Formally, what is the plus/minus $\pm$ object? My guess is this:
$$
\pm: F \to F\times F
$$
where $F$ is a field.
For instance, $\pm 2$ maps $2 \in \mathbb{R}$  to the tuple $(2,-2) \in \mathbb{R}\times \mathbb{R}$.
Is it correct to think of the $\pm$ in this manner?
 A: Often it is used to code two different equations.  One with $+$ the other with $-$.  [That is easier than trying to imagine equations involving addition of numbers to pairs of numbers.]  
For example,
$$
x_{\pm} = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
is a short way to write two equations:
$$
x_{+} = \frac{-b + \sqrt{b^2-4ac}}{2a},\qquad
x_{-} = \frac{-b - \sqrt{b^2-4ac}}{2a}
$$
A: If you want a formalism, it's better to replace the $\pm$ symbol with something else altogether, because it's hard to make sense of expressions like $1 \pm \sqrt5$ if you regard $\pm$ as an operator.
For example, instead of writing $1 \pm \sqrt5$ you could write
$$1 + \varepsilon \sqrt5 \quad\mathrm{where}\quad \varepsilon \in \{-1,1\}.$$
This can be useful when dealing with formulas where at least one of the symbols $\pm$ or $\mp$ occurs multiple times. It lets the formula itself provide an answer to the questions, "If one of the $\pm$ signs is positive, must they all be positive? If one of the $\pm$ signs is negative, must they all be negative?"
If you use the same $\varepsilon$ throughout the formula, the answer to both questions is "yes";
if you define multiple factors like $\varepsilon_1 \in \{-1,1\}$ and $\varepsilon_2 \in \{-1,1\}$ then the answer to both questions is "no."
I have seen this technique used in formal proofs.
A: Strictly speaking it's not correct. When we say the solution to the equation $x^2-1$ is $\pm 1$, that's not the same thing as $(1,-1)$, which cannot be a solution owing to the fact that it lives in the wrong space.
Closer might be $\{1,-1\}$, but that's still not really correct -- think of the $\pm$ that occurs in the quadratic formula, for instance. Best to think of $\pm$ as shorthand with no formal meaning.
