explain why any number (except 0) to the zero power always equals 1

Question

nikvist · Answer

$$a^0=a^{b-b}=a^b\cdot a^{-b}=a^b\cdot\frac{1}{a^b}=1$$

nowhereman · Answer

Because 1 is the neutral element of the multiplicative group of real numbers except 0.

anonymous · Answer

please use smaller words "nowhereman" i am going to write this on thetest and i cant sound overlly smart

anonymous · Answer

nikvist's algebraic interpretation is correct.

nowhereman · Answer

The above calculation is quite good already. You could also write: $$a^0 \cdot a^n = a^{0+n} = a^n$$ so $$a^0 = 1$$

anonymous · Answer

explain why 0 to the 0 power is not one

nowhereman · Answer

Because it is undefined. For the exponential function 0^x to be continuous it must be 1 but for x^0 to be continuous it must be 0. So you can't define it consistently (e.g. so that all power-rules hold)

anonymous · Answer

Or you can go back to nikvist's explanation and realize that to get $$0^0$$ you'd have to divide by 0 which isn't defined.

anonymous · Answer

"The choice whether to define 0^0 is based on convenience, not on correctness"

nowhereman · Answer

Well, what is correctness anyway. After all you choose which axioms you rely on.

anonymous · Answer

Don't shoot the messenger :(

nowhereman · Answer

Whos quote was it then?

anonymous · Answer

Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999.