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Date: 09/26/2000 at 11:13:38
From: Dina Zeliger
Subject: Paradox in the zero power

Hi,

My question is not about solving a question, but more of proving the

I know that n^0 equals one, and there are a lot of ways to prove it.
I understand those ways of proving the problem, but it doesn't seem
logical, and I don't think it fits the definition of power.

A power is the product of a number multiplied by itself a given
number of times. i.e. n^k = n*n*...*n (k times).

This definition in right when k > 0. But, if k = 0, why does the
product of multiplying a number by itself zero times equal 1?

Zero is basically nothing. Its value is not a quantity, i.e. zero
times something is nothing - zero. in that case, how is it possible
that n^0 = 1? Multiplying n by itself zero times equals nothing, since
you don't multiply it at all.

That theory also suggests that 0^0 = 0, and that this expression is
not indefinite. If (n^0 = 0) and (0^n = 0), then (0^0 = 0).

I hope you can help me solve my problem. It is very disturbing; I
have been thinking about it for a week now.

Thank you,
Dina Zeliger
```

```
Date: 09/26/2000 at 14:32:29
From: Doctor Ian
Subject: Re: Paradox in the zero power

Hi Dina,

If you haven't yet seen the explanations in the Dr. Math FAQ, you
should take a look at them:

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

In the end, you need to understand that the interpretation of n^0
isn't something that can be (or needs to be) proven.  We're free to
choose what we want it to be, so long as that choice is consistent

Basically, n^0 is _defined_ to be 1 for the simple reason that any
_other_ definition would cause things to break. For example, if it's
going to be true that

n^a * n^b = n^(a+b)

then n^0 can't be anything _but_ 1. Choosing any other value would
cause this rule (and others like it) to stop working.

It's true that this approach leads to some pretty goofy definitions,
such as

0! = 1

and

0^0 = 1

But as Winston Churchill once said, democracy is the worst possible
form of government... except for all the others. You could make a
similar comment about these definitions.  They are the worst possible
choices... except for all the others.

or anything else.

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

```
Date: 09/26/2000 at 17:01:42
From: Doctor Peterson
Subject: Re: Paradox in the zero power

Hi, Dina.

I'd like to add something to what Dr. Ian wrote. He explained this the
same way I would, in terms of defining the zeroth power to avoid
breaking the rules we already have. I like to supplement that with an
explanation which, while not mathematically profound, feels more
emotionally satisfying, which may be what you need. Your complaint is
that it just seems wrong that multiplying something zero times should
give a non-zero result. Well, here's a way to think about that
multiplication that makes this seem a little more reasonable:

First, let's get some practice by trying to explain why anything TIMES
zero is zero. We think of multiplication as repeated addition. If you
add together 5 3's, you find that

5 * 3 = 3 + 3 + 3 + 3 + 3 = 15
\_______________/
5

Notice that I'm not really "adding 3 to itself 5 times", as is
together 1 * 3 uses NO additions. How can you use this concept to
explain that 0 * 3 = 0? Would you have -1 additions??

5 * 3 = 0 + 3 + 3 + 3 + 3 + 3 = 15
\_________________/
5

This is neater; now I can actually say "the product is the result of
adding a number TO ZERO some number of times." And by starting with
zero, we can better understand why 0 * 3 = 0; you just start with 0

0 * 3 = 0     = 0
\_/
0

I think you are sort of assuming that exponents work the same way: you
start with 0, and if you multiply nothing, you end up with zero. But
you get

3 ^ 5 = 3 * 3 * 3 * 3 * 3 = 243
\_______________/
5

But if we started with zero, and multiplied it by 3 five times, we
would end up with zero. Instead, when we multiply we have to start
with 1, and multiply that by 3 five times:

3 ^ 5 = 1 * 3 * 3 * 3 * 3 * 3 = 243
\_________________/
5

So three to the fifth power is the result of multiplying 1 by 3 five
times. That's a cleaner definition than "multiplying 3 by itself five
times."

Now, what do you get if you start with 1 and don't multiply it by
anything? You get 1, which must be the value of 3^0!

3 ^ 0 = 1     = 1
\_/
0

So the choice of 1 as the definition of a number raised to the zero
power really fits well with the cleanest definition we can come up
with of exponentiation in general. Again, this isn't a "proof," but a
further reason to be satisfied with the definition.

Now, what do you get if you multiply 1 by 3 -2 times? If division is
the opposite of multiplication, then maybe you would start with 1 and
divide it by 3 twice.

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