Is x^pi Real for Negative Values of x?

[cstvlr]

The plot of x^(pi) looks like an odd function, does that make pi an odd number?

http://www.wolframalpha.com/input/?i=x^pi

same goes for x^e

http://www.wolframalpha.com/input/?i=x^e

 

[Mentallic]

Those functions are real for x\geq 0 only. If you take a look at that interval for both odd and even functions, you'll realize that you can't really tell a difference between their shape.
And no it doesn't make pi an odd number.

 

[camilus]

the definition of an odd number n is a number that's able to be expressed as n=2k+1 for some integer k. use that definition to see if pi is an odd number.

 

[mxbob468]

Those functions are real for x\geq 0 only. If you take a look at that interval for both odd and even functions, you'll realize that you can't really tell a difference between their shape.
And no it doesn't make pi an odd number.

why is that true?

 

[Mentallic]

why is that true?

Because if we take some positive number x, then (-x)^{\pi}=(-1)^{\pi}\cdot x^{\pi}
Since x is positive, x^{\pi}>0 so we just have to deal with the (-1)^{\pi} factor. It is complex, but if you want a proof of this, simply convert it into its complex form:

e^{i\pi}=-1 therefore (-1)^{\pi}=e^{i\pi ^2}=cos(\pi ^2)+isin(\pi ^2) so if it is to be a real number, then the sin of \pi ^2 needs to be equal to 0, but this isn't the case.

In fact we can take any power x^{\alpha}, and deduce the circumstances whether it will be real or complex for negative values of x by following a similar process. But be wary, it is a little more complicated dealing with all rational values.