Linked plusminus notation $\pm$

Question

Given an expression with multiple $\pm$ where some of the $\pm$ have to behave the same. What is the correct notation?

Colours?

$$1 \color{red}\pm (x \color{red} \pm 3) \pm 4$$ Here the two red $\pm$ must either both be $+$ or both be $-$ and the black $\pm$ can behave independently. Using colours seems fairly self-explanatory but cannot then be printed in black and white.

Variables?

$$1 \pm_{1} (x \ \pm_{1} 3) \pm 4$$

Attaching a variable or number to the plus-minus looks a bit messy and could be misread:

Any suggestions / most widely used notation would be appreciated.

Answer 1

To the best of my knowledge, there is no common or standard notation which conveys the idea in the question. Because there is a lack of such notation, I would suggest that if new notation is going to be introduced, then it should be very clearly defined somewhere before it is used. I would also strongly suggest that color not be used—not all journals print in color, colorblindness is a thing, etc. The notation $\pm_1$, $\pm_2$, etc. might work, though I find it unaesthetic (personally).

As I see it, there are two pre-existing and commonly used notations which are relevant:

1. The symbols $\pm$ and $\mp$ can be used. This is somewhat limited: in the expression $1 \pm (x \pm 3) \pm 4$, it is not clear if the three signs must be the same or can be different; while in the expression $1 \pm (x \pm 3) \mp 4$ it must be the case that the first two signs are the same, and the third is different. However, there are contexts in which $\pm$ and $\mp$ are very useful.

2. Dummy variables may be used. For example, as noted in the comments by JMoravitz, it would be very reasonable to write $$ 1 + c_1(x + 3 c_1) + 4 c_2, \qquad c_1, c_2 \in \{\pm 1\}. $$ That is, instead of subscripting the operators, introduce new variables, and subscript them. One could go so far as to introduce a class of dummy variables towards the beginning of a paper, e.g.

   The variables $s_i$ will be used throughout to indicate the sign of a term. As such, $s_i \in \{\pm 1\}$ for any index $i$. For example, $$ 1 + s_1(x+3s_1) + 4s_2 $$ expands to $$ 1 + (x+3) - 4, \quad 1 + (x+3) + 4, \quad 1 - (x-3) - 4, \quad\text{or}\quad 1 - (x-3) + 4. $$