Abstract
People who excel at mathematics use better strategies than the rest of us, they don’t necessarily have better brains. We teach simple strategies that can have you multiplying large numbers in your head, doing mental long division, even squaring and finding square roots of numbers off the top of your head.
And here is a secret. People equate intelligence with mathematical ability. In other words, if you are able to do lightning calculations in your head, people will think you are intelligent in other areas as well.
Here is one of my most important rules of mathematics. It is an unfair rule, but it is a rule just the same.
The easier the method you use to solve a problem, the faster you will solve it and there is less chance of making a mistake. The more complicated the method you use, the longer you take to solve it and greater the chance of making a mistake.
So, the people who use better methods are faster at getting the answer and make fewer mistakes. Those who use poor methods are…
Speed Math Lessons
“The number you have dialed is imaginary. 
The following is a rewrite and reorganization of math
multiplication lessons I initially presented to friends.
=====
Multiplication =====
This short course is designed to teach the reader an
alternative multiplication system to the one they may have learned in school
(that being rote memorization of the “times tables,” most likely
resulting in areas of more and less confidence in their ability) and its
application.
This system has been designed with the overall goal of making it possible
for the student to confidently multiply any size number by from between one and
twelve without having to carry any value higher than two. It is aimed at making
it possible for students whose knowledge of mathematics includes only addition,
doubling, and subtraction to solve complicated multiplication problems.
Therefore, the lessons are organized such that each presents you with only
one new step for solving each new problem, building up from the simple to the
somewhat more complex (combinatory) rules for solving any sized multiplication
problem.
Terms 

Carry 
the value of the “tens” place of the number you 
Half 
a simplified half, throw out fractions. (ex. half of 5 is 
Multiplicand 
the value to be multiplied. 
Multiplier 
the value by which to multiply. 
Neighbor 
the digit of the multiplicand which is to the right of the 
Number 
when working right to left to achieve the product, this is 
Product 
the answer to a multiplication problem. 
Lesson I
Multiplication by One
The Rule:
write the number
Simple enough, 25 times 1 is 25. Below are the steps toward achieving the
solution to the problem; not to insult your intelligence but to illustrate the
format used throughout this series.
0 2 5 x 1  5 
write the number. You should always write your answers 
0 2 5 x 1  2 5 
write the number. 
0 2 5 x 1  0 2 5

writing the number from under the zero position will keep 
Multiplication by
Ten
The Rule
write the neighbour
This is another way of looking at the commonly taught ‘add a zero to the
multiplicand,’ which better fits in with the simplified rules of this
multiplication system. Take a look at the simple example of 425 times 10.
0 4 2 5 x 10  0 
remember that the “neighbour” when one is 
0 4 2 5 x 10  5 0 
write the neighbour. 
0 4 2 5 x 10  2 5 0 
write the neighbour. 
0 4 2 5 x 10  4 2 5 0

write the neighbour. 
Multiplication by
Eleven
The Rule:
add the neighbour
Notice that this rule says ‘add’ instead of ‘write’ meaning ‘add the neighbour
to the number. A simple combination of the rules used for multiplication by one
and by ten. Once again this is a simplified restatement of a method you may
have already learned which is to ‘multiply the multiplicand by ten and add the
multiplicand.’ So now let’s try the simple equstion 75632 times 11.
0 7 5 6 3 2 x 11  2

see the 2, say [ 2 ], add the neighbour (zero). 
0 7 5 6 3 2 x 11  5 2

see the 3, say [ 3 ], add the neighbour (2) and say [ 5 ]. 
0 7 5 6 3 2 x 11  9 5 2 
see the 6, say [ 6 ], add the neighbour (3) and say [ 9 ]. 
0 7 5 6 3 2 x 11  `1 9 5 2 
see and say [ 5 ], add the neighbour (6) and say [ 11 ]. 
0 7 5 6 3 2 x 11  `3`1 9 5 2

see the carry and the 7, and say [ 8 ], add the neighbour 
0 7 5 6 3 2 x 11  8`3`1 9 5 2

see the carry and the zero, and say [ 1 ], add the 
Thus each figure of the multiplicand is used twice; once as a number and
once as a neighbour. There is a special case when multiplicands beginning with
9 followed by another large figure, the product may have a 10 in the last step.
Try the next one yourself; multiply 98,325 times 11. The answer is 1,081,575.
Multiplication by Two
The Rule:
double the number
Another simple concept, but one well used in this system and worth your
attention. Whenever the rule says ‘double’ for a step, you should try to look
at the value represented and double it instantaneously (ex. if you are to
double 6, do not say to yourself ‘six times two is twelve’ or ‘six and six is
twelve,’ instead say ‘six, twelve.’) at first you may wish to practice by
saying the value and then its double, but your goal should be to reduce even
that to the single step of saying the double (look at the 6 and say ‘twelve’).
Once again I have provided a sample problem, this time to illustrate the proper
mental method with regards to the carry. Try 984 times 2.
0 9 8 4 x 2  8 
double the number. 
0 9 8 4 x 2  `6 8 
double the number. 
0 9 8 4 x 2  `9`6 8

double the number and add the carry. Do not say [ 9, 18, 
0 9 8 4 x 2  1`9`6 8

double the number (zero) and add the carry. 
Multiplication by
Twelve
The Rule:
Double the number
and add the neighbour
This is the same as multiplying by 11 except that now you use the new rule from
multiplication by two and double the number before adding its neighbour. Take
an easy example 252 times 12:
0 2 5 2 x 12  4 
see the number (2) and say its double [ 4 ], and add the 
0 2 5 2 x 12  `2 4 
see the 5 and say its double [ 10 ], then add the 2 and 
0 2 5 2 x 12  `0`2 4

see the 2 and the carry, say [ 5 ], then add the 5 and say 
0 2 5 2 x 12  3`0`2 4

