4
3
2015] Mental Multiplication Made Easy 13
Now multiply vertically (up and down) the numbers in the right-
most column. Multiplying −3 × −4 = 12. The 12 goes in the right
bottom box, but only one digit will fit there. So put the 2 in the right
bottom box and carry the 1 over and add it to the 3. This will give you
4 in the left bottom box and 2 in the right bottom box, like this:
7 −3
× 6 −4
3 𝟒 𝟐
Put the numbers together and you have your answer. 7 × 6 = 𝟒𝟐. And
you’re done.
Let’s do one more and then we’ll step it up. Let’s do 9 × 8,
which you already know is 72. Set up the equation like before.
9 −1
× 8 −2
The −1 is next to the 9 since that is the distance from the base 10. It’s
the same for the −2 and the 8.
Next, add crosswise. Remember you can add 9 and −2 or 8 and
−1. They both give you the same result, which is 7. Place the 7 in the
bottom left box.
9 −1
× 8 −2
7
Now multiply vertically on the right. Since −1 × −2 = 2, put the 2 in
the bottom right box.
9 −1
× 8 −2
𝟕 𝟐
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Combine the numbers and you have your answer: 9 × 8 = 72.
Let’s do the same thing but instead of 10, let’s use 100 as our
base. The equation is 98 × 97. The major difference between using base
10 and base 100 is that the bottom right box only has room for one digit
with base 10, but the box must have two digits with base 100.
The first step is to figure out how far away each number is from
your base. 98 is −2 from the base 100, and 97 is −3 away. The next
step is to set up your table (on paper or in your mind) like before, only
using two digit numbers instead of one.
98 −02
× 97 −03
The next step is to add crosswise (you can pick), which gives you 95.
Put the 95 in the bottom left box.
98 −02
× 97 −03
𝟗𝟓
Lastly, multiply vertically. Since −02 × −03 = 06, put the 06 in the
bottom right box.
98 −02
× 97 −03
𝟗𝟓 𝟎𝟔
And you’re done. 98 × 97 = 𝟗, 𝟓𝟎𝟔.
You probably notice that all of our examples so far dealt with
numbers below the base. Let’s look at a few equations with numbers
above the base. We’ll start again with base 10 and solve the equation
12 × 13. Like before, we need to determine how far each number if
from the base, but since the numbers are above the base the differences
2015] Mental Multiplication Made Easy 15
will be positive instead of negative. 12 is 2 away from base 10 and 13
is 3 away.
Next, set up your table as before, but this time remember that
the differences on the right are positive and not negative.
12 +2
× 13 +3
Not add crosswise. Bothe 12 + 3 and 13 + 2 give you 15, so put the 15
in the bottom left box.
12 +2
× 13 +3
15
Multiply the differences and you’ll get 2 × 3 = 6. Place the 6 in the
bottom right box. Since we’re only in base 10, the bottom right box can
only have one digit.
12 +2
× 13 +3
𝟏𝟓 𝟔
And we’re done. 12 × 13 = 𝟏𝟓𝟔.
Now let’s multiply two numbers that are above base 100. We’ll
try103 × 107. The difference between 103 and 100 is 3 and the
difference between 107 and 100 is 7. Now you can set up your table,
but remember the bottom right box must have two digits since the base
is 100.
103 +03
× 107 +07
Add crosswise to get 110, which you place in the bottom left box.
16 J. T. Manhire [Vol. XX:XXX
103 +03
× 107 +07
110
Multiply the differences vertically to get 03 × 07 = 21. Place the 21 in
the lower right box and we’re done.
103 +03
× 107 +07
𝟏𝟏𝟎 𝟐𝟏
103 × 107 = 𝟏𝟏, 𝟎𝟐𝟏.
If you use base 100 and the number in the bottom right box has
more than three digits, put the two right-most digits in the box and carry
the left-most digit over to be added to the number in the bottom left box.
What is one number is above and the other below the base? Then
what? Well, it’s pretty much the same method. It is probably best to
explain with an example. Let’s go to base 10 and solve 13 × 9. 13 is 3
away from 10 and 9 is −1 away from 10. Like before, set up your table
and add crosswise. 13 − 1 = 12 and 9 + 3 = 12, so 12 goes in the
bottom left box. +3 × −1 = −3, so −3 goes in the bottom right box.
