Zero and the Commutative Property of Multiplication

Date: 04/18/2003 at 05:25:29
From: Ong Jia Jun Edmund
Subject: Zero

If we take 0 x 1, we get 0, and if we take 0 x 2, we still get 0, for 
it is still 0 + 0 + 0. But if we do it the other way round, why must 
it still be 0, if any number to the power of 0 = 1? 

Any number to the power of 0 = 1, but why will any number times 0 
always = 0, and not at least 1?

Times means adding a few times, like 1 x 2 = 1 + 1. But with 2 x 0, 
why can't it just stay as at least 1, just like when we raise a 
number to the power of 0?

Date: 04/18/2003 at 11:56:46
From: Doctor Rick
Subject: Re: Zero

Hi, thanks for writing to Ask Dr. Math.

Are you letting new knowledge about powers push out what you knew 
before about multiplication? Multiplication has a commutative 
property: if you switch the order of the factors, the product doesn't 
change. If 0 times anything = 0, then anything times 0 also equals 
ZERO.

I can guess what you're thinking. You think of a number to a power 
(such as 2^3) as the number times itself as many times as the power:

  2^3 = 2*2*2 = 8

Because of this, you expect that 2^0 would be 0 because you're not 
multiplying anything, and then you're told no, the answer is 1.

Then you wonder if what you thought was obvious about multiplication 
could be just as wrong. Multiplying 2 by 3 means adding 2 three times:

  2*3 = 2+2+2 = 6

In the same way you expect that 2*0 would be 0 because you're not 
adding anything. But if your expection about 2^0 was wrong, how can 
you be sure about 2*0?

Here is the answer. Don't think of 2^0 as doing NO multiplications, 
and don't think of 2*0 as doing NO additions. In each case we have to 
start somewhere. 

When we add, we start at zero (the "additive identity". From this 
starting point, to calculate 2*3, we add 2 three times:

  2*3 = 0 + 2 + 2 + 2 = 6

To calculate 2*0, we add 2 no times, so all we have is:

  2*0 = 0             = 0

When we multiply, we start at one (the "multiplicative identity"). 
From this starting point, to calculate 2^3, we multiply 2 three times:

  2^3 = 1 * 2 * 2 * 2 = 8

To calculate 2^0, we multiply by 2 no times, so all we have is:

  2^0 = 1             = 1

I've used the same method to show that 2^0 = 1 (contrary to our first 
expectation) and that 2*0 = 0 (confirming what we thought). So don't 
worry, things work the way you've been taught.

I hope this helps. I'll be happy to discuss it further if you have 
questions about what I've said. We like to hear from thinkers who 
want to understand WHY things work!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/