see the zero and the carry, say [ 1 ], then add the 2 and 
The product of 252 times 12 is 3,024. Now try 65535 x 12 on your own. The
answer is 786,420.
Some practice in
proper mental methods
Practice looking at a number (8 for example) and saying its double… try that
with each of the numbers in the following list:
2, 5, 3, 8, 5, 3, 6,
9, 5, 3, 6, 8, 9, 6, 4, 2,
9, 8, 1, 0, 8, 6, 5, 4, 2, 6, 7, 4, 3
Great! Now practice looking at the number, and adding the neighbour… this is
how you multiply by eleven.
2 5 
3 8 
5 3 
6 9 
5 1 
4 8 
9 6 
1 2 
4 0 
9 5 
8 6 
5 4 
2 6 
8 9 
7 4 
3 1 
6 3 
5 8 
3 5 
1 6 
6 5 
9 4 
5 2 
3 6 
Finally, using the same set of numbers, practice seeing and doubling the
number, and adding the neighbour… which is, as you know, how you multiply by
twelve.
enjoy…
Lesson II
Multiplication by Five —————–
========================================
The Rule:
take half the neighbour; add 5 if
the “number” is odd.
This is another simple lesson, note that there is now one
added twist “add 5 if the ‘number’ is odd.” This is another frequently used method in
this multiplication system, and it always and only refers to the
“number” (see definitions) in its raw state… and has nothing to do
with the result of doubling or halving the “number.” Note that if the Rule were written in the
proper order of execution, it would read:
if the “number” is odd
add 5; either way add half the
neighbour.
Obviously that version is not as easy to understand, so keep
in mind that the semicolon in the rule separates steps that may actually be
executed in a different order, but are written in that order for clairity. Now, take an easy example of 6381 times 5.
0 6 3 8 1 x 5
———
5 – look
at the number and see that it is odd, say [ 5 ] then look for the neighbour (assume zero since
there is no neighbour) and add
“half” of that (zero again) to the five… Your mental stops should have been: [ 5 ] answer: 5.
0 6 3 8 1 x 5
———
0 5 – look
at the number (8) and see that it is not odd,
then look for the neighbour (1)
and add half of that (“half” of 1 is zero)… Your
mental stops should have been: [ 0 ]
answer:
0
0 6 3 8 1 x 5
———
9 0 5 – look
at the number (3) see that is it odd, say [ 5 ] then add “half” of the neighbour to
the 5, and say [ 9 ].
Your mental stops should have been: [ 5,
9 ] answer: 9
continue the pattern… I will
continue to give abbreviated commentary.
0 6 3 8 1 x 5
———
1 9 0 5 –
even, add half of 3… say [ 1 ].
Your mental stops should have been: [ 1 ]
answer:
1
0 6 3 8 1 x 5
———
3 1 9 0 5 – even, add half of 6… say [ 3 ]. Your
mental stops should have been: [ 1 ]
answer:
1
See… what did I tell you?
Now try the next one on your own.
0 4 5 8 9 7 2
x 5
————
the product is 2,294,860.
You’ve already learned my usage of “number”,
“neighbour”, and “carry”.
Now I’ll introduce you to to the concept of “half” a number. “Half” with quotation marks because
it is a simplified half… throw out fractions.
Thus “half” of 5 would be 2, “half” of 3 is 1, and
“half” of 1 is zero. Of course
half of 4 is still 2, and so on with all even numbers.
Try to do this instantly… don’t look at 6 and say
“half of 6 is 3”, instead look at 6 and say 3. Try doing that now on the following digits.
8,6,4,5,3,0,9,2,1,5,3,7,4,9,0,8,1,7,2,6
Multiplication by Six ——————
=========================================
The Rule:
To each
“number” add “half” of the neighbour;
plus 5 if the
“number” is odd.
Note that you do not add 5 if the “neighbour” is
odd… you only have to do that if the “number” is odd. Take an easy example,
2563 times 6.
0 2 5 6 3 x 6
———
8 – add
5 to the 3 because the “3” is odd; there is no
neighbour, so 8 is the answer.
0 2 5 6 3 x 6
———
7 8 – 6
plus “half” of 3 is 7; the “number”, 6, is even so
no further
steps are taken…
0 2 5 6 3 x 6
———
`3 7 8 – 5
(odd) plus 5 (the number) plus “half” of 6 is 13.
0 2 5 6 3 x 6
———
5`3 7 8 – the
“carry” plus 2 plus “half” of 5 is 5.
0 2 5 6 3 x 6
———
1 5`3 7 8 – 0 plus “half” of 2 is 1.
Of course all this explanation is only to make this simple
method as crystal clear as possible.
With a reasonable amount of practice in your daily life this method will
become second nature. Try these two yourself:
0 1 2 5 3 x 6
———
0 2 1 3 4 8 9 6 x 6
—————
The answer to the first problem is 7,518. And the answer to the second is 12,809,376. Thus far all of the
numbers I have used as multiplicands have been long numbers, but you can use
the same methods for single digit multiplicands. Try this one, many people had problems
memorizing it for some reason (but there will be no need for your children to
spend many unhappy hours in memorization if you teach them what I’m teaching
you) :
0 9 x 6
—
`4 – 5
(odd) plus 9 is 14; no “neighbour”
0 9 x 6
—
5`4 – the “carry” plus 0 plus
“half” of 9 is 5
Some
comments on proper mental methods ——————
==========================================================
It is important that you start off with the proper mental
methods of calculation. Just as in
learning to speed read you must learn to see whole words and lines of text
rather than o n e l e t t e r at a time, so too should you
learn to look at a number and see its “half”
rather than have to think about it.
The methods I’m teaching you become second nature to
accountants and mathematicians whose livelyhood involves numbers, but have not yet
become so for you… that will be the most difficult part of
what you are going to learn.
So you must practice, practice, practice.Another important practice is
only saying only the result of adding a number or taking half of its neighbour,
like this:
0 1 2 8 x 6
——
6 8
The 6 is the 2 plus half the 8. but do not say “half of 8 is 4, and 2
and 4 is 6.” Instead, look at the 2
and the 8, see that hald of 8 is 4, and say to yourself “2,6.” At first this will be difficult, so it may be better to say to
yourself “2,4,6.”
Another point to practice is the step of adding the 5 when
the number (not the neighbour) is odd.
Take this case:
0 2 3 6 x 6
——
`1 6
The 1 is the 1 of 11, as the (`) shows, and the 11 is 3 plus
5 (because 3 is odd) plus 3 (half of 6).
The correct procedure, at first, is to say “5,8,3,11.” After some practice it should be cut down to
simply “8,11.” The 5 that come
sin because 3 is odd should be added first, otherwise you might forget it.
In the same way, when there is a (`) for a carried 1 (or
more rarely (“) for a carried 2), this should be added before the neighbour
(for times 11) is added or half the neighbour (for times 6). Once again, if it
is left for after this step it may be completely forgotten.
One final example to bring the rest of the mental steps together:
0 8 3 4 x 6
——
5`0`0 4
Look at the 8 and say “9,”, adding the dot (`);
then say “10,” adding “half” the 3. At first it is better to look at the 8 and say
“9,” adding the dot, then say “1” for “half” of
3, then “10,”, and write the zero and “carry” dot.
When there is a dot and also a 5 to be added (because of
oddness), say “6” instead of “5” and then add the number
itself. This cuts out a step and is easy
to get used to. Try some problems of
your
own.
Multiplication by Seven —————–
=========================================
The Rule:
Double
the number and add half the neighbour;
add 5 if the number is odd.
Just like multiplication by six only different right? I will continue to present the solutions in
the same manner you should be solving them yourself. Practice solving them in this manner yourself
before looking at the solution as this helps to develop your concentration…
the secret to learning. Take the easy
example of 452 times 7.
0 4 5 2 x 7
——
4 –
(even), 4 (double 2), (no neighbour), 4.
0 4 5 2 x 7
——
`6 4 – 5
(odd), 15 (double 5), 1 (half of 2), 16.
0 4 5 2 x 7
——
`1`6 4 –
(even), 9 (double 4 plus the carry), 2 (half of 5), 11.
0 4 5 2 x 7
——
3`1`6 4 – (even), 1 (double zero plus the carry), 2
(half of 4), 3.
Like I said, easy.
Now you try the next one on your own:
0 8 3 1 5
x 7
———
0 4 1 2 0 5 9 3 x 7
—————
You didn’t forget that half of 1 is zero did you? The solutions are 58,205 and 28,844,151
aren’t they.. Of course… Now I’ll help you practice some of the mental
steps you have learned.
Some
practive in proper mental methods ——————
==========================================================
You’ve already practiced taking half of a number, but though
it should be simple for you, you haven’t practiced looking at a number (8 for
example) and saying its double… do try that now:
2,5,3,8,5,3,6,9,5,3,6,8,9,6,4,2,
1,0,3,5,8,6,5,4,2,6,8,9,7,4
Great! Now practice
looking a number, saying its double and adding the neighbour… this is how you
multiply by 12.
2 5 3 8
5 3 6 9 5 1
4 8 9 6 1 2
4 0 9 5
8 6 5 4 2 6
8 9 7 4 3 1
6 3 5 8
3 5 1 6
6 5 9 4 5 2
3 6
Almost done… now, using the same pairs of numbers practice
looking at the number, saying its double and adding half the neighbour. This is used in “times 7” problems.
Finally look at the following numbers, and practice saying
“5,” then say 5 plus double the number.
5, 3, 9, 7,
1, 9, 5, 7, 3, 1, 3, 5, 1, 9, 5, 3
There are all kinds of ways you can practice what you have learned
without having to actually sit at a table with pencil and paper… try multiplying the price of gas by
6, 7, 11, or 12 with the grease of your finger on the dust on your back window
while filling your tank… try doing so with your birthdate, ssn, home or work
phone numbers, or drivers liscence number in your head… while waiting at a
stoplight do so with liscense plate numbers…see if you can think of some
more… and practice, practice, practice.
enjoy…
Addition of two numbers:
2166
9327

