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Paradox in the Zero Power

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


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:   

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 
with the choices we've already made. 

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


  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.  

I hope this helps.  Write back if you have more questions, about this 
or anything else. 

- Doctor Ian, The Math Forum   

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

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

Instead, I like to think about what you had BEFORE you started adding; 
you had nothing to start with, then added 3 to it five times:

    5 * 3 = 0 + 3 + 3 + 3 + 3 + 3 = 15

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 
and add nothing.

    0 * 3 = 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 
multiplication can't start with zero! If you multiply together 5 3's, 
you get

    3 ^ 5 = 3 * 3 * 3 * 3 * 3 = 243

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

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 

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

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.

Does that help you feel better about this?

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Number Theory

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