Commutative Property of Multiplication

[jpanknin]

I'm trying to get a better understanding of the why's behind the commutative property of multiplication. I came across the following paragraph in a discussion about the commutative property and it does seem like a miracle (as stated in the excerpt below). I get the commutative property of addition pretty clearly, but multiplication still seems vague to me.

After all, if you decide to multiply together two numbers such as 395 and 428 using long multiplication, then the calculations you do will depend very much on whether you work out 395 428s or 428 395s. In the first case you will find yourself adding 128400, 38520 and 2140 while in the second you will add 158000, 7900 and 3160. Why isn't it a miracle that both triples of numbers add to 169060?

I've seen some proofs from analysis (not there yet) and the geometric explanation with rectangles (and smaller numbers than those used in the excerpt above), but it seems like there should be a more intuitive way of looking at it. Why does 3 x 5 or 3 + 3 + 3 + 3 + 3 = 5 + 5 + 5 in a more general way?

 

[fresh_42]

You cannot "prove" that multiplication is commutative. It is an axiom, a kind of definition since we want to have width times length to be the same area as length times width. There are other multiplications (other than ordinary numbers, e.g. functions) that are not commutative.

Your example says: five rows of three things is the same as three rows of five things. We can rearrange those things without adding one or taking away one thing, or simply look at it from another side.

 

[jpanknin]

You cannot "prove" that multiplication is commutative. It is an axiom, a kind of definition since we want to have width times length to be the same area as length times width. There are other multiplications (other than ordinary numbers, e.g. functions) that are not commutative.

Your example says: five rows of three things is the same as three rows of five things. We can rearrange those things without adding one or taking away one thing, or simply look at it from another side.

Ok, then if we restrict the terms to something like positive integers is there is a proof? Is there a way to show the example from the excerpt in my first post, for example? That ab = ba?

 

[fresh_42]

It all depends on what multiplication is. If you start with natural numbers, then the picture of arranging things works. Say we have [imath] a\cdot b [/imath] many apples arranged in [imath] a [/imath] many rows, each row containing [imath] b [/imath] many apples. Then turn the table by 90 degrees and you get [imath] b [/imath] many rows, each row containing [imath] a [/imath] many apples. Since we only moved the table and did not touch the apples, we still have [imath] a\cdot b =b\cdot a[/imath] many apples total.

 

[eipi45]

 

[Agent Smith]

Here's a little proof:

The reciprocal of [imath]a[/imath] is [imath]1 \div a[/imath]
The reciprocal of [imath]\frac{a}{b}[/imath] is [imath]1 \div \frac{a}{b} = 1 \times \frac{b}{a} = \frac{b}{a}[/imath]

A number multiplied by its reciprocal gives 1.

So [imath]\frac{a}{b} \times \frac{b}{a} = \frac{ab}{ba} = 1[/imath] (by fraction multiplication rule)
[imath]\frac{ab}{ba} = 1 \implies ab = ba[/imath]