16
$\begingroup$

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 ?

$\endgroup$
3
  • 2
    $\begingroup$ It has the same meaning as $\pm$, but as you noted, when used in conjunction, they have "opposite" meanings $\endgroup$
    – M Turgeon
    May 30, 2012 at 13:57
  • 5
    $\begingroup$ Sometimes it is used to indicate alternating signs in a series, starting with a minus, as in $x-\frac{x^3}{3!}+\frac{x^5}{5!} \mp \ldots$ $\endgroup$
    – marlu
    May 30, 2012 at 14:01
  • $\begingroup$ 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. $\endgroup$
    – MJD
    May 30, 2012 at 15:18

3 Answers 3

25
$\begingroup$

$\mp$ 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.

$\endgroup$
6
  • $\begingroup$ 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 ? $\endgroup$
    – Joe King
    May 30, 2012 at 14:38
  • $\begingroup$ 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 -/+. $\endgroup$ May 30, 2012 at 16:13
  • $\begingroup$ @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 $\mp$ $\endgroup$
    – leonbloy
    May 30, 2012 at 16:16
  • $\begingroup$ @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) $\endgroup$ May 30, 2012 at 16:17
  • 3
    $\begingroup$ @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" $\endgroup$ Aug 28, 2012 at 22:02
5
$\begingroup$

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, ${1\over{1\pm x}}\approx 1\mp x$.

$\endgroup$
1
  • 3
    $\begingroup$ Also $x^3 \pm y^3 = (x\pm y)(x^2 \mp xy + y^2)$ $\endgroup$ May 30, 2012 at 14:34
4
$\begingroup$

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 $

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.