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For what $x$ is $f(x)=(-1)^x$ a real number, and when is it a complex number?

When I graph it online, the graph glitches out and has points all over the place.

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  • $\begingroup$ Consider $f(x) = e^{i \pi x}$ $\endgroup$ Feb 3, 2019 at 4:44
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    $\begingroup$ Remember, $(-1) = e^{i \pi}$ $\endgroup$ Feb 3, 2019 at 4:45
  • $\begingroup$ The first thing one should ask, maybe, is the definition of the expression $(-1)^x$. $\endgroup$
    – user587192
    Feb 3, 2019 at 13:44

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This is the same as $\exp(x\pi i)=\cos(x\pi)+i\sin(x\pi)$. For it to be real, $\sin(x\pi)=0$, i.e. $x$ is an integer.

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  • $\begingroup$ That does make sense. The only thing $(-1)^\frac{2}{3}=((-1)^2)^\frac{1}{3}=1$, where is the fallacy in that equation? $\endgroup$
    – Simplex1
    Feb 3, 2019 at 5:14
  • $\begingroup$ @Simplex1 when dealing with complex numbers, numbers can have more than $1$ root. I gave the formula for the principal root, but there are more, given by integer multiples of the argument. Your equation neglects that, and only looks at the real one. $\endgroup$
    – YiFan Tey
    Feb 3, 2019 at 5:17
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If you recall that $(-1) = e^{i \pi}$ then $$f(x)= e^{i \pi x}$$ Which is then $$\cos(\pi x) +i\sin(\pi x)$$

Does this help at all?

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When $-1$ is raised to power of $x$ the complex number so formed has a real part $\cos \pi x $ and imaginary part $\sin \pi x.$ This can be found using Euler formula $ e^{i \pi}=-1$. The two parts are graphed below .. with each part having a period or wave length $\lambda=2, $ the roots are at $x=..,-2,-1,0,1,2,...$ etc.

Real part vanishes for all integer values of $x$ as is obvious from graph and direct check by plugin.

The $f(x)$ thus is always complex for all real values of $x$

Again for complex $x= a+ib ,\, f(x) $ is complex.

enter image description here

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  • $\begingroup$ This is a good answer, but while I agree the phrasing in the question is vague, I think "complex" is supposed to mean "complex and not real" in the question. So even though it's correct that $f(x)$ is always complex, I think this doesn't directly answer the OP's question. $\endgroup$
    – YiFan Tey
    Feb 3, 2019 at 5:50
  • $\begingroup$ I understood OP question as asking: [when is $f(x) $ real and when is it complex?] $\endgroup$
    – Narasimham
    Feb 3, 2019 at 7:13

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