13 +3
× 9 −1
12 −3
Here’s where it gets different from a situation when both numbers are
above or below the base. You clearly can’t just combine 12 and −3, so
what we do is multiply the bottom left number by the base (12 × 10 =
120) and add the negative 3 (120 − 3 = 117). And now we’re done.
13 × 9 = 𝟏𝟏𝟕.
You do the same thing when multiplying one number above and
one number below base 100, except in the last step you would multiply
the bottom left number by 100 instead of 10.
2015] Mental Multiplication Made Easy 17
Now what if your numbers are not close to base 10 or 100. Then
what? Again, it’s pretty much the same idea, except you need to do one
more step if you base is not a 10 or 100 (or 1,000; 10,000; etc.). Again,
let’s take an example and work through it.
Let’s solve 23 × 21. Both numbers are very close to 20, so 20
we’ll use 20 as our base. Like before, find the difference between the
numbers and the base. These numbers are both above the base so the
differences will be positive. 23 is 3 more than base 20, and 21 is 1 more.
Now set up the table, add crosswise, and multiply vertically.
23 +3
× 21 +1
24 3
Here’s the difference since we are in base 20. You have to multiply the
bottom left number by the tens digit of the base and then combine it
with the bottom right number. So in this case the bottom left number is
24 and the tens digit of the base is 2. 24 × 2 = 48. This becomes your
new bottom left number.
23 +3
× 21 +1
24 3
𝟒𝟖 𝟑
Combine the new bottom left number with the bottom right number and
you have your answer. 23 × 21 = 𝟒𝟖𝟑.
You might wonder why there is an extra step for base 20 than
for base 10. Actually, the steps are exactly the same. Before combining
the bottom boxes, you need to always multiply the left bottom number
by the tens digit of the base. It’s just that the tens digit of the base in
base 10 is 1, so multiplying the bottom left number by 1 just gives you
the same number you started with. For simplicity, we just leave that step
off for base 10, but remember that technically it’s still there. Of course,
if the base was 30 you’d multiply the bottom left number by 3, if the
base is 40, multiply by 4, etc.
18 J. T. Manhire [Vol. XX:XXX
Try doing some base method multiplication before moving on
to the next section. When starting out, work on a sheet of paper or a
whiteboard, but after a few exercises, try to do the steps in your head.
Remember the steps:
Find your closest base
Find how far each number is from the base; positive if above the
base, negative if below
Make your table (on paper or in your mind)
Add crosswise to get your bottom left number
Multiply vertically to get your lower right number
Multiply the lower left number by the left-most digit of the base
to get your new bottom left number (skip this step if the left-
most digit is 1)
Combine the bottom left and right numbers, and you’re done!
Once you get used to it, you’ll be able to very quickly multiply
seemingly large numbers at record speed.
C. Star Method
The star method appears more difficult at first than it actually is.
Once you understand the star pattern and practice a little, you’ll be
multiplying four- and five-digit numbers in a snap.
Let’s use an example to explain the method. Solve 123 × 456.
Set the problem up as you normally would, like this:
123
× 640
Step 1: Multiply the 3 and the 0 to get 0.
123
× 640
∗∗ 0
2015] Mental Multiplication Made Easy 19
Step 2: Cross multiply the 0 and the 2, cross multiply the 4 and the 3,
and add the two products together to get (0 × 2) + (4 × 3) = 12. Put
the 2 as the next digit of your answer and carry the 1.
123
× 640
∗ 210
Step 3: Cross multiply the 6 and the 3, cross multiply the 1 and the 0,
and vertically multiply the 2 and the 4 (if you draw it out it makes a star
pattern). Add the three products together plus the carried 1 to get
(6 × 3) + (1 × 0) + (2 × 4) + 1 = 27. Put the 7 as the next digit of
your answer and carry the 2.
123
× 640
7220
Step 4: Cross multiply the 6 and the 2, cross multiply the 4 and the 1,
and add the two products together plus the carried 2 to get (6 × 2) +(4 × 1) + 2 = 18. Put the 8 as the next digit of your answer and carry
the 1.
123
× 640
81720
Last Step: Vertically multiply the 6 and the 1, and add the carried 1 to
get (6 × 1) + 1 = 7. Put the 7 as the final digit of your answer.
123
× 640
78720
And you’re done. The answer is 123 × 456 = 𝟕𝟖, 𝟕𝟐𝟎. I know this
looks complicated, but once you understand the star pattern and practice
20 J. T. Manhire [Vol. XX:XXX
it (this one really does take practice), you start to careen through
complicated multiplication problems.
