Zero as an Exponent

Date: Sat, 22 Jun 1996 11:30:52 -0400 (EDT) 
From: Anonymous
Subject: Zero as an exponent

My 7th grade son has a 5^0 question on a study guide and didn't 
know the answer. I told him that I thought that the answer was 1 
based on my math training in years gone by, but I didn't know 
why. I searched the net and found your page which explained it 
well enough for me to understand.

I went on to explain it to him in this way: 

5^1=(5*1)/1, 5^2 = (5*5*1)/1, 5^(-1)=1/(5*1), 5^(-2)=1/(5*5*1). 

Following this flow, 5^0 would be viewed as 1/1 with no 5's. Then 
of course 1/1 = 1 or 5^0 = 1. 

Does this make sense? Have I got it figured correctly? 

- Rick Humphreys

Date: Sat, 22 Jun 1996 17:19:00 -0400 (EDT) 
From: Dr. Ceeks
Subject: Re: Zero as an exponent

Hi,

I do not think your answer is the best answer because it doesn't 
arise out of any natural sequence of ideas. 

I think this is more natural:

First, the exponential was defined as a notational method to 
represent the process of multiplying a given number over and over. 

Thus, 5^n = 5 times 5 times 5 times 5, n times, where n is a 
positive integer. It then follows that 5^(a+b) = 5^a 5^b. 

In mathematics, it often happens that one would like to extend the 
definition of something. How can we extend the definition of the 
exponent to all the integers? What property of the exponential can 
guide us beyond the positive integers? 

We have the beautiful law that 5^(a+b) = 5^a 5^b. Is it possible to 
extend the definition so as to retain the fundamental property that 
5^(a+b) = 5^a 5^b? 

The answer is yes...it can be extended, and it can be extended in 
only one way. 

First, 5^(-1) must be 1/5, because we demand that 5^(n-1)=5^n 5^(-
1). But then we see that 5^0 = 5^(1-1) = 5^1 5^(-1) = 5 * 1/5 = 1.

-Doctor Ceeks, The Math Forum