# Is pi or e an Odd Number?

• cstvlr

#### 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

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 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.

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?

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.