## A 26th way to multiply. Sort of.

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I’m moving offices:   just next door, but I have about 20% less shelf space so I figured it was a good opportunity to see if there were any books that I didn’t really use (answer:  yes).  In looking through one (Man and Number by Donald Smeltzer, which is a distracting title even though I can see that this book was published 50 years ago) I found another way to multiply!  It’s not anything dramatic, but was apparently described by Nichomachus of Alexandria over 1900 years ago in his Introduction to Arithmetic.  It uses the following fact:

$x^2 - y^2 = (x+y) \cdot (x-y)$

Here’s the method:  suppose first that a and b are both odd or both even.  Let x be the average of the two numbers (so x is a whole number, because their sum must be even) and let y be the positive difference between x and a or b.  The radius, as it were.  Then:

$a \cdot b = (x+y) \cdot (x-y) = x^2 - y^2$

The example given in the book is:

$24 \cdot 14 = (19+5) \cdot (19-5) = 19^2 - 5^2 = 361-25=336$

You still have to know your perfect squares, but if you happen to have a table of squares like the Babylonians, that’s no trouble at all.    (Indeed, this is basically Formula (4) here,  so I don’t know if this should count at all.  But I like having an actual citation for the method.)

But what if you have an even and an odd number?  Never fear, just ignore that pesky odd bit and add it on at the end.  For example, if you have 24·15, you know that this is 24·14 plus an additional 24.  So you find 24·14 as above, and add 24 to get 360.

### 2 Responses to “A 26th way to multiply. Sort of.”

1. Samantha Says:

That is a very cute method! Certainly it’s easier to have a list of perfect squares than to have a 100×100 multiplications table.

I wonder if that is idea is part of what sparked Fermat-Euler’s method of the sum of two squares. They said if p is prime and p = 1 mod 4 (or 2) then p is expressible as p = a^2 + b^2 for some integers a, b. The proof is much more complicated than the proof for this formula is and requires imaginary numbers, but that makes sense because we are dealing with addition inside the squares instead of subtraction.

Nice explanation!

2. watchmath Says:

I tutor an 8th grader by using a Singapore Math book. Here is an exercise from that book that need to solved by using the method that you mentioned on your post.

Evaluate
$\displaystyle \frac{236}{236^2-(238\times 234)}$