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### n^0 Power = 1: Defined or Proved?

```Date: 04/18/2003 at 08:28:14
From: Tam
Subject: Number

I have been wondering if a^0 is DEFINED to be 1 or is PROVED to be 1.

I think that we define a^0 = 1, but one of my friends said that a^0
is proved to be 1.

I think that we have to define it first, because as we define a^n =
a.a.a....a (n times) with n a whole number, this definition will not
work for n=0. Therefore, we must define a^0 in another way, which is
a^0 = 1 (with a not equal to 0). As for the reason why we define
a^0 = 1 but not any other number, it is just to keep the fomula
a^m.a^n = a^(m+n), and (a^m)^n = a^(m.n)

But as my friend said, we define a^n first in the way I used above,
but n can be any number including 0. As we can't figure out what a^0
is from this definition, (in other words, it is meaningless to say
a^0 is defined to be a.a...a [0 times]) we have to calculate it, and
he proved to me that 1 = a^n/a^n = a^(n-n) = a^0. Therefore he said
a^0 is PROVED to be 1.

I think before we talk about something, we must first define it, and
the definition must show what that something is (and we don't need
to go through some theorems to figure out or prove what we have
already defined, as with my friend's argument). I think this is just
normal mathematical logic that we have to define something clearly
first before using it.

Am I right?
```

```
Date: 04/18/2003 at 09:27:27
From: Doctor Peterson
Subject: Re: Number

Hi, Tam.

You are correct, but your friend is not too far off. Since we have no
preexisting definition of a^0, we have to choose a definition for it;
but we can prove that this particular definition is the only one that
retains all the important properties of exponents. The only thing
wrong with your friend's approach is that you can't make a definition
that supposedly includes the case n=0, but then say that it can't be
applied in that case. However, if he said that exponentiation was
defined in general by its properties, rather than as repeated
multiplication, then he could prove that a^0 = 1 for all a other than
0 strictly from those properties (which would be taken as axioms).

A definition with a reason given (as in your presentation) is not too
far from a proof; and a proof based on a non-standard definition (as
in my version of your friend's) is not far from a definition.

We discuss these ideas in the Dr. Math FAQ:

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

But it is actually possible to start with a definition that includes
0 (but not negative exponents). Just say

a^n = 1 * a * ... * a
\___________/
n multiplications

and it makes sense to say that a^0 is 1, not multiplied by anything.
You still have to extend this definition to negative, and then
rational, and then real, exponents.

So ultimately, which of you is right depends on what axiomatic system
would say that we are making an extended definition, and proving it
to be consistent.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
Middle School Exponents

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