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Suppose, I want to state that $a$ is less than $b$. However, I do not know $b$ exactly, but only that it is approximately $c$. With other words I want to state that $a$ is less than some value that is approximately $c$. I want to express this using one symbol, since I do not wish to introduce $b$ just for the purpose of making this statement. For example (using $∎$ as the desired symbol):

For $a ∎ 4.2$, the dynamics is chaotic.

Obvious symbols that come to mind for expressing this relation are $\lesssim$ and $\lessapprox$.

Is there any convention on which symbol to use for such a case or are there any good argument for or against either alternative regarding consistency and avoiding confusions? Related as well as an example of what kind of information I am looking for: I consider using $\sim$ instead of $\approx$ for approximately equal a bad idea, since it is used for mathematical equivalence as well as for proportional to (despite the existence of $\propto$) and thus it is often unclear what is meant by this symbol.

Wrzlprmft
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    I've seen $\approx$ and $\simeq$ for approximately equal in NSA contexts. I'd think that $≲$ and $⪅$ are as different as $\leq$ and $\leqq$, it's more a question of taste if you want the big or smaller stack of symbols. – Lutz Lehmann Feb 12 '14 at 17:44
  • I've seen $\lesssim$ in NSA contexts, too. So, $a\gtrsim 0$ means that $a$ is either positive or infinitesimal. – Akiva Weinberger Jun 07 '15 at 03:21
  • I am confused. "With other words I want to state that $a$ is lesser than some value which is approximately $c.$" So you know that $a<c$ and you know that $b\approx c,$ and you want to claim that $a<b$? Either you know $a<b$ (because you have enough information about your approximation) in which case you can write "hence $a<b$," or you don't, in which case you must surely write words to the effect of "assume $a<b$" (perhaps prefixed with "on these grounds" or "it is safe to" or some such phrase). I don't see why another symbol is necessary or useful. – Will R Feb 13 '18 at 22:24
  • @WillR: I added an example of usage to my question. – Wrzlprmft Feb 14 '18 at 07:11
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    The meaning of no symbol is unclear if you are explicit about what you mean by it. – Mariano Suárez-Álvarez Feb 14 '18 at 07:32
  • I want to bring to your attention that I've added another option to my answer. – celtschk Feb 14 '18 at 07:35
  • For reference: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4442573/ notates approximate ranges like "270⪅Ts⪅290 K" while https://arxiv.org/pdf/astro-ph/0606134.pdf uses "6≲z≲10" – endolith May 17 '18 at 21:35
  • Just being pedantic, but "dynamics are choatic" would be correct instead of "dynamic is chaotic", right? – Varad Mahashabde Jan 03 '21 at 08:07
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    @VaradMahashabde: Following the fine English tradition of messing with the grammatical number of loanwords, dynamics is typically used as a singular word in dynamical-system science and similar. – Wrzlprmft Jan 03 '21 at 09:43

2 Answers2

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In Unicode, the symbol $\lessapprox$ is named “LESS-THAN OR APPROXIMATE”, which describes pretty much the meaning you are after. It also is built from the symbols $<$ (less than) and $\approx$ (approximately equal) in the very same way as $\leqq$ (less or equal) is built from $<$ (less) and $=$ (equal).

Given that commonly for "less than equal" a single line is used instead of a double line (i.e. $\leq$ instead of $\leqq$), using $\lesssim$ instead of $\lessapprox$ seems justified as well.

Indeed, in the comments to your question, Akiva Weinberger has confirmed that the latter symbols is actually used with that meaning in "NSA contexts" (which probably means non-standard analysis).

So in summary, I'd say for both symbols there's strong evidence that their use with that meaning is appropriate.

Another option might be to write $a<b+\epsilon$. Commonly $\epsilon$ is understood as a small positive quantity, so I'd argue that you are not really introducing a new variable there.

celtschk
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  • I'd always assumed that the symbol meant "less-or-equal-than AND approximate"... – leonbloy Feb 13 '18 at 22:12
  • Does that also cover a slightly different case: I have a value a and a value b. If a << b something happens. If a >> b something else happens and if a is smaller than but CLOSE to b another thing happens, which is very different from what happens when a is bigger than but close to b and yet different from what happens when a = b – Walter Lars Lee Oct 07 '20 at 11:07
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'I want to state that $a$ is lesser than some value which is approximately $c$.'

Then write $$a<b\approx c.$$

Allawonder
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    Please see the edit to my question (the same information existed in a comment for about a year). – Wrzlprmft Feb 14 '18 at 07:18
  • This is the exact answer to the question. But I think @Wrzlprmft means a different thing. I think there is no "b" as shown by the example stated in the question. – Walter Lars Lee Oct 07 '20 at 11:03