Square roots and solutions to equations with squares. Why are there negatives and positives?

Question

So it seems that my textbook is making some assumptions that I do not fully understand.

So I understand that this symbol $\sqrt()$ refers to the principle square root. So the $\sqrt(4)$ = 2 and not -2 because that is the definition of principle square root.

In complex numbers, $\sqrt(-4)$ = $2i$ and not $-2i$ and I'm fine with this.

But why is that:

$x^2$ = $-9$

$x = \pm\sqrt{-9}= \{3i,-3i\}.$

Can someone point me to a good reason why this is the case?

Answer 1

The equation $x^2=-9$ has two solutions : $-3i$ and $3i$, but $3i$ is the pinciple square-root, therefore $\pm \sqrt{-9}$ because one solution is $\sqrt{-9}=3i$ and the other $-\sqrt{-9}=-3i$

Answer 2

If you factorise the equation $$x^2+9=0$$which is a quadratic, you naturally get two linear factors $$(x+3i)(x-3i)=0$$this equation holds if either of the factors is equal to zero.

---

Interlude

This happens because you are in a context where there are no (non-trivial) zero divisors. In other circumstances, different results occur. eg $x^2-1\equiv 0 \bmod 8$ has four solutions modulo $8$. These are (the classes of) $1,3,5,7 \bmod 8$.

---

Back to the main story:

If you are extracting a square root, you normally want to define one of the two values as the principal value, so you can keep track of what is going on. If you are solving an equation, you generally want to find all the possible solutions, and these may not be restricted to the principal value.

Answer 3

I think that there is a slightly deeper reason here, actually two reasons: one mathematical and one practical.

The mathematical reason is as follows: real and complex numbers have an important difference:

• You cannot ‘shuffle’/‘relabel’ real numbers in such way that all operations are preserved. (In math, we say that “the field $\mathbb R$ does not admit a nontrivial automorphism”, for details see a related post.
• However, you can do that with complex numbers: the mapping of $a+bi\mapsto a-bi$ preserves all properties of complex numbers. Thus, $i$ and $-i$ are indistinguishable (unlike e.g. $2$ and $-2$), it is impossible to tell which is which.

The bottom line is that it makes much less sense to define a “principal” square root of a negative number than of a positive number. So even if some books would still refer to $2i$ as a “principal” square root of $-4$, this is just done for pedagogical reasons rather than being motivated by math.

The practical reason is, obviously, that the formulas with the $\pm$ sign work, in the sense that they convey the desired meaning. E.g. in the context of solving the equation $x^2+9=0$, the solutions are $3i$ and $-3i$, and convention to express them as $\pm\sqrt{-9}$ just manages to show both solutions in a compact manner. It is just language, maybe imprecise, but good enough.