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When you have a single plus-or-minus symbol, the meaning is clear: $a±b = (a+b) OR (a-b)$

When you have plus-or-minus and minus-or-plus symbols, the meaning is also clear, as described in many places, such as this post (https://math.stackexchange.com/a/160479/1238228): $a±b∓c = (a+b-c) OR (a-b+c)$

Here's my question, what is the meaning when you have multiple plus-or-minus symbols? I see two possible potential meanings, as described in the following two expressions:

  1. $a±b±c = (a+b+c) OR (a-b-c)$
  2. $a±b±c = (a+b+c) OR (a+b-c) OR (a-b+c) OR (a-b-c)$

If multiple plus-or-minus symbols mean that all symbols take the same sign (as shown in the first potential meaning), there should be a way to clearly indicate the behavior shown in the second potential meaning (and visa versa).

Technically, there is a way to clearly show the first potential meaning: $a±b∓(-c) = (a+b-(-c)) OR (a-b+(-c)) = (a+b+c) OR (a-b-c)$ However, this way is clunky, and there should be another way to write this.

Adamimoka
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    I think the simplest option is to just clarify in words whether the signs are independent or not. – Qiaochu Yuan Jun 06 '24 at 01:00
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    Symbols in math (and not only) are used to clarifying things: if a formula is so confused that we have to supplement it with a verbal clarification, it means that it is WRONG. – Mauro ALLEGRANZA Jun 06 '24 at 13:39

3 Answers3

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I would say that $a \pm b \pm c$ is ambiguous and should be clarified in the text, since both conventions exist. Some other rules of thumb:

  • If there is a $\mp$ sign anywhere, all the $\pm$ and $\mp$ should be coordinated.
  • If there are $\pm$ signs on both sides of an equation then all of them should be coordinated, especially if there's a different number of them on each side.
  • If you need anything in between "all independent" and "all coordinated", a possibility is to use $\pm_1$, $\pm_2$ and so on, where the same index is used for those that are coordinated. See eg https://en.wikipedia.org/wiki/Quartic_equation#Ferrari's_solution.
ronno
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  • +1. I never saw the $\pm_1$ version, and the version $\pm_{s}$ and $\pm_t$ used in the Wikipedia article would confuse me at first. Maybe worth a clarification such as "$\pm_1$ and $\pm_2$ are independent", in which case using $\varepsilon_1$ and $\varepsilon_2$ with $\varepsilon_i$ explicitly defined to be in ${-1,1}$ seems better. – Taladris Jun 06 '24 at 00:58
  • To be fair, the wikipedia article does explain the meaning after each use of that notation. I'm sure I've seen it without explanation somewhere, but can't track down an example. – ronno Jun 06 '24 at 05:14
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I think most mathematicians would interpret $$ a \pm b \pm c $$ as allowing the four options listed in option (2).

If you want option (1), write $$ a \pm (b+c). $$

There is no uniformly accepted convention for multiple $\pm$ and $\mp$ signs. It's the author's responsibility to make the intended meaning clear in any particular context.

Ethan Bolker
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  • Likewise for $x\pm y\mp z$ you could write $x\pm(y-z)$ and have no need for the $\mp$ symbol. And yet we have $\mp.$ – David K Jun 04 '24 at 20:20
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    @DavidK You need both $\pm$ and $\mp$ when the signs are coordinated in separate places in the expression, as in the trigonometry examples the OP links to. – Ethan Bolker Jun 04 '24 at 21:16
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    So the examples in the question and this answer aren't the best examples. Note that in the sine formula in the link we have coordinated $\pm$ signs. In any case, the "convention" is not universally recognized or followed. See https://math.stackexchange.com/q/2538374/139123 – David K Jun 05 '24 at 03:07
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    And in the formula $\sin (x\pm y) = \sin x \cos y \pm \cos x \sin y$ (from the same link) you have coordinated $\pm$ signs. I don't think there's a general rule that covers all cases. – David K Jun 05 '24 at 03:20
  • @DavidK Fair points. Best not to rely on any convention when there's potential ambiguity. – Ethan Bolker Jun 05 '24 at 11:03
  • Definitely not most mathematicians. – 温泽海 Jun 06 '24 at 02:22
  • It seems this answer is attracting some downvotes signifying disagreement, which is unfortunate. But this may be a result of the question being featured on the Hot Network Questions list and/or the answer starting by seemingly speaking for "most mathematicians". – ronno Jun 06 '24 at 05:03
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I interpret this as all the $\pm$ represent the same sign, all the $\mp$ represent the same sign, and the $\mp$ is the sign opposite to $\pm$.

So each such line has two valid interpretations:

(1) $\pm$ means $+$ and $\mp$ means $-$;

(2) $\pm$ means $-$ and $\mp$ means $+$.

marty cohen
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