The Zero Power of Two

Date: 12/10/98 at 10:51:56
From: David Burns
Subject: Exponents/powers of two

Dear Dr. Math,

In fifth grade we've learned that 2 to the third power = 8, two squared 
= 4, 2 to the first power = 2, and 2 to the zero power = 1. Could you 
please explain how 2 the zero power = 1 because I'm having trouble 
understanding this. For example, 2 cubed means that you multiply 2 by 
itself 3 times. How do you multiply 2 by itself 0 times in 2 to the 
zero power?

I understand the pattern of 2 cubed, squared, to the first power, and 
to the zero power (8, 4, 2, 1), but I'm still having trouble with this 
idea.

Could you help? I looked through your elementary archives and found 
nothing on this subject.

David Burns

Date: 12/10/98 at 13:01:51
From: Doctor Rick
Subject: Re: Exponents/powers of two

Hi, David. Good question! Actually we do have material on why a number 
to the zero power is 1, but I'm not surprised that it isn't in the 
Elementary Archives. Questions about why numbers behave as they do are 
best answered when you get to study algebra.

Here is our FAQ (Frequently Asked Questions) page about this question:

  http://mathforum.org/dr.math/faq/faq.number.to.0power.html   

You will see some things there that you won't understand, but some of 
it may help you convince yourself.

You know, there was a time when the only numbers people knew were the
counting numbers 1, 2, 3, .... Zero hadn't been invented yet, so nobody 
could ask your question. Then zero and negative numbers were invented, 
and fractions and decimals, and even more that you probably haven't 
heard of yet.

Each time new numbers were invented, mathematicians had to figure out 
how those numbers behave. You don't want to have a whole new set of 
rules for the new numbers - you want them to follow the same old rules, 
but to take them where no number has gone before. 

This is what happened with powers. When zero is added to the counting
numbers, you need to figure out what 2^0 (2 to the 0 power) is. The old 
definition doesn't help you, because as you say, multiplying zero 2's 
together doesn't make sense. But you want powers to keep working the 
same way they always did, and one rule is this: if you divide a number 
to a power by the same number to a different power, the answer is the 
same number raised to the difference of the first two powers. 
For example,

     3
    2      (3-2)    1
   ---- = 2      = 2
     2
    2

What happens when the powers in the numerator and denominator are the 
same?

     3
    2      (3-3)    0
   ---- = 2      = 2
     3
    2

But you know that 8/8 = 1. So 2^0 must equal 1.

You can do the same sort of thing to figure out what 2^(-1) should be, 
or what 2^(1/2) should be.

I hope this helps you. Keep asking those "why" questions, and you will 
be all set for algebra, and more!

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