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Why does the result only equal the principal root? For example, why is the square root of -16 just 4i, instead of positive and negative 4i?

When I think about this, I see sqrt -16 as sqrt 16 * sqrt -1. The sqrt 16 is positive and negative 4 and square root -1 is i, I get the result pos and neg 4i. All my textbooks just list 4i. Why is that?

Forgive my syntax, but when the question is stated: simplify sqrt -16, I still believe the answer is + and - 4i. Yet, my texts show only 4i and do not include -4i. Is this simply for convention sake? Or a higher reason than I am missing? The solutions that I have seen thus far indicate, that it is convention, rather than pure mathematics that this is so.

Speaking with colleagues, I have been told that unless a positive/negative symbol is placed before the square root of -16 the expression can only be equivalent to 4i. Is this accurate? And if so, why?

Nikki
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  • There is no "the square root" of a complex number, as a nonzero complex number has two square roots. I'd say both "$4i$ is a square root of $-16$" and "$-4i$ is a square root of $-16$". – Angina Seng Jan 13 '18 at 04:56
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    "The sqrt 16 is positive and negative 4" Huh? Any reference for this meaning of $\sqrt{16}$? – Did Jan 13 '18 at 07:08
  • I thought that that what was I was trying to convey, that both 4i and -4i are square roots of -16. Am I still stating that idea incorrectly? – Nikki Jan 13 '18 at 07:51
  • You have put @Did in your comment if you want him pinged. – spaceisdarkgreen Jan 13 '18 at 08:02
  • @Nikki I think what he meant is that $\sqrt{16}=4$ is the usual definition full stop. Saying things like "16 has two square roots, $\pm 4$" is also common, but when we write $\sqrt{16}$ we mean $4,$ not $-4$ or some ambiguous combination of the two. – spaceisdarkgreen Jan 13 '18 at 08:09
  • @spaceisdarkgreen So, if I say sqrt 16, that must be 4, no other answer. But if I ask what are the roots of 16, the answer is +/- 4? Yet, when solving equations and actual use of sqrt 16 we need to account for -4. I suppose my issue is more with the appropriate use of the principal square root and why we even use it at all then. – Nikki Jan 13 '18 at 08:16
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    @Nikki I mean, expressions evaluate. Like https://www.wolframalpha.com/input/?i=sqrt(-16)*sqrt(-25) . It's a nice feature, no? It's true that you often have to think about both roots, but it's nice to be able to say something like: "clearly the root that gives us our attracting fixed point is $-\frac{b}{2} + \sqrt{(b/2)^2-4c}$ because (reasoning)" and actually know which one you're talking about. – spaceisdarkgreen Jan 13 '18 at 08:40
  • Related: Why do we assume principal root for the notation $\sqrt{}$ –  Jan 16 '18 at 01:03

1 Answers1

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You are correct that both $4i$ and $-4i$ are square roots of $-16,$ just as $4$ and $-4$ are both square roots of $16.$

Nonetheless we like sometimes to pretend that "the square root" is a function. One convention for defining a function on the complex numbers that results in a square root is to write a complex number $z$ in polar coordinates as $$ z = x+iy = r\cos\theta + i r\sin\theta$$ for $r\ge 0$ and $0 \le \theta < 2\pi$ and then define $$\sqrt{z} = \sqrt{r}\cos(\theta/2) + i\sqrt{r} \sin(\theta/2),$$ which you can check has the property that $(\sqrt{z})^2 = z.$ This results in every square root being in the upper half-plane.

By this definition, $$\sqrt{-16} = 4i. $$ However, this function has some odd properties. For one, it is discontinuous at the positive real axis since a point just below the positive real axis will get mapped to a point just above the negative real axis whereas points on the positive real axis are mapped to points on the positive real axis. Also, it doesn't have many of the arithmetical properties we are used to, and this makes calculation tricky. For instance, we have $$ \sqrt{-4} \sqrt{-9}=(2i)(3i) = -6 \ne \sqrt{(-4)(-9)} = 6.$$

Generally speaking, it is better to think of the square root as a multi-valued function, but it can be nice to use the single-valued version for the sake of writing down arithmetical expressions with unambiguous meaning. Either way, thought and care are required.

spaceisdarkgreen
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  • I feel that this answer fits what I was looking for. I especially like the use of polar coordinates, because I think that is what my colleagues were trying to convey (Without actually bringing them up), while I was looking at it from the perspective of a possible multi valued function. With my students, I tend not to use the pricipal square so much and rather choose to show why certain values or results don't make sense in the context of the problem. – Nikki Jan 13 '18 at 07:56