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In my life¹, I have never seen a symbol other than ≈ used in handwriting to express “approximately equal”. The symbol ~ was only used for more mathematical purposes such as equivalence, proportionality, or “is distributed as”. Yet, I often see the symbol ~ used for “approximately equal” in the scientific literature, in particular in fields where TeX use is scarce.

I do not want to debate what is right™ here, but I am just curious whether this usage is a result of the limitations of typewriters and predominant keyboard layouts and encodings from the early days of computing, which make ≈ difficult to produce or impossible to encode, while ~ is readily available. By contrast, I would expect that in moveable type (and similar systems that do not allow superimposing characters), you either had both or neither glyphs available, as I am not aware of any relevant application for the ~ glyph besides equations in such systems². Also, in TeX, both symbols are comparably difficult to use (\approx and \sim).

Is there any harder evidence that the usage of ~ for “approximately equal” is a consequence of it being a makeshift solution in light of technical restrictions? For example:

  • Examples of ~ being used for this purpose in systems that do not technically favour this symbol.
  • Works of the same author using ~ only when technical restrictions applied.
  • Historical guidelines on symbol usage.

¹ I went to school and studied physics and math in Germany.
² On a typewriter, the ~ character could be superimposed with, e.g., n to obtain ñ.

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  • $\begingroup$ I thought the typewriter way was an equal sign with a period imposed above, or maybe it was just an alternative $\endgroup$ – Michael E2 May 17 at 19:27
  • $\begingroup$ In my field, physics, we use both $\sim$ and $\approx$, and they have distinct meanings. We use $\sim$ to mean "is on the order of," and $\approx$ to mean "is approximately equal to." For example, the population of the US is $\sim10^8$. Also, it's grammatical to write $\sim$ with nothing on the left, but not $\approx$. $\endgroup$ – Ben Crowell May 30 at 0:20
  • $\begingroup$ Note that the Bachmann-Landau notation gives a very specific and completely formal meaning to the symbol $\sim$, i.e., $f(n) \sim g(n)$ means $\lim_{n\to\infty} f(n)/g(n) = 1$. Maybe what happened is simply that people who do not know or understand the Bachmann-Landau notation saw this symbol used and adopted it with a non-formal meaning. $\endgroup$ – Federico Poloni Jun 6 at 13:22
  • $\begingroup$ @FedericoPoloni: Isn’t that just a special case of equivalence classes? $\endgroup$ – Wrzlprmft Jun 6 at 13:26
  • $\begingroup$ @Wrzlprmft Yes, in the end it's a special case of equivalence classes, but it's also a notation that is often used without explicit reference to equivalence classes and that is very easy to misunderstand as just a vague "is approximately equal to", which is my point above. $\endgroup$ – Federico Poloni Jun 6 at 13:33
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There is evidence that the story is more complicated than adoption of pre-existing paper and pencil practice being obstructed by keyboard limitations. Cajori makes some interesting remarks on the symbols for “nearly equal” in History Of Mathematical Notations (1928):

Among the many uses made in recent years of the sign $\sim$ is that of “nearly equal to,” as in “$e\sim\frac14$” [sic!]; similarly, $e\cong\frac14$ is allowed to stand for “equal or nearly equal to.” [reference to Kratzer] A. Eucken lets $\simeq$ stand for the lower limit, as in “$J\simeq45\cdot10^{-40}$ (untere Grenze),” where J means a mean moment of inertia. Greenhill denotes approximate equality by two parallel sawtooth lines. An early suggestion due to Fischer was the sign $\asymp$ for “approximately equal to.” This and three other symbols were proposed by Boon who designed also four symbols for “greater than but approximately equal to” and four symbols for “less than but approximately equal to.”

Some comments: Cajori published in 1928, which means that symbols for “nearly equal” were recent then, with the exception of Fisher’s (1829) and Greenhill’s (1892). Second, the now dominant $\approx$ is not even among them (Greenhill’s two parallel sawtooth lines would be the closest). Third, it seems that all these symbols were proposed/designed rather than adopted from pre-existing paper and pencil usage.

This said, we do generally see the rise of new symbology towards the end of 19th century (think of Boole, Frege and Peano, for example), which does correlate with the commercial adoption of typewriters, and their standardization in 1893–1910. The trend is not exclusive to mathematicians, Kratzer’s $\sim$ appears in a journal for physics, and Eucken’s $\simeq$ in a journal for physical chemistry. And it would have been hard for paper and pencil usage to spread without typographic support. So it seems fair to infer that the increase in typographic flexibility spurred development and proliferation of new symbols, rather than distortion of existing ones to fit what was available. Of course, this does not mean that at a later time, after the use of $\approx$ was established, keyboard or ASCII did not motivate authors to make pragmatic substitutions.

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  • $\begingroup$ Re "And it would have been hard for paper and pencil usage to spread without typographic support", I have seen a number of papers (I think mainly from the 1960s/70s) in journals (so not self-published) which were typewritten and where the typist left a gap for symbols which they didn't have available and added them afterwards in pen. $\endgroup$ – Peter Taylor May 20 at 8:10
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In my experience, the tilde , "~" , is used in mathematics to indicate a loose relationship. In geometry, it specifically means "similar," i.e. same angles, different length sizes.
$\approx$ is used only for items "close to equality" .

So far as I know, there was never a case when either math or physics chose a symbol based on "ease of typing." (The only case I've seen where 'ease of typing' was involved is a couple computer languages that used " <-" as the replacement operator, because once upon a time some computer keyboards had a single key for that symbol pair!)

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  • $\begingroup$ I am sorry, but I fail to see how this addresses my question apart from: “So far as I know, there was never a case when either math or physics chose a symbol based on ‘ease of typing.’” And that’s not very strong evidence. (Also, I would argue that at least using the middle dot (·) as a decimal separator may be a counterexample.) $\endgroup$ – Wrzlprmft May 17 at 16:43
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    $\begingroup$ @Wrzlprmft until very recently, all work was done with pencil and paper, and what existed or did not exist on a keyboard was irrelevant. $\endgroup$ – Carl Witthoft May 17 at 18:12
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    $\begingroup$ @CarlWitthoft Actually, typewriters existed since 16th century, and became common since 19th. Aside from that, when authors prepared papers for publication they had to be mindful of typographic restrictions. Frege's original logical notation was dropped largely because it was hard to typeset. $\endgroup$ – Conifold May 18 at 5:24

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