Here’s the most common mistake I’ve seen with those starting
out: you forget to carry. You need to practice your mental carrying for
this method to produce accurate results. I recommend working through
a number of problems on paper until you get the method and carrying
down.
D. Multiplying Any Pair of Two-Digit Numbers
This approach is extremely simple, but many people don’t
bother learning it because it requires memorizing what looks like an
ugly algorithm. However, if you take the time to memorize it, the
formula provides a very quick mental multiplication technique. Here’s
the formula defining 𝑎 as the tens digit of the first number, 𝑏 as the ones
digit of the first number, 𝑐 as the tens digit of the second number, and 𝑑
as the ones digit of the second number:
⟨𝑎𝑏⟩ × ⟨𝑐𝑑⟩ = 100𝑎𝑐 + 10(𝑏𝑐 + 𝑎𝑑) + 𝑏𝑑.
For example:
37 × 62 = 100(3 × 6) + 10[(7 × 6) + (3 × 2)] + (7 × 2).
This reduces to:
1800 + 480 + 14 = 𝟐, 𝟐𝟗𝟒.
Again, it takes a little practice to memorize the formula and get
comfortable using it, but once you do this method can be a less
cumbersome alternative to the vertical and cross method for two-digit
multiplication.3
3 There are other techniques for multiplying a pair of two-digit numbers that
are based on more formal derivations of quadratics equations. For example, recall from
high school algebra that (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2. If 𝑎 is the tens digit and 𝑏 the
ones digit, one can imagine the equation 64 × 56 thereby defining 𝑎 as 60 and 𝑏 as 4.
Using the quadratic factors yields a solution to the multiplication problem since 64 ×56 = (60 + 4)(60 − 4) = (602 − 42) = 3600 − 16 = 3,584. Again, these formal
expressions can be helpful if you wish to spend the time learning and practicing them.
2015] Mental Multiplication Made Easy 21
E. Squares
Any number squared is simply an operation of multiplying a
number by itself. For this reason, it’s important to include techniques
for mentally squaring any number in a discussion on rapid mental
multiplication.
It is probably obvious that since squaring a number is a
multiplication operation at its heart, we can use the techniques we’ve
already learned to solve any equation involving a square. For example,
if we want to solve the equation 112, we can simply imagine the
equation as 11 × 11 and apply any of the methods we’re already
familiar with. Since it involves multiplying by 11, we can split the
number (1 __ 1), add the integers (1 + 1 = 2) and place the sum in the
middle of the split numbers (1 2 1). We’d arrive at the correct answer
of 112 = 𝟏𝟐𝟏.
Since 11 is close to base 10, we can also use our base method by
finding the distance of 11 from base 10 (+1), adding 11 to +1 to get 12,
placing the 12 in the bottom left box, and multiplying +1 × +1 = 1 to
find the number that goes in the bottom right box. Of course, we get the
same answer, 121.
And lastly we can use the star method to multiply the right digits
vertically (1 × 1 = 𝟏), multiply crosswise and add the products ((1 ×
1) + (1 × 1) = 𝟐), and multiply the left digits vertically (1 × 1 = 𝟏).
Putting them all together gives us the same answer: 121.
But squares are a special form of multiplication, and as such
there are a few special tricks that allow us to solve for them faster than
with our now-familiar mental multiplication tricks. We’ll first discuss
solving for any square ending in the digit “5,” then we’ll discuss the
method for solving any square in just seconds.
1. Squares Ending in “5”
Recall the 𝑟2 problem from the introduction. We used the
percentage 0.45 for 𝑟, but to make this example simpler, let’s now
Since such advanced methods are beyond the scope of this brief introduction to mental
multiplication, this paper does not go into these formulae further.
22 J. T. Manhire [Vol. XX:XXX
simply solve for the square of 45. Here’s something you need to
remember: All squares with the last digit “5” will have an answer
ending with the last two digits “25.” This is true for all squares ending
in “5.” Therefore, we know that 452 = ___ 2 5. To get the first part of
the answer, all we do is add 1 to the digit in the ten’s place and multiply
the answer times the number in the ten’s place. So in this example since
4 is in the ten’s place, we add 1 to it to get 4 + 1 = 5. We then take the
5 and multiply it by the 4 to get 5 × 4 = 20. We then use the 20 as the
first part of our answer (2 0 2 5), and we’re done. The answer is 452 =
𝟐, 𝟎𝟐𝟓. If you’re not sure, check the calculator app.