Let’s add these two numbers in our heads (i.e., without paper). Can you do
that? Our first attempt is to do it like most of us do on paper: 6+7=3 carry
the 1 (13), 1+6+2=9, 1+3=4, 2+9=11. The answer is . . . now what were those
numbers? The problem here is memory, not mental arithmetic. That’s why people
use paper (or an abacus, or their fingers), to help out their memories.
Memory is why speed arithmetic experts (I call them
“arithmetickers”) usually add numbers like these from left to right.
It is a little more complicated that way (you have to back track). But, you end
up saying the answer from left to right, just as it is normally said. It is
easier to remember a number from left to right.
Let’s try again: 2+9=11, 1+3=4, 6+2=8, 6+7=3 and that previous 8 should have
been a 9 (because of the carry). I actually remembered the answer, 11493, that
time. It’s still a test of my memory, but not bad. It may take you a little
practice to be able to do that.
Addition of columns:
29
37
15
21
32
85
44

How about this addition problem? Add these up in your head. It’s not too
tough to add up the right column, remember the carry, and add up the left
column, just as you would do with a pencil.
In the right column, a speed arithmetic person might group the 9 and the 1
(10), the two 5’s (10), and then the 7+2+4 (13) to get 33. Something similar
works for the left column. A few people group elevens instead of tens
Instead, what I do is add 29+37=66, then 66+15=81, then 81+21=102,
102+32=134, 134+85=219, and 219+44=263. Isn’t that slower? Maybe. But I have
little to remember, just the sum so far. It becomes very fast, if you practice
doing it that way. That is how a person with an abacus (or a person counting
with fingers, as in Chisanbop) would do it. And an abacus is just a way of
remembering the latest sum. That’s something you can easily do without an
abacus.
Multiplication:
24
x48

Speed arithmetic people multiply from left to right, too. No matter how you
do it, there’s plenty to remember here. Let me show you how this is done. It
takes practice to get good at it. But, just watch me do this:
2x4 = 8
2x8 = 16

96
4x4 = 16

112
4x8 = 32

1152
See how that is done? The order in which this is done is the trick: left
digits, opposite corners, right digits. Bigger numbers can be done in a
similar, but more complicated way.
Why?
Why should a person learn to do this? Is it just a way to show off? No, it
helps a person in math and sciences. It makes a person more accurate (even when
using paper or a calculator). You won’t get bogged down doing the arithmetic.
You can get on to other, even more interesting ideas.
Addendum (tricks):
Older speed arithmetic books dwelt almost exclusively on tricks. Here are
some of those tricks (which you can deduce on your own, instead of memorizing
this table):
 Multiply by 5: Multiply by 10
and divide by 2.  Multiply by 6: Sometimes
multiplying by 3 and then 2 is easy.  Multiply by 9: Multiply by 10
and subtract the original number.  Multiply by 11: See The Trachtenberg System of
Speed Arithmetic for a very easy method.  Multiply by 12: Multiply by
10 and add twice the original number.  Multiply by 13: There is no
easy trick method. Multiply by 3 and add 10 times original number.  Multiply by 14: Multiply by 7
and then multiply by 2 (or vice versa, whichever seems easier).  Multiply by 15: Multiply by
10 and add 5 times the original number, as above.  Multiply by 16: You can
double four times, if you want to. Or you can multiply by 8 and then by 2.  Multiply by 17: There is no
easy trick method. Multiply by 7 and add 10 times original number.  Multiply by 18: Multiply by
20 and subtract twice the original number (which is obvious from the first
step).  Multiply by 19: Multiply by
20 and subtract the original number.  Multiply by 24: Multiply by 8
and then multiply by 3. A similar method works for other numbers that can
be factored, like 32 or 45 or many others.  Multiply by 27: Multiply by
30 and subtract 3 times the original number (which is obvious from the
first step).  Multiply by 45: Multiply by
50 and subtract 5 times the original number (which is obvious from the
first step).  Multiply by 90: Multiply by 9
(as above) and put a zero on the right.  Multiply by 98: Multiply by
100 and subtract twice the original number.  Multiply by 99: Multiply by
100 and subtract the original number.
There are a lot more tricks for multiplication, division (divide by 5 by
multiplying by 2 and dividing by 10), addition, subtraction, and squaring.
Square a twodigit number that ends in 5 (like 85) by multiplying the left
digit by the next highest number (8×9=72), and tack on “25” on the
right (7225). Square any two digit number (like 87) by squaring the left digit
and tacking on two zeros (6400), double the product of the two digits and tack on
one zero (1120), add those (6400+1120=7520), square the right digit and add
(7520+49=7569). You don’t need to tack on zeros, if you can keep the columns
straight. This is just an application of the multiplication method described
above.
When you multiply a number by two, you can do it just as you do it on paper,
but do it in your head. You just double each digit, remembering to add a carry
when that comes up. You should be able to double almost anything in your head.
Dividing by two should be almost as easy.
Taking Square Roots
© Copyright 1999, Jim Loy
When I was in high school, I learned a mysterious method for taking square
roots. I tried to find out why it worked. And I ended up finding methods of
taking cube roots and fifth roots (to take a fourth root, just take the square
root twice). Now, with modern calculators, you can do all of this with the
touch of a few buttons. Do you long for the good old days when you took square
roots by hand? Me neither.
It is surprisingly easy to take square roots (and other roots) by trial and
error. Let’s take the square root of 19:
guess squared
4.0 16 < 19
4.5 20.25 > 19
4.4 19.36 > 19
4.3 18.89 < 19
4.35 18.9225 < 19
4.36 19.0096 > 19
4.355 18.966025 < 19
So, 4.36 is correct to 3 places. That is not too bad. You may actually have
to do something like that, when you need to take a cube root, if your
calculator does not have power functions on it. Of course, other methods are
faster, if you can figure out how.
Here’s
how we were taught in school. We draw a diagram much like we do with long
division. In fact, the whole method is very much like long division. We mark
off every other digit of the number under the square root sign, starting at the
decimal point. Every two digits of that number produces one digit of the
answer.
We estimate the first digit of the answer: greater than 4 and less than 5,
so we choose 4. Now we put that 4 to the left of the square root sign. We
multiply 4×4 and get 16, which we put under the 19. We subtract and get 3, and
we bring down two digits (00 in this case), making 300.
Here’s the part that is different from long division (where the divisor
stays the same): We double the four on the left, making 8, and put that to the
left of the 300. Now we estimate what times 80+what is just less than 300. 4×84
is greater than 300, so 3×83 is what we want. 3 is the next digit of the
answer, and we put that 3 to the right of the 8 (making 83 (or 80+what)) on the
left side. 3×83=249, which we put under the 300 and subtract, getting 51. We
bring down 00 from above.
Now we get 86 (twice 43) on the left side. And we repeat the operation,
getting 865×5 and 8708×8 and 87169×9. I did that last one to see which way to
round the previous answer. So the answer is 4.359. That process may look
daunting. But it is fairly fast.
That is not the way computers and calculators do square roots. They use a
very fast method called Newton’s Method, or Newton’s Iteration.
Estimating the square root of n:
x(new)=(x+n/x)/2
The x(new) on the left (x_{k} or “x sub k” in technical
terms) is the new x. The other x’s (x_{k1} or “x sub k1”)
are the old x. The new x is an improved estimate of the answer. Let’s try to
estimate the square root of 19. We can start with any estimate except 0. So,
let’s start with 1:
 1
 10
 5.95
 4.571638655462
 4.363848830052
 4.358901750853
 4.358898943542
 4.358898943541
That is correct out to the last digit. It took a little while to get
started, so you may want to start with a better estimate than 1. For an eight
digit number you would start with a four digit estimate. After a while, this
method is very fast, faster than the method we learned in school. For example,
step 7 (above) gained 8 digits of accuracy over the estimate in step 6. Step 8
would have been even more impressive (about 16 more digits), had we not
exceeded the accuracy of my calculator.
Newton’s
Method has other uses, besides square roots. [I originally showed an erroneous
formula for cube roots, here.] Calculators and computers do not use Newton’s method for cube
roots. They would use a power function, which is based on natural (base e)
logarithms. The calculator uses series to estimate these. And the process is
much slower, and less accurate than Newton’s
Method. Newton’s
method is not always applicable.
Sometimes you can be very accurate just by making a quick estimation of a
square root. I can tell that the square root of 1.02 is about 1.01. Let’s say
that 1.01 is 1+x. Squared, that is 1+2x+x^{2} (or 1+2x+x^2). Here, x
squared is fairly insignificant. Try the square root of 1.001. That is about
1.0005 (x is .0005 here).
. Here is the Newton method formula for finding the pth
root of n. We can easily derive a cube root formula from that:
x(new) = x – (x^pn)/(px^(p1))
With a computer, you don’t want to take something to the p power, you
multiply it times itself, p times.
A New Look at Subtraction
Subtraction is pretty simple, right?
Well, no matter how simple it seems to you, you may enjoy this article.
What is 13788?
Traditional method: The following is the traditional way to do it. I
used dots (over the 1 and 3) to indicate borrows. You may mark them
differently, or not mark them at all.
..
137
88
49
I started at the right column. 7 is less than 8, so I borrowed from the three
(making it 2) and changing the 7 into 17. 178=9, which I wrote as the right
digit of the answer. In the next column, our 3 is now a 2, and it is less than
8. So I had to borrow from the 1 (making it 0) and the two became 12. 128=4.
Our answer is 49.
Second method: Another method (slightly easier, in my opinion) is to
add when we borrow:
137
.
88
49
Here I put the dot over the left 8. Instead of making the 3 above it into a
2, I made the 8 into a 9. We still have 178=9 in the right column. But now, we
have 139=4 in the second column. This method is easier when some of the top
digits are zero.
Third method: Now let’s look at a method which involves addition, but
very little actual subtraction:
137 +2 = 139 +10 = 149
88 +2 = 90 +10 = 100
49
Normally, most of this would be done mentally, without writing it down. But
this also works well if you do write it down, in some manner (you may leave out
the +2’s and +10’s). This method works because adding the same number to both
numbers of a subtraction problem doesn’t change the answer. We add numbers
which make the eventual subtraction easier. We could have stopped after step #1
(13990), as that is fairly easy to subtract. Personally, I think that this
method is the easiest of the three methods, especially when I’m doing the work
in my head. You may disagree.
Calculators: So why do we even want to know how to subtract, in a
world filled with calculators? I can think of three reasons:
 You may not be able to find
your calculator. This happens to me often.  Some subtractions are so easy
that you can do them in your head faster. Try it. 13788 turned out to be
fairly easy with the above third method.  What if you make a mistake
while using your calculator. If you are skilled at subtraction, you can
say, “That doesn’t look right,” and recalculate the answer.
Of course, we should use calculators. They are more accurate. And they
simplify difficult arithmetic. But we do need to know how to do arithmetic without
calculators.
Addendum:
I received email asking if I was actually recommending the use of
calculators in early grades. I thought it was obvious that I was recommending
learning arithmetic skills. Can this be done with calculators around? Maybe. But
I must leave that up to the teachers who have to teach these skills.
Calculators are important and useful to our society, and I think that kids need
to learn about them, eventually. But, universal use of calculators probably
prevents a student from learning arithmetic skills. So what? I don’t need
skills; I have a calculator. But, what happens when I multiply 17×37 (using a
calculator) and get 663? If I have arithmetic skills, I can see that the answer
is wrong (the right digit should be a 9). I pushed the wrong button, happens
often.
Also, it is important to check your work, even when using a calculator.
Mistakes happen. We skip a number; we misread a number; we put the decimal
point in the wrong place; we do the wrong operation. So we still need to check
our work.
This addendum is really a message to the student, not to the teacher. I wish
people, including myself, would do what is good for them, even when it is a
bother. You should see that practicing arithmetic skills (and many other
skills) is good for you. You see that you have a weakness in mathematics.
Instead of complaining or even bragging about it, it makes more sense to
correct the situation. Work work work. Sorry. We are willing to work work work
in athletics, but not mathematics. We should be smarter than that.
Speed Math Lessons
“The number you have dialed is imaginary. 
The following is a rewrite and reorganization of math
multiplication lessons I initially presented to friends.
=====
Multiplication =====
This short course is designed to teach the reader an
alternative multiplication system to the one they may have learned in school
(that being rote memorization of the “times tables,” most likely
resulting in areas of more and less confidence in their ability) and its
application.
This system has been designed with the overall goal of making it possible
for the student to confidently multiply any size number by from between one and
twelve without having to carry any value higher than two. It is aimed at making
it possible for students whose knowledge of mathematics includes only addition,
doubling, and subtraction to solve complicated multiplication problems.
Therefore, the lessons are organized such that each presents you with only
one new step for solving each new problem, building up from the simple to the
somewhat more complex (combinatory) rules for solving any sized multiplication
problem.
Terms 

