Function x^(pi)

Main Question or Discussion Point

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
Homework Helper
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.

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

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
Homework Helper
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.