Is $\mp a$ actually different than $\pm a$?

Question

So, the way I understand $\pm a$ as a general concept is basically as follows: $\pm a$ is really just two numbers, functions, or whatever $a$ represents, but the catch is that one of the $a$'s is positive, and the other is negative.

All of that makes sense to me. Mathematicians like to be efficient, but also precise, so they created a way to represent two (or more) entirely different objects, simply by using a special symbol.

These, then, are my questions:

• Is $\pm a$ actually different than $\mp a$?
• Why aren't there more these, if you will, Frankenstein symbols?
• If the answer to my previous question is, "There are," then why aren't they as common?

Answer 1

On its own, $\mp a$ means the same thing as $\pm a$.

However -- and this is a big however -- you almost never see $\mp$ unless it occurs in an expression with $\pm$ being used as well. And then it means "the opposite of whatever sign $\pm$ is currently."

For example, the sum-or-difference of cube factorizations can be fit into one formula $$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2),$$

where it should be understood that the sign in $(a \pm b)$ needs to be different from the sign of $ab$.

I suspect $\pm$ and $\mp$ are atypical because addition and subtraction are such well-behaved mirror-images of each other. More importantly, there are many situations in which you'll "either do one, or the other."

I can't really think of any pair of operations that work like that, to they point that we'd 'Frankenstein' their symbols together. That doesn't mean there aren't any, but even multiplication and division don't tend to crop up like that, and they'd be obvious second candidates. (Although we could write "either $ab$ or $\frac{a}{b}$" as $ab^{\pm 1}$ if we wanted to, and still not need a comparable hybrid symbol).

Answer 2

Standing by themselves $\pm a$ and $\mp a$ mean the same thing; usually the set $\{ a, -a \}$, such as solutions to some equation.

The notation $\mp a$ is a handy abbreviation in certain contexts. For instance $$b_\pm = c \mp d$$ means $b_+ = c -d$ and $b_- = c + d$.

Answer 3

Yes, there are functions of the form $a\pm b\mp c$. Which is the two functions $a-b+c$ and $a+b-c$.

Answer 4

$\pm a$ and $\mp a$ represent the same thing. However consider expressions:

$x \pm y \pm z$

and:

$x \pm y \mp z$

First one stands for either $x+y+z$ or $x-y-z$ while the second stands for $x+y-z$ or $x-y+z$. That's why we have two symbols.