Carry 
the value of the “tens” place of the number you 
Half 
a simplified half, throw out fractions. (ex. half of 5 is 
Multiplicand 
the value to be multiplied. 
Multiplier 
the value by which to multiply. 
Neighbor 
the digit of the multiplicand which is to the right of the 
Number 
when working right to left to achieve the product, this is 
Product 
the answer to a multiplication problem. 
Lesson I
Multiplication by One
The Rule:
write the number
Simple enough, 25 times 1 is 25. Below are the steps toward achieving the
solution to the problem; not to insult your intelligence but to illustrate the
format used throughout this series.
0 2 5 x 1  5 
write the number. You should always write your answers 
0 2 5 x 1  2 5 
write the number. 
0 2 5 x 1  0 2 5

writing the number from under the zero position will keep 
Multiplication by
Ten
The Rule
write the neighbour
This is another way of looking at the commonly taught ‘add a zero to the
multiplicand,’ which better fits in with the simplified rules of this
multiplication system. Take a look at the simple example of 425 times 10.
0 4 2 5 x 10  0 
remember that the “neighbour” when one is 
0 4 2 5 x 10  5 0 
write the neighbour. 
0 4 2 5 x 10  2 5 0 
write the neighbour. 
0 4 2 5 x 10  4 2 5 0

write the neighbour. 
Multiplication by
Eleven
The Rule:
add the neighbour
Notice that this rule says ‘add’ instead of ‘write’ meaning ‘add the neighbour
to the number. A simple combination of the rules used for multiplication by one
and by ten. Once again this is a simplified restatement of a method you may
have already learned which is to ‘multiply the multiplicand by ten and add the
multiplicand.’ So now let’s try the simple equstion 75632 times 11.
0 7 5 6 3 2 x 11  2

see the 2, say [ 2 ], add the neighbour (zero). 
0 7 5 6 3 2 x 11  5 2

see the 3, say [ 3 ], add the neighbour (2) and say [ 5 ]. 
0 7 5 6 3 2 x 11  9 5 2 
see the 6, say [ 6 ], add the neighbour (3) and say [ 9 ]. 
0 7 5 6 3 2 x 11  `1 9 5 2 
see and say [ 5 ], add the neighbour (6) and say [ 11 ]. 
0 7 5 6 3 2 x 11  `3`1 9 5 2

see the carry and the 7, and say [ 8 ], add the neighbour 
0 7 5 6 3 2 x 11  8`3`1 9 5 2

see the carry and the zero, and say [ 1 ], add the 
Thus each figure of the multiplicand is used twice; once as a number and
once as a neighbour. There is a special case when multiplicands beginning with
9 followed by another large figure, the product may have a 10 in the last step.
Try the next one yourself; multiply 98,325 times 11. The answer is 1,081,575.
Multiplication by Two
The Rule:
double the number
Another simple concept, but one well used in this system and worth your
attention. Whenever the rule says ‘double’ for a step, you should try to look
at the value represented and double it instantaneously (ex. if you are to
double 6, do not say to yourself ‘six times two is twelve’ or ‘six and six is
twelve,’ instead say ‘six, twelve.’) at first you may wish to practice by
saying the value and then its double, but your goal should be to reduce even
that to the single step of saying the double (look at the 6 and say ‘twelve’).
Once again I have provided a sample problem, this time to illustrate the proper
mental method with regards to the carry. Try 984 times 2.
0 9 8 4 x 2  8 
double the number. 
0 9 8 4 x 2  `6 8 
double the number. 
0 9 8 4 x 2  `9`6 8

