I have $f(x)=\sin(x)$ so it has a period of $2\pi$, and I have $g(x)=\sin (\sqrt{2}\,x)$ so it has a period of $\sqrt{2}\pi$. I also know that a function $f(x)$ is periodic if $f(x)=f(x+p)$, now I want $h(x)=f(x)+g(x)$. How do I check whether $h(x)$ is periodic or not?
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Since $\mathbb{N}(2\pi) \cap \mathbb{N}(\sqrt{2}\pi) = \emptyset$, it's not periodic. – TorsionSquid Dec 14 '16 at 03:01
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Hint: assume that it is periodic, with period $P$. Then $P$ must be an integer multiple of both $\sqrt{2}\pi$ and $2\pi$.
Ashwin Trisal
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1This isn't necessarily true. The sum might have a period unrelated to the periods of $f$ and $g$, just by accident. – Eric Wofsey Dec 14 '16 at 03:14