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ThePerfectHacker · #3 November 7th, 2006, 10:56 AM

Quote:

Originally Posted by Judi

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.