double the number and add the carry. Do not say [ 9, 18, 
0 9 8 4 x 2  1`9`6 8

double the number (zero) and add the carry. 
Multiplication by
Twelve
The Rule:
Double the number
and add the neighbour
This is the same as multiplying by 11 except that now you use the new rule from
multiplication by two and double the number before adding its neighbour. Take
an easy example 252 times 12:
0 2 5 2 x 12  4 
see the number (2) and say its double [ 4 ], and add the 
0 2 5 2 x 12  `2 4 
see the 5 and say its double [ 10 ], then add the 2 and 
0 2 5 2 x 12  `0`2 4

see the 2 and the carry, say [ 5 ], then add the 5 and say 
0 2 5 2 x 12  3`0`2 4

see the zero and the carry, say [ 1 ], then add the 2 and 
The product of 252 times 12 is 3,024. Now try 65535 x 12 on your own. The
answer is 786,420.
Some practice in
proper mental methods
Practice looking at a number (8 for example) and saying its double… try that
with each of the numbers in the following list:
2, 5, 3, 8, 5, 3, 6,
9, 5, 3, 6, 8, 9, 6, 4, 2,
9, 8, 1, 0, 8, 6, 5, 4, 2, 6, 7, 4, 3
Great! Now practice looking at the number, and adding the neighbour… this is
how you multiply by eleven.
2 5 
3 8 
5 3 
6 9 
5 1 
4 8 
9 6 
1 2 
4 0 
9 5 
8 6 
5 4 
2 6 
8 9 
7 4 
3 1 
6 3 
5 8 
3 5 
1 6 
6 5 
9 4 
5 2 
3 6 
Finally, using the same set of numbers, practice seeing and doubling the
number, and adding the neighbour… which is, as you know, how you multiply by
twelve.
enjoy…
Lesson II
Multiplication by Five —————–
========================================
The Rule:
take half the neighbour; add 5 if
the “number” is odd.
This is another simple lesson, note that there is now one
added twist “add 5 if the ‘number’ is odd.” This is another frequently used method in
this multiplication system, and it always and only refers to the
“number” (see definitions) in its raw state… and has nothing to do
with the result of doubling or halving the “number.” Note that if the Rule were written in the
proper order of execution, it would read:
if the “number” is odd
add 5; either way add half the
neighbour.
Obviously that version is not as easy to understand, so keep
in mind that the semicolon in the rule separates steps that may actually be
executed in a different order, but are written in that order for clairity. Now, take an easy example of 6381 times 5.
0 6 3 8 1 x 5
———
5 – look
at the number and see that it is odd, say [ 5 ] then look for the neighbour (assume zero since
there is no neighbour) and add
“half” of that (zero again) to the five… Your mental stops should have been: [ 5 ] answer: 5.
0 6 3 8 1 x 5
———
0 5 – look
at the number (8) and see that it is not odd,
then look for the neighbour (1)
and add half of that (“half” of 1 is zero)… Your
mental stops should have been: [ 0 ]
answer:
0
0 6 3 8 1 x 5
———
9 0 5 – look
at the number (3) see that is it odd, say [ 5 ] then add “half” of the neighbour to
the 5, and say [ 9 ].
Your mental stops should have been: [ 5,
9 ] answer: 9
continue the pattern… I will
continue to give abbreviated commentary.
0 6 3 8 1 x 5
———
1 9 0 5 –
even, add half of 3… say [ 1 ].
Your mental stops should have been: [ 1 ]
answer:
1
0 6 3 8 1 x 5
———
3 1 9 0 5 – even, add half of 6… say [ 3 ]. Your
mental stops should have been: [ 1 ]
answer:
1
See… what did I tell you?
Now try the next one on your own.
0 4 5 8 9 7 2
x 5
————
the product is 2,294,860.
You’ve already learned my usage of “number”,
“neighbour”, and “carry”.
Now I’ll introduce you to to the concept of “half” a number. “Half” with quotation marks because
it is a simplified half… throw out fractions.
Thus “half” of 5 would be 2, “half” of 3 is 1, and
“half” of 1 is zero. Of course
half of 4 is still 2, and so on with all even numbers.
Try to do this instantly… don’t look at 6 and say
“half of 6 is 3”, instead look at 6 and say 3. Try doing that now on the following digits.
8,6,4,5,3,0,9,2,1,5,3,7,4,9,0,8,1,7,2,6
Multiplication by Six ——————
=========================================
The Rule:
To each
“number” add “half” of the neighbour;
plus 5 if the
“number” is odd.
Note that you do not add 5 if the “neighbour” is
odd… you only have to do that if the “number” is odd. Take an easy example,
2563 times 6.
0 2 5 6 3 x 6
———
8 – add
5 to the 3 because the “3” is odd; there is no
neighbour, so 8 is the answer.
0 2 5 6 3 x 6
———
7 8 – 6
plus “half” of 3 is 7; the “number”, 6, is even so
no further
steps are taken…
0 2 5 6 3 x 6
———
`3 7 8 – 5
(odd) plus 5 (the number) plus “half” of 6 is 13.
0 2 5 6 3 x 6
———
5`3 7 8 – the
“carry” plus 2 plus “half” of 5 is 5.
0 2 5 6 3 x 6
———
1 5`3 7 8 – 0 plus “half” of 2 is 1.
Of course all this explanation is only to make this simple
method as crystal clear as possible.
With a reasonable amount of practice in your daily life this method will
become second nature. Try these two yourself:
0 1 2 5 3 x 6
———
0 2 1 3 4 8 9 6 x 6
—————
The answer to the first problem is 7,518. And the answer to the second is 12,809,376. Thus far all of the
numbers I have used as multiplicands have been long numbers, but you can use
the same methods for single digit multiplicands. Try this one, many people had problems
memorizing it for some reason (but there will be no need for your children to
spend many unhappy hours in memorization if you teach them what I’m teaching
you) :
0 9 x 6
—
`4 – 5
(odd) plus 9 is 14; no “neighbour”
0 9 x 6
—
5`4 – the “carry” plus 0 plus
“half” of 9 is 5
Some
comments on proper mental methods ——————
==========================================================
It is important that you start off with the proper mental
methods of calculation. Just as in
learning to speed read you must learn to see whole words and lines of text
rather than o n e l e t t e r at a time, so too should you
learn to look at a number and see its “half”
rather than have to think about it.
The methods I’m teaching you become second nature to
accountants and mathematicians whose livelyhood involves numbers, but have not yet
become so for you… that will be the most difficult part of
what you are going to learn.
So you must practice, practice, practice.Another important practice is
only saying only the result of adding a number or taking half of its neighbour,
like this:
0 1 2 8 x 6
——
6 8
The 6 is the 2 plus half the 8. but do not say “half of 8 is 4, and 2
and 4 is 6.” Instead, look at the 2
and the 8, see that hald of 8 is 4, and say to yourself “2,6.” At first this will be difficult, so it may be better to say to
yourself “2,4,6.”
Another point to practice is the step of adding the 5 when
the number (not the neighbour) is odd.
Take this case:
0 2 3 6 x 6
——
`1 6
The 1 is the 1 of 11, as the (`) shows, and the 11 is 3 plus
5 (because 3 is odd) plus 3 (half of 6).
The correct procedure, at first, is to say “5,8,3,11.” After some practice it should be cut down to
simply “8,11.” The 5 that come
sin because 3 is odd should be added first, otherwise you might forget it.
In the same way, when there is a (`) for a carried 1 (or
more rarely (“) for a carried 2), this should be added before the neighbour
(for times 11) is added or half the neighbour (for times 6). Once again, if it
is left for after this step it may be completely forgotten.
One final example to bring the rest of the mental steps together:
0 8 3 4 x 6
——
5`0`0 4
Look at the 8 and say “9,”, adding the dot (`);
then say “10,” adding “half” the 3. At first it is better to look at the 8 and say
“9,” adding the dot, then say “1” for “half” of
3, then “10,”, and write the zero and “carry” dot.
When there is a dot and also a 5 to be added (because of
oddness), say “6” instead of “5” and then add the number
itself. This cuts out a step and is easy
to get used to. Try some problems of
your
own.
Multiplication by Seven —————–
=========================================
The Rule:
Double
the number and add half the neighbour;
add 5 if the number is odd.
Just like multiplication by six only different right? I will continue to present the solutions in
the same manner you should be solving them yourself. Practice solving them in this manner yourself
before looking at the solution as this helps to develop your concentration…
the secret to learning. Take the easy
example of 452 times 7.
0 4 5 2 x 7
——
4 –
(even), 4 (double 2), (no neighbour), 4.
0 4 5 2 x 7
——
`6 4 – 5
(odd), 15 (double 5), 1 (half of 2), 16.
0 4 5 2 x 7
——
`1`6 4 –
(even), 9 (double 4 plus the carry), 2 (half of 5), 11.
0 4 5 2 x 7
——
3`1`6 4 – (even), 1 (double zero plus the carry), 2
(half of 4), 3.
Like I said, easy.
Now you try the next one on your own:
0 8 3 1 5
x 7
———
0 4 1 2 0 5 9 3 x 7
—————
You didn’t forget that half of 1 is zero did you? The solutions are 58,205 and 28,844,151
aren’t they.. Of course… Now I’ll help you practice some of the mental
steps you have learned.
Some
practive in proper mental methods ——————
==========================================================
You’ve already practiced taking half of a number, but though
it should be simple for you, you haven’t practiced looking at a number (8 for
example) and saying its double… do try that now:
2,5,3,8,5,3,6,9,5,3,6,8,9,6,4,2,
1,0,3,5,8,6,5,4,2,6,8,9,7,4
Great! Now practice
looking a number, saying its double and adding the neighbour… this is how you
multiply by 12.
2 5 3 8
5 3 6 9 5 1
4 8 9 6 1 2
4 0 9 5
8 6 5 4 2 6
8 9 7 4 3 1
6 3 5 8
3 5 1 6
6 5 9 4 5 2
3 6
Almost done… now, using the same pairs of numbers practice
looking at the number, saying its double and adding half the neighbour. This is used in “times 7” problems.
Finally look at the following numbers, and practice saying
“5,” then say 5 plus double the number.
5, 3, 9, 7,
1, 9, 5, 7, 3, 1, 3, 5, 1, 9, 5, 3
There are all kinds of ways you can practice what you have learned
without having to actually sit at a table with pencil and paper… try multiplying the price of gas by
6, 7, 11, or 12 with the grease of your finger on the dust on your back window
while filling your tank… try doing so with your birthdate, ssn, home or work
phone numbers, or drivers liscence number in your head… while waiting at a
stoplight do so with liscense plate numbers…see if you can think of some
more… and practice, practice, practice.
enjoy…
Speed Math Lessons
“The number you have dialed is imaginary. 
The following is a rewrite and reorganization of math
multiplication lessons I initially presented to friends.
=====
Multiplication =====
This short course is designed to teach the reader an
alternative multiplication system to the one they may have learned in school
(that being rote memorization of the “times tables,” most likely
resulting in areas of more and less confidence in their ability) and its
application.
This system has been designed with the overall goal of making it possible
for the student to confidently multiply any size number by from between one and
twelve without having to carry any value higher than two. It is aimed at making
it possible for students whose knowledge of mathematics includes only addition,
doubling, and subtraction to solve complicated multiplication problems.
Therefore, the lessons are organized such that each presents you with only
one new step for solving each new problem, building up from the simple to the
somewhat more complex (combinatory) rules for solving any sized multiplication
problem.
Terms 

