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Was reading about waves in my Physics textbook and a mathematical fact was invoked which I was curious about:

If we combine an infinitely large number of sinusoidal component waves, each with infinitesimally different reciprocal wavelength drawn from the same range K = 9 to 15, we obtain a central group quite similar to the one shown in Figure 3-9, but the auxiliary groups will not be present. The reason is that in such a case there is no length of the x axis into which an exactly integral number of wavelengths fits for every one of the infinite number of components

(from Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles by R Eisberg and R Resnick (p. 74) in Chapter 3: Wave Properties of Matter)

In other words: there exists no real $x$ such that $x = nA$ for any $A$ within a range where $n$ is an integer. I'm not sure if this fact is restricted to just real numbers, but from the context around this quote I know it must at least apply to them. Why is this?

physBa
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  • If $\mathbb Q$ denotes the rational numbers, then we can say that $\mathbb R=\mathbb Q\cup \mathbb Q’$. Now, if x is rational then x=nA has no integer solution for irrational A. Eg. 42=n$\sqrt2$ has no integer solution. If x is irrational, then x=nA has no solution for integral n if A is rational, except of course, n=0, which is however trivial. – insipidintegrator Jul 02 '22 at 19:24
  • If there was, then the reals would be countable. – copper.hat Jul 02 '22 at 19:29
  • @insipidintegrator Thank you! And this fact is simple to prove too. This made it clear (: – physBa Jul 02 '22 at 19:30
  • https://math.stackexchange.com/questions/1914009/classify-all-rings-with-cyclic-additive-group#:~:text=An%20infinite%20ring%20with%20cyclic,of%20d%20yield%20nonisomorphic%20rings. – Asinomás Jul 02 '22 at 19:31
  • @copper.hat what would the bijection be? Pleeeeeease I’m very curious – insipidintegrator Jul 02 '22 at 19:35
  • @insipidintegrator Since it does not exist I am not sure what you are asking? – copper.hat Jul 02 '22 at 19:36
  • ‘If there was’… then what would the bijection be? – insipidintegrator Jul 02 '22 at 19:40
  • Since there ISN"T such a bijection there is utterly no way to describe what one would be if it existed. .... Suppose I said "There is no way to turn banana peels into gold; If there were then everyone would be rich". ANd you say, "Wow, now I'm curious. If there was a way for everyone to be rich by turning banana peels into gold, I really want to now what way that would be. Please tell me what the way to turn banana peels into gold would be if there was a way". We can't tell you that because there isn't any way, there can't be any way, and we had no way in mind when we said that. – fleablood Jul 02 '22 at 20:21

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You can't generate $x/2$ as an integer multiple of $x$ if $x\neq 0$, and for $x=0$ you don't get all of them either.

Asinomás
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