power of zero

Judi · #1 $Old$ November 7th, 2006, 11:09 AM

Why any number to zero power is 1?
I know one of the reasons but, I want to know more of ways to look at it

Thanks,

CaptainBlack · #2 $Old$ November 7th, 2006, 11:21 AM

Quote:

Originally Posted by Judi $View Post$

Why any number to zero power is 1?
I know one of the reasons but, I want to know more of ways to look at it

Thanks,

Except for $0^0$ , which usually leads to a fight (between those
who think it is undefined and those who want it defined as - what do they
want it defined as - I forget).

RonL

ThePerfectHacker · #3 $Old$ November 7th, 2006, 11:56 AM

Quote:

Originally Posted by Judi $View Post$

Why any number to zero power is 1?
I know one of the reasons but, I want to know more of ways to look at it

Thanks,

If $a$ is a number.
And $n,m$ positive integers.
Then,
$a^n\cdot a^m=\underbrace{a\cdot...\cdot a}_n\cdot \underbrace{a\cdot...\cdot a}_m=\underbrace{a\cdot ... \cdot a}_{n+m}=a^{n+m}$

Now, mathemations define for $a\not = 0$
$a^{-n}=\frac{1}{a^n}$
Therefore,
$a^n\cdot a^{-n}=\frac{a^n}{a^n}=1$
Now if we use the exponential addition law above (eventhough we only proved it for positive integers it will still work because of the way we defined exponents for negatives)
$a^{n-n}=a^0$
Which we take to mean as,
$a^0=1$
Now, in this demonstration I said $a\not = 0$ .
In fact, some people define it and some leave it undefined.

I like to define $0^0=1$ because Calculus there is something called "Power Series" and the pattern looks much nicer when you you let power of zero be 1.

topsquark · #4 $Old$ November 7th, 2006, 12:51 PM

Similar to TPH's post, you can look at it in terms of a series:
$a^4,a^3,a^2,a^1,a^0$ ( $a \neq 0$ ).

Each term in the series is one power greater than the one before. ie. we are dividing each term by "a" to get the next term. So what should the final term in the series be? It should be the term before it "a" divided by "a." What's a/a? It's 1. So $a^0=1$ .

Note: This is a purely logical argument and in many cases mathematical logic does not always follow from this. (Because of various convenient definitions that Mathemeticians come up with.) In this case it not only works, but gives a way to remember it.

-Dan