Carry 
the value of the “tens” place of the number you 
Half 
a simplified half, throw out fractions. (ex. half of 5 is 
Multiplicand 
the value to be multiplied. 
Multiplier 
the value by which to multiply. 
Neighbor 
the digit of the multiplicand which is to the right of the 
Number 
when working right to left to achieve the product, this is 
Product 
the answer to a multiplication problem. 
Lesson I
Multiplication by One
The Rule:
write the number
Simple enough, 25 times 1 is 25. Below are the steps toward achieving the
solution to the problem; not to insult your intelligence but to illustrate the
format used throughout this series.
0 2 5 x 1  5 
write the number. You should always write your answers 
0 2 5 x 1  2 5 
write the number. 
0 2 5 x 1  0 2 5

writing the number from under the zero position will keep 
Multiplication by
Ten
The Rule
write the neighbour
This is another way of looking at the commonly taught ‘add a zero to the
multiplicand,’ which better fits in with the simplified rules of this
multiplication system. Take a look at the simple example of 425 times 10.
0 4 2 5 x 10  0 
remember that the “neighbour” when one is 
0 4 2 5 x 10  5 0 
write the neighbour. 
0 4 2 5 x 10  2 5 0 
write the neighbour. 
0 4 2 5 x 10  4 2 5 0

write the neighbour. 
Multiplication by
Eleven
The Rule:
add the neighbour
Notice that this rule says ‘add’ instead of ‘write’ meaning ‘add the neighbour
to the number. A simple combination of the rules used for multiplication by one
and by ten. Once again this is a simplified restatement of a method you may
have already learned which is to ‘multiply the multiplicand by ten and add the
multiplicand.’ So now let’s try the simple equstion 75632 times 11.
0 7 5 6 3 2 x 11  2

see the 2, say [ 2 ], add the neighbour (zero). 
0 7 5 6 3 2 x 11  5 2

see the 3, say [ 3 ], add the neighbour (2) and say [ 5 ]. 
0 7 5 6 3 2 x 11  9 5 2 
see the 6, say [ 6 ], add the neighbour (3) and say [ 9 ]. 
0 7 5 6 3 2 x 11  `1 9 5 2 
see and say [ 5 ], add the neighbour (6) and say [ 11 ]. 
0 7 5 6 3 2 x 11  `3`1 9 5 2

see the carry and the 7, and say [ 8 ], add the neighbour 
0 7 5 6 3 2 x 11  8`3`1 9 5 2

see the carry and the zero, and say [ 1 ], add the 
Thus each figure of the multiplicand is used twice; once as a number and
once as a neighbour. There is a special case when multiplicands beginning with
9 followed by another large figure, the product may have a 10 in the last step.
Try the next one yourself; multiply 98,325 times 11. The answer is 1,081,575.
Multiplication by Two
The Rule:
double the number
Another simple concept, but one well used in this system and worth your
attention. Whenever the rule says ‘double’ for a step, you should try to look
at the value represented and double it instantaneously (ex. if you are to
double 6, do not say to yourself ‘six times two is twelve’ or ‘six and six is
twelve,’ instead say ‘six, twelve.’) at first you may wish to practice by
saying the value and then its double, but your goal should be to reduce even
that to the single step of saying the double (look at the 6 and say ‘twelve’).
Once again I have provided a sample problem, this time to illustrate the proper
mental method with regards to the carry. Try 984 times 2.
0 9 8 4 x 2  8 
double the number. 
0 9 8 4 x 2  `6 8 
double the number. 
0 9 8 4 x 2  `9`6 8

double the number and add the carry. Do not say [ 9, 18, 
0 9 8 4 x 2  1`9`6 8

double the number (zero) and add the carry. 
Multiplication by
Twelve
The Rule:
Double the number
and add the neighbour
This is the same as multiplying by 11 except that now you use the new rule from
multiplication by two and double the number before adding its neighbour. Take
an easy example 252 times 12:
0 2 5 2 x 12  4 
see the number (2) and say its double [ 4 ], and add the 
0 2 5 2 x 12  `2 4 
see the 5 and say its double [ 10 ], then add the 2 and 
0 2 5 2 x 12  `0`2 4

