Plus or Minus = Minus or Plus It is the same?

Question

Are "Plus or Minus" or "Minus or Plus" the same ?

$$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

$$x = \frac{-b\mp \sqrt{b^2-4ac}}{2a}$$

$$x^2=9 \implies x = \pm 3 \quad \text{or}\quad x = \mp3$$

These signs can we use a normal mathematics lesson, or can we use the lesson of algebra?

Answer 1

Sometimes we can write for example

$$x=\pm 1 \quad y=\mp 1$$

to indicate that

• the value $x=+1$ corresponds to the value $y=-1$
• the value $x=-1$ corresponds to the value $y=+1$

otherwise they are equivalent symbols.

Answer 2

logistically $\pm a$ means "either $a$ or $-a$" and $\mp a$ means "either $-a$ or $a$". Logistically they are exactly the same.

However $\mp a$ looks unnatural and lopsided so by convention we always use $\pm a$.

So if we ever do see $\mp a$ there usually is some other reason; that elsewhere in the expression there is an indication that whether we chose $a$ or $-a$ is dependant on something else. For instance: If I saw: $k = a \pm \sqrt {b \mp c}$, I would interpret it as there being two cases: Either $k = a + \sqrt{b - c}$ or that $k = a - \sqrt{b + c}$.

But this is unavoidably ambiguous. I would also inconsistantly interpret a statment $k = a \pm \sqrt{b \pm c}$ as having four possible cases: $k = a + \sqrt{b + c}; k = a -\sqrt{b+c}; k = a +\sqrt{b-c}; k = a-\sqrt{b-c}$.

Care should be given to avoid potential ambiguity.