The Power of 0: Is It Just An Agreement?

[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?

 

[LeonhardEuler]

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?

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

 

[HallsofIvy]

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?

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 xⁿx^m= x^n+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= x^{n+ 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).

 

[Buckshot23]

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 xⁿx^m= x^n+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= x^{n+ 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).

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

 

[debeng]

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 xⁿx^m= x^n+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= x^{n+ 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).

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

 

[Zurtex]

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

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.

 

[StatusX]

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

 

[Buckshot23]

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

That works also.

 

[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.