2

So I am doing an exercise on khan academy and I answer this question and ask for the explanation and I don't understand this Step. Why does changing the placement of the i change the fraction from negative to positive?

Cuppa Milk
  • 21
  • 3
    It doesn't. The notation is dreadful, but what they're doing is just consolidating the minus inside the $\pm$, like this: $\pm(-a)=\mp a$ – Sassatelli Giulio Aug 21 '22 at 09:54

1 Answers1

2

It didn't. The expression is $$\pm \color{red}{-}\frac{i \sqrt{3}}{2},$$ where I have highlighted the extra $-$ sign in red. This is what changes the $\pm$ symbol to $\mp$, not the placement of $i$ after $\sqrt{3}/2$. For instance, we have $$\pm (-1) = \mp 1,$$ although in the absence of another $\pm$ symbol in the expression, the choice of $\mp$ or $\pm$ is not important. What I mean is that we typically only write $\mp$ if there is another choice of sign elsewhere in the expression; e.g., in the identity

$$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b,$$

we must choose on both sides of the equation the top sign $+$ or the bottom sign $-$, but cannot choose $+$ on the left and $-$ on the right. So in the corresponding cosine identity

$$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b,$$

the use of $\mp$ on the right is necessary in order for the sign choice to be correct: if we choose the top symbol $+$ on the left, we must also choose the top symbol $-$ on the right.

But when there is no such restriction, e.g.,

$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$ then the expression $$\frac{-b \mp \sqrt{b^2 - 4ac}}{2a}$$ is equivalent, and $\pm$ is typically preferred instead of $\mp$, because the use of $\mp$ implies that there exists some other $\pm$ in the expression for which the choice of sign must be made "top-top" or "bottom-bottom" as exemplified by the cosine identity above.

Note that on occasion, some authors might violate the "top-top"/"bottom-bottom" convention when writing an expression, for instance

$$\pm 1 \pm x \pm x^2 \pm x^3 \pm \cdots$$ could imply that the choice of sign is independent for each term in the sum. For if the restriction were to apply, the author would probably have written instead

$$\pm (1 + x + x^2 + x^3 + \cdots),$$

which would force all signs to be the same. As you can see, the use of $\pm$ and $\mp$ is sometimes not clear, in which case it may be necessary for the author to specify or explain what is meant.

heropup
  • 135,869