zero to the power zero is one

0^0 = x implies log (0^0)=log x

but log (0^0)=0*log(0)=0

so log x=0
so x=1

You can't prove it. [math] 0^0 [/math] is not defined. However [math]\displaystyle \lim_{x\rightarrow 0} x^x [/math] can be shown to be 1.

>>8769169
this implies, first, that you can take the log of 0^0, second that 0*log0=0, assuming, again, that log0 is defined.

This is the proper way:

let x>0, x^x=y>0. Then log(x^x)=log(y). So xlogx=logy. Take the limit x to 0. Since polynomial decrease is faster than logarithmic decrease (can show this via Taylor series, well known result), then the limit is 0. Hence logy=0 so e^0=y=1.
Q.E.D.

>>8769178
0^0 is defined as 1.

>>8769185
according to whom?
Certainly not anyone who has actually thought about it for more than a fraction of a second

Is there really no value that can be assigned to division by 0 that has consistent properties?

>>8769187
It's the convention used by mathematicians.

>>8769192
Are you sure you're not mixing it up with factorials?

It's not defined, but most mathematicians define it such that x^0=1 for all x.

It's not defined because x^0 and 0^x approach different values as x-> 0.

>>8769208
No it is the convention, but for no good reason it seems. Mathematicians are aware that it's undefined, but they want to keep their math pretty.

http://www.math.utah.edu/~pa/math/0to0.html

File: maclaurin_ex03[1].png (3KB, 462x134px) Image search: [Google]

3KB, 462x134px

>>8769208
0^0 needs to be 1 or taylor series' doesn't work when x=0.

>>8769228
>http://www.math.utah.edu/~pa/math/0to0.html
Nowhere there does it say that it is the convention.

It shows the two possible values but because those two are contradictory, none of them can really be accepted.

>>8769220
One seldom needs limit 0^x but limit x^0 is widely used so we can 'define' 0^0=1

>>8769241
>>8769242

Likewise for the Binomial Theorem. If we assume x=1 and y=0, it will only work if we assume 0^0=1.

[math] (1+0)^n = 1 = \sum_{k=0}^{n} \binom {n} {k}1^k0^{n-k} = \binom {n} {n}1^n0^0 = 1 [/math]

>>8769250
Those are good points but if this was convention I would have heard of it before this thread.

>>8769245
As Knuth said, "the function 0^x is quite unimportant."

An empty product is always the neutral element of multiplication.

>>8769253
>if this was convention I would have heard of it before this thread
It never arose as a problem for the work you had to do, and that's because teachers avoid those types of problems.