Prove the Commutative Law

Date: 4/1/96 at 5:23:36
From: Anonymous
Subject: The Commutative Law

Hi Dr. Math!

I'm a pupil in the 10th grade here in Israel.  I have a question 
that may sound very silly: Can anyone PROVE the Commutative law?

Is a * b = b * a ?

It sounds silly, but a * b means a TIMES b, which is b+b+b... a 
times, and b * a means a+a+a... b times. How can you prove it's 
the same result?

I have no clue how even to start solving this question. It seems 
so obvious, but so difficult to solve.

Hope you can help me,
Lior Barenboim

Date: 5/30/96 at 11:39:26
From: Doctor Alain
Subject: Re: The Commutative Law

Your question is not silly at all. Many people know things that 
they can't prove, and this sometimes leads to many people 
"knowing" things which are false.

Commutativity of multiplication can be proved by using the fact 
that a * b is the area of a rectangle whose sides are of length 
a and b.

3 * 4 = 12

So 3 rows of 4 Xs have 12 Xs. But obviously there is the same 
number of Xs in 4 columns of 3 Xs, so 4 * 3 = 12 also.

 X X X X
 X X X X
 X X X X

So a * b is equal to the number of Xs in "a" rows containing 
"b" Xs each.  It's also equal to the number of Xs in "b" columns 
containing "a" Xs each.

-Doctor Alain,  The Math Forum