see the 2 and the carry, say [ 5 ], then add the 5 and say 
0 2 5 2 x 12  3`0`2 4

see the zero and the carry, say [ 1 ], then add the 2 and 
The product of 252 times 12 is 3,024. Now try 65535 x 12 on your own. The
answer is 786,420.
Some practice in
proper mental methods
Practice looking at a number (8 for example) and saying its double… try that
with each of the numbers in the following list:
2, 5, 3, 8, 5, 3, 6,
9, 5, 3, 6, 8, 9, 6, 4, 2,
9, 8, 1, 0, 8, 6, 5, 4, 2, 6, 7, 4, 3
Great! Now practice looking at the number, and adding the neighbour… this is
how you multiply by eleven.
2 5 
3 8 
5 3 
6 9 
5 1 
4 8 
9 6 
1 2 
4 0 
9 5 
8 6 
5 4 
2 6 
8 9 
7 4 
3 1 
6 3 
5 8 
3 5 
1 6 
6 5 
9 4 
5 2 
3 6 
Finally, using the same set of numbers, practice seeing and doubling the
number, and adding the neighbour… which is, as you know, how you multiply by
twelve.
enjoy…
Lesson II
Multiplication by Five —————–
========================================
The Rule:
take half the neighbour; add 5 if
the “number” is odd.
This is another simple lesson, note that there is now one
added twist “add 5 if the ‘number’ is odd.” This is another frequently used method in
this multiplication system, and it always and only refers to the
“number” (see definitions) in its raw state… and has nothing to do
with the result of doubling or halving the “number.” Note that if the Rule were written in the
proper order of execution, it would read:
if the “number” is odd
add 5; either way add half the
neighbour.
Obviously that version is not as easy to understand, so keep
in mind that the semicolon in the rule separates steps that may actually be
executed in a different order, but are written in that order for clairity. Now, take an easy example of 6381 times 5.
0 6 3 8 1 x 5
———
5 – look
at the number and see that it is odd, say [ 5 ] then look for the neighbour (assume zero since
there is no neighbour) and add
“half” of that (zero again) to the five… Your mental stops should have been: [ 5 ] answer: 5.
0 6 3 8 1 x 5
———
0 5 – look
at the number (8) and see that it is not odd,
then look for the neighbour (1)
and add half of that (“half” of 1 is zero)… Your
mental stops should have been: [ 0 ]
answer:
0
0 6 3 8 1 x 5
———
9 0 5 – look
at the number (3) see that is it odd, say [ 5 ] then add “half” of the neighbour to
the 5, and say [ 9 ].
Your mental stops should have been: [ 5,
9 ] answer: 9
continue the pattern… I will
continue to give abbreviated commentary.
0 6 3 8 1 x 5
———
1 9 0 5 –
even, add half of 3… say [ 1 ].
Your mental stops should have been: [ 1 ]
answer:
1
0 6 3 8 1 x 5
———
3 1 9 0 5 – even, add half of 6… say [ 3 ]. Your
mental stops should have been: [ 1 ]
answer:
1
See… what did I tell you?
Now try the next one on your own.
0 4 5 8 9 7 2
x 5
————
the product is 2,294,860.
You’ve already learned my usage of “number”,
“neighbour”, and “carry”.
Now I’ll introduce you to to the concept of “half” a number. “Half” with quotation marks because
it is a simplified half… throw out fractions.
Thus “half” of 5 would be 2, “half” of 3 is 1, and
“half” of 1 is zero. Of course
half of 4 is still 2, and so on with all even numbers.
Try to do this instantly… don’t look at 6 and say
“half of 6 is 3”, instead look at 6 and say 3. Try doing that now on the following digits.
8,6,4,5,3,0,9,2,1,5,3,7,4,9,0,8,1,7,2,6
Multiplication by Six ——————
=========================================
The Rule:
To each
“number” add “half” of the neighbour;
plus 5 if the
“number” is odd.
Note that you do not add 5 if the “neighbour” is
odd… you only have to do that if the “number” is odd. Take an easy example,
2563 times 6.
0 2 5 6 3 x 6
———
8 – add
5 to the 3 because the “3” is odd; there is no
neighbour, so 8 is the answer.
0 2 5 6 3 x 6
———
7 8 – 6
plus “half” of 3 is 7; the “number”, 6, is even so
no further
steps are taken…
0 2 5 6 3 x 6
———
`3 7 8 – 5
(odd) plus 5 (the number) plus “half” of 6 is 13.
0 2 5 6 3 x 6
———
5`3 7 8 – the
“carry” plus 2 plus “half” of 5 is 5.
0 2 5 6 3 x 6
———
1 5`3 7 8 – 0 plus “half” of 2 is 1.
Of course all this explanation is only to make this simple
method as crystal clear as possible.
With a reasonable amount of practice in your daily life this method will
become second nature. Try these two yourself:
0 1 2 5 3 x 6
———
0 2 1 3 4 8 9 6 x 6
—————
The answer to the first problem is 7,518. And the answer to the second is 12,809,376. Thus far all of the
numbers I have used as multiplicands have been long numbers, but you can use
the same methods for single digit multiplicands. Try this one, many people had problems
memorizing it for some reason (but there will be no need for your children to
spend many unhappy hours in memorization if you teach them what I’m teaching
you) :
0 9 x 6
—
`4 – 5
(odd) plus 9 is 14; no “neighbour”
0 9 x 6
—
5`4 – the “carry” plus 0 plus
“half” of 9 is 5
Some
comments on proper mental methods ——————
==========================================================
It is important that you start off with the proper mental
methods of calculation. Just as in
learning to speed read you must learn to see whole words and lines of text
rather than o n e l e t t e r at a time, so too should you
learn to look at a number and see its “half”
rather than have to think about it.
The methods I’m teaching you become second nature to
accountants and mathematicians whose livelyhood involves numbers, but have not yet
become so for you… that will be the most difficult part of
what you are going to learn.
So you must practice, practice, practice.Another important practice is
only saying only the result of adding a number or taking half of its neighbour,
like this:
0 1 2 8 x 6
——
6 8
The 6 is the 2 plus half the 8. but do not say “half of 8 is 4, and 2
and 4 is 6.” Instead, look at the 2
and the 8, see that hald of 8 is 4, and say to yourself “2,6.” At first this will be difficult, so it may be better to say to
yourself “2,4,6.”
Another point to practice is the step of adding the 5 when
the number (not the neighbour) is odd.
Take this case:
0 2 3 6 x 6
——
`1 6
The 1 is the 1 of 11, as the (`) shows, and the 11 is 3 plus
5 (because 3 is odd) plus 3 (half of 6).
The correct procedure, at first, is to say “5,8,3,11.” After some practice it should be cut down to
simply “8,11.” The 5 that come
sin because 3 is odd should be added first, otherwise you might forget it.
In the same way, when there is a (`) for a carried 1 (or
more rarely (“) for a carried 2), this should be added before the neighbour
(for times 11) is added or half the neighbour (for times 6). Once again, if it
is left for after this step it may be completely forgotten.
One final example to bring the rest of the mental steps together:
0 8 3 4 x 6
——
5`0`0 4
Look at the 8 and say “9,”, adding the dot (`);
then say “10,” adding “half” the 3. At first it is better to look at the 8 and say
“9,” adding the dot, then say “1” for “half” of
3, then “10,”, and write the zero and “carry” dot.
When there is a dot and also a 5 to be added (because of
oddness), say “6” instead of “5” and then add the number
itself. This cuts out a step and is easy
to get used to. Try some problems of
your
own.
Multiplication by Seven —————–
=========================================
The Rule:
Double
the number and add half the neighbour;
add 5 if the number is odd.
Just like multiplication by six only different right? I will continue to present the solutions in
the same manner you should be solving them yourself. Practice solving them in this manner yourself
before looking at the solution as this helps to develop your concentration…
the secret to learning. Take the easy
example of 452 times 7.
0 4 5 2 x 7
——
4 –
(even), 4 (double 2), (no neighbour), 4.
0 4 5 2 x 7
——
`6 4 – 5
(odd), 15 (double 5), 1 (half of 2), 16.
0 4 5 2 x 7
——
`1`6 4 –
(even), 9 (double 4 plus the carry), 2 (half of 5), 11.
0 4 5 2 x 7
——
3`1`6 4 – (even), 1 (double zero plus the carry), 2
(half of 4), 3.
Like I said, easy.
Now you try the next one on your own:
0 8 3 1 5
x 7
———
0 4 1 2 0 5 9 3 x 7
—————
You didn’t forget that half of 1 is zero did you? The solutions are 58,205 and 28,844,151
