# What is the purpose of the $\mp$ symbol in mathematical usage?

Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please correct me if I am wrong). Is there any other mathematical usage for the $\mp$ symbol, particularly on its own ?

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It has the same meaning as $\pm$, but as you noted, when used in conjunction, they have "opposite" meanings – M Turgeon May 30 '12 at 13:57
Sometimes it is used to indicate alternating signs in a series, starting with a minus, as in $x-\frac{x^3}{3!}+\frac{x^5}{5!} \mp \ldots$ – marlu May 30 '12 at 14:01
I upvoted @marlu's comment, but then I got worried that it was not actually correct. The example is a little bit wrong, and when I tried to fix it I was not aple to support the point I thought was being made. All I could come up with were things like ${(x\pm y)}^n = x^n \pm x^{n-1}y + x^{n-2}y^2 \pm \cdots$ where there is already a $\pm$ outside to refer to. – MJD May 30 '12 at 15:18

$\mp$ really only has a use when written in the same expressions as $\pm$.

The one that comes to mind is $\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$.

But I suppose if you really wanted to, you could write things like $\sin(\alpha \mp \beta) = \sin \alpha \cos \beta \mp \cos \alpha \sin \beta$... if you really wanted to.

On a more humorous vein, it wouldn't surprise me if someone overloaded the symbol to have a different meaning too. Most likely someone like Conway (as in, Combinatorial Game Theory Conway, not Complex Analysis Conway), who thought $+_n$ was a perfectly good name for a state of a game (not an operation).

an aside

On a non-mathematical note, $\pm$ denotes an advantageous position for white in chess. $\mp$ denotes a position for black.

If we really go for it, $\mp$ looks like (干) wiki page, which means 'to dry' in Japanese and might mean 'to do' in Mandarin. $\pm$ looks like (士)wiki page, which might mean 'gentleman' in Japanese and is used in the symbols for doctorate and doctor's thesis.

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Thank you, that is helpful. @marlu wrote in a comment to my OP that "Sometimes it is used to indicate alternating signs in a series, starting with a minus, as in $x-\frac{x^3}{3!}+\frac{x^5}{5!} \mp \ldots$" - is that not standard usage ? – Joe King May 30 '12 at 14:38
I am a high-level chess player and have read dozens of chess books, and I have never once seen the notation $\pm$ or $\mp$ used in chess. Everywhere I have ever read, an advantage for white is written +/-, while an advantage for black is written -/+. – BlueRaja - Danny Pflughoeft May 30 '12 at 16:13
@BlueRaja-DannyPflughoeft: I've been an amateur chess player, I've read quite many books (with descriptive notation instead of algebraic!) and I'm pretty sure (not totally) of having seen $\pm$ and $\mp$ – leonbloy May 30 '12 at 16:16
@leon: Wikipedia seems to back that. Perhaps we are just reading different books :) (then again, wikipedia also makes a distinction between +/- and +-, a distinction I doubt is in wide use) – BlueRaja - Danny Pflughoeft May 30 '12 at 16:17
@BlueRaja-DannyPflughoeft : it is somewhat surprising that a high-level chess player is unfamiliar with the Chess Informant notation (or am I a bit too old? :) ). In CI $\pm$ ($\mp$) stands for "White (Black) stands clearly better", whereas $+-$ ($-+$) stands for "White (Black) has a decisive advantage" – Andrea Mori Aug 28 '12 at 22:02

You are correct; $\mp$ only makes sense in a formula that already has $\pm$.

One simple and useful example is that when $x$ is small, ${1\over{1\pm x}}\approx 1\mp x$.

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Also $x^3 \pm y^3 = (x\pm y)(x^2 \mp xy + y^2)$ – Dilip Sarwate May 30 '12 at 14:34

Like the other answerer, I've only seen it used in the same line as a $\pm$, to mean "positive when the other term is negative and negative when the other term is positive." So, for instance, if we were to say

$\pm a = \mp b$

that would imply that

$a = -b$

and

$-a = b$

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