I have following paragraph taken from the Stanford's study material.
Question: Is the sum of two periodic functions periodic?
Answer: I guess the answer is no if you are Mathematician, yes if you are an Engineer i.e. no if you believe in irrational numbers and
leave it at that, and yes if you compute things and work with
approximation.
This sounds something interesting to me. As an engineer, can I always assume that sum of periodic functions is always periodic?
Best Answer
It's more because for engineers, periods tends to have common divisors and hence the sum of two functions of periods $n x$ and $m x$ with $n,m∈ℕ$ is then $\mathrm{lcm}(n,m)x$.
For instance, in maths the usual counterexample is $\sin(x)$ and $\sin(x\sqrt 2)$ and that to get yourself into that situation in real life is difficult.
Another example, that happen to be highly strange and will never happen in practise: there exist two periodic functions $f$ and $g$ such that their sum is the identity function on $\mathbb R$ (yes, $∀x∈ℝ~~f(x)+g(x)=x$). But this time, even in math it is difficult to find yourself into this situation. (See this for how to do it, but it is a spoiler, it is really fun to look into it yourself.)