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There seems to be confusion/incoherence around the ‘Approximately Equal To’ symbol in Unicode, LaTeX, on Wikipedia, and elsewhere.

Let me summarise:

$$≈ \tag{double tilde}$$

Learnt in school (UK) and see in many places for ‘approximately equal to’ including for example Wp: Simpson's rule. However it is U+2248 ALMOST EQUAL TO in Unicode, and \approx in LaTeX.

$$≅ \tag{single tilde over double bar}$$

U+2245 APPROXIMATELY EQUAL TO. However it is \cong in LaTeX, and Wp: Approximation#Unicode describes it as “isomorphism or sometimes congruence”.

$$≊ \tag{double tilde over single bar}$$

LaTeX \approxeq. However it is U+224A ALMOST EQUAL OR EQUAL TO in Unicode, and Wp: Approximation#Unicode describes it as “equivalence or approximate equivalence”.

I've also seen a single tilde over a single bar ($≃$) be used for ‘approximately equal to’ (by one Maths teacher who taught me at school). This is U+2243 ASYMPTOTICALLY EQUAL TO in Unicode, \simeq in LaTeX.

I'm sure there's some history behind it. Which is the preferred symbol? Please discuss this incoherence.

While on the topic of the notation of approximation, what is the symbol (or symbols) for an approximate expression? I sometimes precede an expression with a tilde to show that it's approximate, or use an ellipsis at the end of a rounded decimal (e.g. $(3.142\ldots)(1.414\ldots)^2$) to show that a substitution has occurred but imply that I'm actually calculating with higher precision (i.e. I had stored the result of a previous calculation on the calculator or computer program, and am recalling that result / using that result as a parameter).

What are your thoughts?

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Why should their be a preferred symbol? How can this question generate anything other than a list of each respondent's preference? As you observed, different people use different notations. If you want others to understand what you write, you should explain it first! –  GEdgar May 26 '13 at 18:55
    
Maybe you could ask this (in a less subjective, poll-type version; so it doesn't get closed as well) on the TeX SE. –  JMCF125 Dec 6 '13 at 22:54
    
This is a good question which shouldn't have been closed: there are many people who are similarly confused by the assortment of approximately-equal-to symbols and wonder what the precise meanings are. The answer is: it's a mess! A good answer would have cited a couple of textbooks which define the same symbol in two different ways. Though I suspect you're unlikely to find a crisp definition of "approximately", and if you do, it probably doesn't reflect the intuitive meaning (either that or its definition will be longer than a page of text)... –  Evgeni Sergeev Mar 25 at 13:20
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closed as not constructive by Qiaochu Yuan, O.L., Amzoti, Julian Kuelshammer, Henry T. Horton May 26 '13 at 21:49

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