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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?

user551850
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  • Is the question regarding the way to use the Plus-minus sign ? – Mauro ALLEGRANZA Apr 18 '18 at 14:02
  • It is only a convention to write ± instead if reversing them. – Mauro ALLEGRANZA Apr 18 '18 at 14:03
  • The technically mean the same thing. The difficulty arises when there are two of these ($\pm a; \pm b$). If they are independent there are four options ($(a,b),(a,-b),(-a,b),(-a,-b)$ but if the are dependent there are only two options $((a,b),(-a,-b))$ which of these interpretations is meant is hopelessly ambiguous. If the latter is should be written $\pm(a,b)$ IMO. As $\mp$ is not standard, if we do see it it probably means dependence. $(\pm a, \mp b)$ probably means $(a,-b)$ or $(-a,b)$. But this should be made clear by $\pm(a,-b)$ IMO. – fleablood Apr 18 '18 at 16:57
  • @user551850 Please remember that you can choose an answer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Apr 19 '18 at 19:42

2 Answers2

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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.

fleablood
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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.

user
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