The power of 0

1. Dec 21, 2005

TSN79

We all know that a number to the power of 0 equals 1. Bit I heard this is only a definition, iow something that has just been agreed on. But how can such a thing be done? If one had agreed that it would equal 0 instead, I suppose many mathematical proofs and stuff would be different, things that can't be done suddenly would be possible. Is it really the case that this is just an agreement?

2. Dec 21, 2005

LeonhardEuler

It is according to the definition of raising a number to a power that a nonzero number to the zero power is equal to one, just like it is according to this definition that a x to the second power is x times x. It is really not an unnatural definition. We know that $x^n\times x^m=x^{m+n}$. If we want this to be true then $x^n\times x^0=x^n \rightarrow x^0=1$. Alternitively, powers can be defined by series. In this case $b^x=e^{x\ln{b}}$ and
$$e^x=1+\sum_{n=1}^{\infty}\frac{x^n}{n!}$$
So
$$b^x=e^{0\ln{b}}=1+\sum_{n=1}^{\infty}\frac{0^n}{n!}=1$$

Last edited: Dec 21, 2005
3. Dec 21, 2005

HallsofIvy

Staff Emeritus
Drop the word "just" and I will agree with you. In mathematics, everything in mathematics is a definition but not "just" a definition! As Leonhard Euler said, it is not "an unnatural definition".

From the simplest case, xn with n a positive integer, where we can think of xn as meaning "x multiplied by itself n times" (which is, after all, "just" a definition), we find that xnxm= xn+m because "n times" followed by "m times" is the same as "n+ m times".

If we want that very nice property to continue to be true even if m is 0, we must have xnx0= xn+ 0. But n+0= 0 so we have xnx0= xn. As long as x is not 0, we can divide both sides by xn and get x0= 1.

That is why we define x0 to be 1 (and only define it for x not equal to 0).

4. Dec 22, 2005

Buckshot23

Excellent explanation thank you. However N+0=N not N+0=0.

5. Dec 22, 2005

debeng

i was confused if 1^0 = 1 and 1^n = 1 (n= 1,2,3.....)
then o = 1,2,3........

6. Dec 22, 2005

Zurtex

That's like saying 0*0 = 0 and 0*n = 0 (n= 1,2,3...) then 0 = 1,2,3...

Think about why that isn't true for a moment.

7. Dec 22, 2005

StatusX

This is the example I usually think of when this comes up: 1=xn/xn=xn-n=x0.

8. Dec 22, 2005

Buckshot23

That works also.

9. Dec 23, 2005

xcoder66@yahoo.com

Another way I like to think of it as a notation with xn, where you factor out an x each time n decreases. If x = 3,

x3 = 27

if you factor out three 27/3

x2 = 9

factor out three again 9/3

x1 = 3

and factoring three again 3/3

x0 = 1

I really had trouble visualing "raised to a zero power" and "negative exponents" until I "just" thought of it as a notation.