aren’t they.. Of course… Now I’ll help you practice some of the mental
steps you have learned.
Some
practive in proper mental methods ——————
==========================================================
You’ve already practiced taking half of a number, but though
it should be simple for you, you haven’t practiced looking at a number (8 for
example) and saying its double… do try that now:
2,5,3,8,5,3,6,9,5,3,6,8,9,6,4,2,
1,0,3,5,8,6,5,4,2,6,8,9,7,4
Great! Now practice
looking a number, saying its double and adding the neighbour… this is how you
multiply by 12.
2 5 3 8
5 3 6 9 5 1
4 8 9 6 1 2
4 0 9 5
8 6 5 4 2 6
8 9 7 4 3 1
6 3 5 8
3 5 1 6
6 5 9 4 5 2
3 6
Almost done… now, using the same pairs of numbers practice
looking at the number, saying its double and adding half the neighbour. This is used in “times 7” problems.
Finally look at the following numbers, and practice saying
“5,” then say 5 plus double the number.
5, 3, 9, 7,
1, 9, 5, 7, 3, 1, 3, 5, 1, 9, 5, 3
There are all kinds of ways you can practice what you have learned
without having to actually sit at a table with pencil and paper… try multiplying the price of gas by
6, 7, 11, or 12 with the grease of your finger on the dust on your back window
while filling your tank… try doing so with your birthdate, ssn, home or work
phone numbers, or drivers liscence number in your head… while waiting at a
stoplight do so with liscense plate numbers…see if you can think of some
more… and practice, practice, practice.
enjoy…
MULTIPLICATION
(PART ONE)
How well do you know your
basic multiplication tables?
What would you think if I
told you that you could master your tables up to the ten times table in less
than 15 minutes? And your tables up to the twenty times table in less than half
an hour? You can, using the methods we explain in this chapter. We only assume
you know the two times table reasonably well, and that you can add and subtract
simple numbers.
Let’s go straight to the
method. Here is how you multiply numbers up to ten times ten.
We’ll take 7 X 8 as an
example.
Write 7 X 8 = down on a piece
of paper and draw a circle below each number to be multiplied.
Now go to the first number to be multiplied, 7. How many more do you need to
make 10? the answer is three. Write 3 in the circle below the 7. Now go to the
eight. What do we write in the circle below the eight? How many more to make
10? The answer is two. Write 2 in the circle below the eight. Your work should
look like this.
We now take away diagonally. Take either one of the circled numbers (3 or 2)
away from the number, not directly above, but diagonally above, or crossways.
In other words, you either take 3 from 8 or 2 from 7. Either way, the answer is
the same, 5. This is the first digit of your answer. You only take away one
time, so choose the subtraction you find easier. Now you multiply the numbers
in the circles. 3 times 2 is 6. This is the last digit of your answer. The
answer is 56. This is how the completed sum looks.
Let’s try another, 8 times 9.
How many more to make ten? The answer is 2 and 1. We write 2 and 1 in the
circles below the numbers. What do we do now? We take away diagonally. 8 – 1 =
7 or 9 – 2 = 7. 7 is the first digit of your answer. Write it down. Now
multiply the two circled numbers together. 2 X 1 = 2, the last digit of the
answer. The answer is 72. Isn’t that easy? Here are some problems to try by
yourself.
Do all of the problems, even if you know your tables well. This is the basic
strategy we will use for almost all of our multiplication.
How did you go? The answers
are 81, 64, 49, 63, 72, 54, 45 and 56. Isn’t this much easier than chanting
your tables for 15 minutes every school day?
Does this method work for
multiplying large numbers? It certainly does. Let’s try it for 96 times 97.
What do we take these numbers up to? How many more to make what? One hundred.
So we write 4 under 96 and 3 under 97.
What do we do now? We take
away diagonally. 96 minus 3 or 97 minus 4 equals 93. Write that down as the
first part of your answer. What do we do next? Multiply the numbers in the
circles. 4 times 3 equals 12. Write this down for the last part of the answer.
The full answer is 9,312.
Which method is easier, this
method or the method you learnt in school? This method, definitely, don’t you
agree. Here is my first law of mathematics:
The easier the method you
use, the faster you do the problem and the less likely you are to make a
mistake. Now, here are some more problems to do by yourself.
The answers are 9,216, 9,215, 9,025, 9,310, 9,212, 9,118, 9,016, 9,021, 7,350.
In the last problem, I hope you remembered to take 75 from a hundred, not
eighty. Did you get them all right? If you made a mistake, go back and find
where you went wrong and do it again.
MULTIPLICATION
(PART ONE)
How well do you know your
basic multiplication tables?
What would you think if I
told you that you could master your tables up to the ten times table in less
than 15 minutes? And your tables up to the twenty times table in less than half
an hour? You can, using the methods we explain in this chapter. We only assume
you know the two times table reasonably well, and that you can add and subtract
simple numbers.
Let’s go straight to the
method. Here is how you multiply numbers up to ten times ten.
We’ll take 7 X 8 as an
example.
Write 7 X 8 = down on a piece
of paper and draw a circle below each number to be multiplied.
Now go to the first number to be multiplied, 7. How many more do you need to
make 10? the answer is three. Write 3 in the circle below the 7. Now go to the
eight. What do we write in the circle below the eight? How many more to make
10? The answer is two. Write 2 in the circle below the eight. Your work should
look like this.
We now take away diagonally. Take either one of the circled numbers (3 or 2)
away from the number, not directly above, but diagonally above, or crossways.
In other words, you either take 3 from 8 or 2 from 7. Either way, the answer is
the same, 5. This is the first digit of your answer. You only take away one
time, so choose the subtraction you find easier. Now you multiply the numbers
in the circles. 3 times 2 is 6. This is the last digit of your answer. The
answer is 56. This is how the completed sum looks.
Let’s try another, 8 times 9.
How many more to make ten? The answer is 2 and 1. We write 2 and 1 in the
circles below the numbers. What do we do now? We take away diagonally. 8 – 1 =
7 or 9 – 2 = 7. 7 is the first digit of your answer. Write it down. Now
multiply the two circled numbers together. 2 X 1 = 2, the last digit of the
answer. The answer is 72. Isn’t that easy? Here are some problems to try by
yourself.
Do all of the problems, even if you know your tables well. This is the basic
strategy we will use for almost all of our multiplication.
How did you go? The answers
are 81, 64, 49, 63, 72, 54, 45 and 56. Isn’t this much easier than chanting
your tables for 15 minutes every school day?
Does this method work for
multiplying large numbers? It certainly does. Let’s try it for 96 times 97.
What do we take these numbers up to? How many more to make what? One hundred.
So we write 4 under 96 and 3 under 97.
What do we do now? We take
away diagonally. 96 minus 3 or 97 minus 4 equals 93. Write that down as the
first part of your answer. What do we do next? Multiply the numbers in the
circles. 4 times 3 equals 12. Write this down for the last part of the answer.
The full answer is 9,312.
Which method is easier, this
method or the method you learnt in school? This method, definitely, don’t you
agree. Here is my first law of mathematics:
The easier the method you
use, the faster you do the problem and the less likely you are to make a
mistake. Now, here are some more problems to do by yourself.
The answers are 9,216, 9,215, 9,025, 9,310, 9,212, 9,118, 9,016, 9,021, 7,350.
In the last problem, I hope you remembered to take 75 from a hundred, not
eighty. Did you get them all right? If you made a mistake, go back and find
where you went wrong and do it again.