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I just noticed that the Unicode character set contains an entire block (256 characters) of math symbols. I've got no idea how some of them should be used. For example:

  • U+22DA (⋚) and U+22DB (⋛): less-than-equal-to-or-greater-than, and greater-than-equal-to-or-less-than. Why not just use the good old equals sign = ?
  • U+2295 through U+2298 (⊕⊖⊗⊘): circled + − × ⁄ symbols
  • U+2235 (∿): sine wave; why do we need a character for this? does anyone use this?
  • U+2268 (≨): less-than-but-not-equal-to; isn't this redundant? if x < y then isn't it always true the x ≠ y?

I got these from here (you can click "more").

Thanks!

wchargin
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    $\oplus$ is a special symbol, indicating "direct sum"; it is different from $+$. $\otimes$ is the tensor product, different from $\times$. Both are extremely common. The "sine wave" is not a sine wave, it's a symbol used to denote weak equivalence (often used for equivalence relations, or for stating that two functions are of the same order $\sim$. – Arturo Magidin Sep 11 '11 at 19:55
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    The symbol $\lt$ is used for more than simple order symbols. For example, it is used to denote substructures (subspaces, subgroups). In those cases, you need a symbol that specifies that they are proper substructures, i.e., not equal to the original substructures. Again, the symbol is used quite a lot. I don't see your first two used much, but that doesn't mean they aren't, just like the fact that you don't see the others used somehow imply they are "useless". – Arturo Magidin Sep 11 '11 at 19:57
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    Maybe you could use the first for something like: if $x ⋚ y$ then $-x ⋛ -y$ (similar to $\pm$ and $\mp$) – TMM Sep 11 '11 at 20:01
  • The "sine wave" is U+223C, not U+2235. If you click on the symbol on the page you linked to, you get a description of it, which among other things says "tilde operator". – joriki Sep 11 '11 at 20:11
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    @joriki: Actually it's U+223F— http://www.fileformat.info/info/unicode/char/223f/index.htm – kahen Sep 11 '11 at 20:55
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    "Some math symbols seem useless"... Yes, of course, most of them are when left unexplained and without context. – t.b. Sep 12 '11 at 04:04
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    One book I have on group theory mentions an alternative name for ⊕ (and the like) — "hot cross plus", etc. This is highly appropriate, since the lot of them can then be referred to as bunnery operators. Badoom tish. – detly Sep 12 '11 at 05:05
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    I'd rather have a plethora of unused math symbols than reuse symbols which could cause confusion, even with some context in its use. I can't think of any examples currently, but I'm sure there are at least a few out there. (Although this is more of a comment, I don't have the authority to make one.) – Dustin Tran Sep 12 '11 at 07:44
  • Wow, thanks so much everyone! I never knew most of this. I wish I could accept all the answers :) I'm giving it to Michael Hardy solely because he taught me the most, but everyone taught me something today. – wchargin Sep 13 '11 at 04:03
  • ⊘ can be used for the measure of the diameter of a telscope – Elements In Space Jan 05 '13 at 17:07
  • "- U+22DA (⋚) and U+22DB (⋛): less-than-equal-to-or-greater-than, and greater-than-equal-to-or-less-than. Why not just use the good old equals sign = ?"

    Because that's not what the equals sign means. The equals sign means it has to be exactly equal, and cannot be greater than or less than. I think you mean to ask why anyone would use the symbol at all since the symbol seemingly could apply to any two numbers.

     – Kyle Delaney Dec 11 '15 at 01:52
  • @KyleDelaney Heh, you're right, of course. (I now imagine that the intended use is within the context of non-total orderings—so while it can apply to any two numbers, we can partially compare other things than those!) Thanks for your correction, but I've left the post as is to preserve the context. – wchargin Dec 11 '15 at 04:24
  • The sine wave ∿ (U+223F: SINE WAVE) IS the Unicode sine wave symbol. You are talking about ∼ (U+223C: TILDE OPERATOR). In general, it is important not to confuse similar-looking Unicode characters. For instance, – (U+2013: EN DASH) and − (U+2212: MINUS SIGN) are different, as are ° (U+00B0: DEGREE SIGN) and º (U+00BA: MASCULINE ORDINAL INDICATOR), and β (U+03B2: GREEK SMALL LETTER BETA) and ß (U+00DF: LATIN SMALL LETTER SHARP S). – Andreas Rejbrand Sep 28 '16 at 13:18
  • Ah, I realise now that the OP has made a mistake. The sine wave ∿ is U+223F, but the OP wrote U+2235 which is ∵ (U+2235: BECAUSE). – Andreas Rejbrand Sep 28 '16 at 13:23

