In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"?

The Unicode standard lists all of them inside the Mathematical Operators Block.

  • : ALMOST EQUAL TO (U+2248)
  • : ASYMPTOTICALLY EQUAL TO (U+2243)
  • : APPROXIMATELY EQUAL TO (U+2245)
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    Did anyone else notice this question is basically how different notions of "approximate equality" are only approximately equal? – David H Jul 11 '14 at 21:01
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    @DavidH: God is in the details. ;-) – Lucian Jul 11 '14 at 21:36
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    @Lucian I always took for granted that the "approximately" qualifier acted idempotently, so while we can distinguish between 'exact equality' and 'approximate equality', 'exact approximate equality' is the same thing as 'approximate approximate equality'. A world in which this is not true makes me want to stress vomit. Is it tildes all the way the down!? – David H Jul 11 '14 at 21:51
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    @Lucian: So, approximately, good is in the details? – Scott Jul 12 '14 at 1:18
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    @DavidH OTOH, some people will start using $\stackrel{\approx}{\approx}$ for "approximately approximately" and all hell will break loose. – Adam Hughes Jul 12 '14 at 6:38

The notations $\cong$ and $\simeq$ are not totally standardized. Both are usually used for "isomorphic" which means "the same in whatever context we are." For example "geometrically isomorphic" usually means "congruent," "topologically isomorphic" means "homeomorphic," et cetera: it means they're somehow the "same" for the structure you're considering, in some senses they are "equivalent," though not always "equal:" you could have two congruent triangles at different places in a plane, so they wouldn't literally be "the same" but their intrinsic properties are the same. I've seen colleagues use both for isomorphic, and some (mostly the stable homotopy theorists I hang out with) will use $\cong$ for "homeomorphic" and $\simeq$ for "up to homotopy equivalence," but then others will use the same two symbols, for the same purposes, but reversing which gets which symbol.

The $\approx$ is used mostly in terms of numerical approximations, meaning that the values in questions are "close" to each other in whatever context one is working, and often it is less precise exactly how "close." Topologists also have a tendency to use $\approx$ for homeomorphic.

The main take-away from this answer: notation is not always standardized, and it's important to make sure you understand in whatever context you're working.

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    professionals often use = also for isomorphic. $\approx$ is VERY OFTEN used for homeomorphic. – Zlatan der Zechpreller Jul 11 '14 at 21:27
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    I'll add the $\approx$ one in as well, he didn't ask about =, so I didn't mention it. – Adam Hughes Jul 11 '14 at 21:29
  • When comparing $\cong$ and $\simeq$ and having two kinds of isomorphisms in our context, I think it is more natural to $\cong$ mean “more isomorphic” that $\simeq$. – user87690 Jul 13 '14 at 10:15
  • @user87690 I tend to agree with you, and that's my personal convention, but I have seen algebraists use the latter as their standard, so I didn't want to priviledge my own aesthetic since it's not standard. – Adam Hughes Jul 13 '14 at 10:17

≈ is for numerical data, homeomorphism

≃ is for homotopy equivalence

≅ is for isomorphism, congruence, etc

These are just my own conventions.

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