Let’s try another one: 1252. We know the answer will have the
last two digits 25, so you can set up the answer in your mind like this:
____ 2 5. The remaining number in 125 is 12. Adding 1 to 12 gives us
13. Multiplying them together gives us 13 × 12 = 156 (if necessary,
use the base 10 method discussed earlier to find 13 × 12 = 156). Make
the 156 the first part of the answer (1 5 6 2 5), and you’re done. 1252 =
𝟏𝟓, 𝟔𝟐𝟓. Yes, it really is that simple.
If you want to get faster, try going from left to right. For 1252,
add 1 to the number in front of the 5 and multiply (13 × 12 = 156),
then just tack on the 25 to the end of the answer to get 1252 = 𝟏𝟓, 𝟔𝟐𝟓.
It’s up to you. Again, I encourage you to practice different approaches
and use the method with which you are most comfortable.
2. Base 50 Squares
The mental method for finding squares ending in “5” is almost
too easy, but be very careful…the method does not work if the number
being squared does not end in a 5. For other numbers, we need a
different method. In this section we will discuss squaring numbers that
are close to the base 50. We’ll first discuss squaring numbers below
base 50, and then number above (by the way, 502 = 2,500…you might
want to just remember that one for speed).
If you square a number below base 50, remember to always
work with the number “25.” Consider it your sub-base for these
2015] Mental Multiplication Made Easy 23
problems.4 Working through an example might be helpful here, so let’s
start by solving 472.
Remember how we found the distance of the number from the
base when we discussed the base method? We’ll use the same idea here.
The first step is to figure out how far away from base 50 is 47? Since
50 − 3 = 47, we know that 47 is −3 from 50. Step 2 is to add this
distance (−3) to the sub-base 25. So we get 25 − 3 = 22. This becomes
the first part of our answer (2 2 __ __).
To get the second part of the answer we square the difference of
the number from base 50, which is −3 in this case. Now the second part
of the answer needs to have two digits, so it is helpful to think of −3 as
−03. Since −03 × −03 = 09, just use these two digits as the second
part of your answer to get 2 2 0 9 and you’re done. The answer is 472 =
𝟐, 𝟐𝟎𝟗.
Now let’s try one above base 50. We still need the sub-base 25,
so keep that in mind. The only difference is that the distance from 50
will now be positive instead of negative. All other steps are the same.
Let’s solve 562.
The first step is to determine the distance of 56 from base 50.
Since 50 + 6 = 56, the difference is +06 (again, I suggest using two
digits to remember the answer will require two digits at the end). The
next step is to add the difference to sub-base 25, and you get 25 + 06 =
31. This becomes the first part of your answer (3 1 __ __). Next, square
the difference to get 06 × 06 = 36, make 36 the last two digits of your
answer, and you’re done. The answer is 562 = 𝟑, 𝟏𝟑𝟔. If you got 3,112
as your answer, remember that your are multiplying 6 × 6, not adding
6 + 6.
Of course, the farther you get from 50, the larger the second part
of your answer will be. Always remember that with the base 50 method
for squares, the last part of the answer can only hold two-digit…no less
and no more. If you get a second part of the answer that’s less than two
digits, you must put a zero in front of it. If you get a second part of the
4 How you remember to use 25 as your sub-base is up to you. I always
remember that the sub-base 25 is ½ of base 50. It makes sense to me that the sub-base
is ½ of the base. Again, this is just my trick, but please use the method that works best
for you.
24 J. T. Manhire [Vol. XX:XXX
answer that’s more than two digits, you need to carry the third digit over
to the first part of the answer, as in the following example.
Solve 622. The first step is to determine the distance of 62 from
base 50, which is 12. Next, add sub-base 25 and 12 to get 37. This
becomes the first part of your answer (3 7 __ __). Next, square the
distance 12 to get the second part of your answer. Notice that when you
square 12 you get 144, which is a three digit number. If this happens,
use the 44 as your last two digits and carry the 1 over to the 37 in the
first part of your answer. Since 37 + 1 = 38, the first part of your
answer changes from 37 to 38 (3 78 4 4). And that’s your answer; 622 =
𝟑, 𝟖𝟒𝟒.
3. Algorithmic Approaches to Squaring Numbers
Again, some of the methods for mental multiplication require
learning short algorithms. At first glance, these can overwhelm even a
seasoned mental calculator; however, they are surprisingly simple if you
take the time to learn them and practice them. This section will present
each algorithm and then give a brief example. This section will not
cover each algorithm in as much detail as previous section, but at this
point you probably have enough mental ammunition to work through
the examples quickly.
