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The notation I've used is $a$~$b$, but one of my teachers said that the correct notation was $a\approx b$. Is this notation really used, and if so, are there any uses of this in literature? I haven't found any.

He also says that $a\approx b$ can't be used for, "approximately equal to $b$" for this reason, and instead, $\dot{=}$ should be used, but I don't think this is true ("\approx" is the latex notation for $\approx$), and I can't even find any uses of $\dot{=}$ online! This wikipeda page has a similar symbol at the bottom, saying that a similar one is commonly used in Japan and Korea.

(This was in a class which has absolutely nothing to do with asymptotic anything, so it isn't notation used in-class during the semester.)

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  • $\begingroup$ It could be possible that your school's math department has chosen a convention and taken care that all of your classes are consistent with it, and thus assuming students should be learning that convention as well. $\endgroup$ – Hurkyl Feb 15 '13 at 6:19
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Your teacher has very rigid thinking in this respect. For example, the symbol $\approx$ is named "approx", so a lot of people obviously think it means "approximately". Also, the symbol $\sim$ is used in many places for an asymptotic relation. Specifically,

$$f(x) \sim g(x) \: (x \rightarrow x_0) \implies \lim_{x \rightarrow x_0} \frac{f(x)}{g(x)} = 1$$

See, for example, Bender & Orszag.

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