# How to represent combinations of plus and minus

Just wondering if there is a shortcut for representing the following: $$x=1+i,1-i,-1+i,-1-i$$ I know that you can do: $$x = 1\pm i,-1\pm i$$ But can you use something like: $$x = \pm 1 \pm i$$ NOTE: I know the example above just gives $$x = 1+i, -1-i$$, but I just wanted to provide an example

• I would write $x = 1+i, 1-i, -1+i, -1-i$ (with maybe a line break). No need to complicate things. – talbi Mar 2 at 10:24
• I don't think that $1\pm i, -1\pm i,$ can be regarded as either ambiguous or confusing. However, while technically, a case could be made that $\pm 1 \pm i$ is similarly unambiguous, some may find it confusing/ambiguous. – user2661923 Mar 2 at 10:29
• Overkill: $x\in\{\sqrt{2}e^{i\pi a}\mid a=k+1/4,k\in\mathbb{Z}\}$ – Pixel Mar 2 at 10:44

## 4 Answers

Do you think $$(-1)^x \left( 1 \pm i \right) \\ x\in\{0,1\}$$ is a nice idea?

Suggestion: let $$z=1+i$$ and

$$x \in \{z , \bar z, -z , - \bar z \}.$$

Another variation is \begin{align*} x^4=-4 \end{align*}

• The question asked for a method to combine plus and minus. It never came to me that + and - could sum up to zero ;). Now there is no need of sums! – SteelCubes Mar 3 at 6:06
• @SteelCubes: Yes, thanks to the fundamental theorem of algebra we can encode the wanted information as precisely the zeros of a polynomial of fourth degree. :-) – Markus Scheuer Mar 3 at 6:54

Writing $$x = \pm 1 \pm i$$ is ambiguous as it could either mean $$x \in \{1+i,-1-i\}$$ or the intended $$x \in \{1+i,1-i,-1+i,-1-i\}$$. You can make it unambiguous by specifying (in words) that the $$\pm$$-signs can be chosen independently. I've also seen $$x = \pm 1 \pm' i$$ to make the independence of the $$\pm$$-signs explicit in the notation, but I'm not sure that's very common.

• I've used things like $\pm^a(x\pm^b y)$, but only for personal calculations, not public consumption. ;) – PM 2Ring Mar 2 at 19:40