I will use a special notation here. I will use 𝑥 to denote the
number being squared. This will be for all numbers so 𝑥 can be more
than a one-digit number (e.g., 𝑥2 = 1572). If the algorithm breaks out
the digits of the number being squared, I will use the notation 𝑥1 for the
right-most digit, 𝑥2 for the digit immediately to the left of the right-most
digit, etc. So for the number 6,789, 6 = 𝑥4, 7 = 𝑥3, 8 = 𝑥2, and 9 =
𝑥1 (since 9 is the right-most digit). If there is no subscript attached to 𝑥,
assume 𝑥 represents the entire number being squared.
2015] Mental Multiplication Made Easy 25
a. Any Two-Digit Number:
Algorithm: (𝑥2𝑥1)2 = 100(𝑥2)2 + 20(𝑥2 × 𝑥1) + (𝑥1)2.
Example:
372 = 100(3)2 + 20(3 × 7) + (7)2 = 900 + 420 + 49 = 𝟏, 𝟑𝟔𝟗
b. Numbers 52 to 99:
Algorithm: 𝑥2 = (100 − 𝑥)2 + 100[𝑥 − (100 − 𝑥)].
Example:
932 = (7)2 + 100[93 − (7)] = 49 + 8600 = 𝟖, 𝟔𝟒𝟗
c. Numbers 101 to 148:
Algorithm: 𝑥2 = (𝑥 − 100)2 + 100[𝑥 + (𝑥 − 100)].
Example:
1472 = (47)2 + 100[147 + (47)] = 2,209 + 19,400 = 𝟐𝟏, 𝟔𝟎𝟗
d. Numbers Below Base 1,000:
Algorithm: 𝑥2 = (1,000 − 𝑥)2 + 1,000[𝑥 − (1,000 − 𝑥)].
Example:
9942 = (6)2 + 1,000[994 − (6)] = 36 + 988,000 = 𝟗𝟖𝟖, 𝟎𝟑𝟔
e. Numbers Above Base 1,000:
Algorithm: 𝑥2 = (𝑥 − 1,000)2 + 1,000[𝑥 + (𝑥 − 1,000)].
Example:
1,0072 = 72 + 1000[1007 + 7] = 49 + 1,014,000 = 𝟏, 𝟎𝟏𝟒, 𝟎𝟒𝟗
26 J. T. Manhire [Vol. XX:XXX
f. Numbers Ending in 1:
Algorithm: 𝑥2 = (𝑥 − 1)2 + (2𝑥 − 1).
Example:
1472 = (47)2 + 100[147 + (47)] = 2209 + 19400 = 𝟐𝟏, 𝟔𝟎𝟗
g. Numbers Ending in 4:
Algorithm: 𝑥2 = (𝑥 + 1)2 − (2𝑥 + 1).
Example:
142 = (14 + 1)2 − (28 + 1) = (225) − (29) = 𝟏𝟗𝟔
h. Numbers Ending in 6:
Algorithm: 𝑥2 = (𝑥 − 1)2 + (2𝑥 − 1).
Example:
162 = (16 − 1)2 + (32 − 1) = (225) + (31) = 𝟐𝟓𝟔
i. Numbers Ending in 9:
Algorithm: 𝑥2 = (𝑥 + 1)2 − (2𝑥 + 1).
Example:
192 = (19 + 1)2 − (38 + 1) = (400) − (39) = 𝟑𝟔𝟏
There are other methods for squaring numbers. Because the
objective of this paper is to provide a short introduction to mental
multiplication, I will not include these methods here. I do encourage
you to investigate and learn these methods on your own from
publications like those cited in the introductory footnotes.
2015] Mental Multiplication Made Easy 27
IV. CONCLUSION
I hope this brief outline has, at a minimum, convinced you that
mental multiplication is achievable by anyone willing to devote the time
necessary to master it. Again, it takes practice to become one of those
folks you see on TV beating a calculator. But even if you don’t aspire
to be “that guy,” the techniques outlined in this brief introduction should
be sufficient for you to at least do multiplication in your head faster than
before you started reading it. As a social scientist, tax attorney, CPA,
actuary, or any similar profession, just being able to do accurate
multiplication in your head faster than before can be a significant
benefit. It’s also something that’s just really cool.
Good luck, have fun, and keep practicing!