I’m not claiming to have invented “speed math,”
but I did discover it on my own. This was during my senior year of high school. I wasn’t looking for “speed math.”
My
initial curiosity had to do with doubles and squares. Two times two and two squared are both equal to four. But there is no
other whole number whose double and square are equal. Also, the gap widens steadily for larger whole numbers. Why? Is there
a pattern that describes this widening? A principle that explains it?
I set up three columns (see A, B, and C below)
and started comparing the figures. Immediately I noticed that by adding the numbers in columns B and C for a given line, and
then adding one more, you got the number in column B for the next line. In other words (see columns E and G) that x2
+ 2x + 1 = (x + 1)2.
A B
C D
E F
G
x
x2 2x
x2 + 2x x2 + 2x + 1 (x + 1)
(x + 1)2
1 1
2 3
4 2
4
2
4 4
8 9
3 9
3
9 6
15 16
4 16
4
16 8
24 25
5 25
5
25 10
35 36
6 36
6
36 12
48 49
7 49
What value
is there in this knowledge? It occurred to me that if you know the square of one number – say, 20 (202 =
400) – you can easily find the square of the next higher number (21) by a process of addition: 202 + 2 (20)
+ 1 = 441. This is so easy, you can do it in your head. Another example: 1012 = 1002 + 2 (100) + 1 =
10,000 + 200 + 1 = 10,201. Simplicity itself.
Now, the pathway by which I arrived at this method of
mental multiplication is known as induction. I generalized from a number of particular cases and expressed it by
an equation that fits all of them. I could instead have arrived at this equation much more quickly, simply by expanding (x
+ 1)2, as follows:
x + 1
x
+ 1
x + 1
x2
+ x
x2 +
2x + 1
This
might have been sufficient for some people. Those who are better abstract thinkers than I might have grasped this equation’s
usefulness (for “speed math”) without needing to first go through the process of induction I carried out above.
However,
it’s not likely they could have seen its applicability as a shortcut for multiplication without having listed several
particular instances – i.e., to have selected several values for x and plugged them into the equation. Say, x = 10:
112 = (10 + 1)2 = 102 + 2 (10) + 1 = 100 + 20
+ 1 = 121. Or x = 25: 262 = (25 + 1)2 = 252 + 2 (25) + 1 = 625 + 50 + 1 = 676. Grasping this is a process
of deduction: applying a general principle (equation, in this case) to specific instances.
We can generalize
from this equation and get a broader useful speed math technique. If we want to add not 1, but any number a to x and square
that sum, all we need do is square x, add twice the produce of x and a, and add the square of a. In other words, (x + a)2
= x2 + 2ax + a2. For instance, 1022 = (100+2)2 + 2(100) + 22 = 10,000
+ 400 + 4 = 10,404. As an exercise, I invite the reader to derive this equation both inductively (by inspection from a display
of rows and columns of numbers similar to the one above) and deductively (by expanding the expression (x + a)2).
Another
useful equation is (x – 1)2 = x2 - 2x + 1. Grasping this inductively requires a similar insight
to that discussed above. An example of its usefulness: 992 = (100-1)2 = 1002 - 2(100) + 1 = 10,000 - 200 + 1 = 9801. And generalizing,
we get (x-a)2 = x2 – 2xa + a2. For example, 972 = (100-3)2 =
1002 - 2(100)(3) + 32 = 10,000 – 600 +
9 = 9409. Again, it is instructive to derive these equations both inductively by inspecting number arrays and deductively
by expansion of (x-1)2 and (x-a)2.
Yet another interesting avenue to explore is
to compare squares with the products of numbers equally less than and more than the squared numbers. For instance, 1002 = 10,000, while 99(101) = 9999 = 10,000 –
1 = 1002 – 12, and 98(102) = 9996 = 10,000 – 4 = 1002 – 22, and 97(103) = 9991 = 10,000 – 9 = 1002 – 32. In general, (100 – a)(100 + a) = 1002 – a2. Even more generally, (x – a)(x + a) = x2 – a2.
This equation has two useful
applications. First, you can quickly determine, for instance, that 23(27) = 252 – 22 = 625 – 4 = 621. Secondly, you can turn the equation around: x2 = (x – a)(x + a) + a2 and determine things like 952 = (95 – 5)(95 + 5) + 52 = 90(100) + 25 = 9025. Cool, huh! Once again,
I advise the reader to derive the equation inductively by inspecting number arrays and deductively by expanding (x –
a)(x + a).
Deriving
each of these equations deductively is an integral part of the standard high school algebra course, and they follow from a
straightforward adaptation of the rules for long multiplication. But it takes induction or higher-level, speculative deduction
to reveal their usefulness for multiplication, and neither process is given a sufficient focus in math instruction.
That
is, an important part of the practical usefulness of algebra – facilitating arithmetic operations – is simply
overlooked, because students are not encouraged to explore for (induction) or to speculate about (higher-level deduction)
the results they mechanically derive.
[Further references: many more "speed math" techniques
can be found in Short-Cut Math by Gerard W. Kelly (Sterling, 1969; Dover repub., 1984). The importance of inductive
"exploration" is well discussed by George Poly in Induction and Analogy in Mathematics (Princeton
University Press, 1954).]