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$\oplus$ and $\otimes$ are used to denote direct sum and tensor product, which are ubiquitous in mathematics. They also have a pedagogical use as symbols for addition in an abstract abelian group resp. multiplication in an abstract group when you want to make the point that groups are much more general than addition and multiplication of real or complex numbers. $\ominus$ is occasionally used in this abstract way as well.

Qiaochu Yuan
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If $A \subset B$ are linear subspaces of a Hilbert space, $B \ominus A = \{x \in B: (x,y) = 0 \text{ for all }y \in A\}$. $\ominus$ is also used for the symmetric difference of sets.

Robert Israel
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$⋚$ and $⋛$ seem appropriate for use with partial orderings; i.e. $a⋚b$ would mean that $a$ and $b$ are comparable ($a⊥b$ is another convention for symbolising comparability).

David Bevan
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Your "sine wave symbol" looks to me very similar to $\sim$. Certainly that is used: $$ X\sim N(\mu,\sigma^2) $$ $X$ is a normally distributed random variable with expectation $\mu$ and variance $\sigma^2$. $$ X\sim \operatorname{Bin}(n,p) $$ $X$ is a binomially distributed random variable with parameters $n\in\mathbb{N}$ and $p\in[0,1]$.

$$ f(x) \sim \sum_{n=-\infty}^\infty c_n e^{inx} $$ The Fourier series of $f(x)$ is that series. There is no commitment to saying the series converges to $f(x)$ (in some cases it does; in others it doesn't, and it can depend on which kind of convergence is being considered).

$$ a \sim b $$ $a$ is related to $b$ (just which sort of "relation" is referred to depends on the context).

Michael Hardy
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  • No, ∿ (U+223F: SINE WAVE) is not a mathematical operator. It is a symbol for a sine wave, defined by the Unicode standard. You are talking about ∼ (U+223C: TILDE OPERATOR). – Andreas Rejbrand Sep 28 '16 at 13:12
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    In general, it is important not to confuse similar-looking Unicode characters. For instance, – (U+2013: EN DASH) and − (U+2212: MINUS SIGN) are different, as are ° (U+00B0: DEGREE SIGN) and º (U+00BA: MASCULINE ORDINAL INDICATOR), and β (U+03B2: GREEK SMALL LETTER BETA) and ß (U+00DF: LATIN SMALL LETTER SHARP S). – Andreas Rejbrand Sep 28 '16 at 13:16
  • Ah, I realise now that the OP has made a mistake. The sine wave ∿ is U+223F, but the OP wrote U+2235 which is ∵ (U+2235: BECAUSE). – Andreas Rejbrand Sep 28 '16 at 13:23
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The sine wave symbol can be used when typesetting circuit diagrams to indicate that it's an AC circuit.

$\lneqq$ exists (at least if for no other reason) for completion's sake and/or symmetry with $\subsetneqq$ and friends.

kahen
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$\otimes$ is also used to denote the Kronecker product of two matrices.

Sometimes, in analyses of floating-point arithmetic (like this one), $\oplus,\ominus,\otimes,$ and $\oslash$ are used instead of $+,-,\times,$ and $/$ to indicate the inexact arithmetic (i.e. with rounding) is being performed on numbers, instead of the ideal operations.

$\sim$ is also used in the representation of asymptotic expressions, e.g. Stirling's formula here. In there, it means that even though the series given is in fact divergent, truncations of it become a better approximation of the gamma (factorial) function as the argument becomes large.

J. M. ain't a mathematician
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