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My understanding is $\sim$ generally means "on the order of magnitude of" e.g. $T \sim 10^5$ K

$\approx$ is obviously "approximately equal to" so for example one might write $d \approx 400$ pc rather than $d=4 \times 10^2$ pc

$\simeq$ and $\cong$, are where I am more confused as their usage is rarer and less consistent.

The below IAU resolution, for example, seems to use $\simeq$ for truncation $m_{\mbox{bol}} = -26.83199\ldots \simeq -26.832$, but I have also seen $\simeq$ used as "approximately equal to" in other works.

http://arxiv.org/pdf/1510.06262

How should one be using $\sim$, $\approx$, $\simeq$, and $\cong$?

imanorc
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  • I don't think that there's any set standard for using these approximation signs. – Phiteros Mar 22 '17 at 04:35
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    Those are just the pen strokes of hand waving astronomers. :) – LocalFluff Mar 22 '17 at 13:19
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    To point out, you'd likely receive a very different answer if you posted this on the math stack exchange. I think the distinction between these various symbols has more significance in mathematics. Astronomers on the other hand tend to be less strict and will often just use all of these interchangeably, or you'll see different standards being used by different people. There's less uniformity in their meaning in the sciences than there would be in the mathematics groups. – zephyr Mar 22 '17 at 15:01

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$\simeq$ and $\approx$ both mean "approximately equal to". I don't think $\cong$ is used so often, but if I read it, I would interpret is as the same as the two others.

$\sim$ in principle means "of the order of", i.e. correct to within an order of magnitude. However, astronomers tend to be rather slobby sometimes (we call all elements except H and He for "metals", we call it "gas" even when it's ionized and should be called "plasma", we omit factors of order unity if we can get away with it, we mix up terms like "flux", "flux density", and "intensity", and so on), and thus $\sim$ is very often used to mean "approximately equal to". It is even sometimes used as "proportional to", but (usually) only when there's no risk of misunderstanding (e.g. $L_\mathrm{UV}\!~\sim\!~\mathrm{SFR}$, meaning that the UV luminosity is proportional to the star formation rate, omitting a factor of $10^{27}$).

Journals usually have rather strict standards for style and nomenclature, so they might not accept a $\sim$ for a $\propto$, or even for a $\simeq$. But if you read a conference proceeding or lecture notes or some other non-refereed text (even a paper on arXiv waiting to be refereed), you best be prepared for anything.

pela
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Recently, some in the astronomy literature are adopting $\sim$ in the statistics sense to mean "is sampled from" or "has the distribution of" (the last definition on mathworld). For example,

$$ x \sim p(x) $$

would read "$x$ is sampled from the distribution $p(x)$".

ampw
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Wikipedia (where would we be without it) has a handy list of mathematical symbols which is pretty useful in this context.

Note that in particular :

  • $\sim$ will crop up often as meaning asymptotically equivalent which has a very specific meaning and should not be used for as an approximate symbol - they're quite different and some mathematicians will lynch you if you call it approximately.

Last time I checked Carl Bender's lectures on mathematical physics were available on YouTube if it's of interest to anyone and these include asymptotic analysis.

StephenG - Help Ukraine
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    This is definitely not how $\sim$ is used in (observational) astronomy. – pela Mar 22 '17 at 15:23
  • @pela Yes, I read your answer, and I specifically mentioned this mathematical interpretation because it could lead to ambiguity or downright misunderstanding. I'd recommend people stick with the more precisely defined symbols (like $\propto$ for proportional to). – StephenG - Help Ukraine Mar 22 '17 at 19:55
  • Okay, fair enough. But good luck persuading the astronomical community to do that ;-) – pela Mar 22 '17 at 21:23
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As an (ex) professional astronomer, I can tell you that there are no rules that are followed. There are very very few exact measurements in Astronomy, x rarely technically equals anything. In general, $\sim$ means less accurate than $\approx$ which means less accurate than $=$.

Mathematicians (and Chemists) frequently complain about our loose use of mathematics (and chemistry).

Jonathan Twite
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