Complex roots, in a Cartesian sense
So we figured out how to take the square root of \(i\). We did this by assuming that the square root of \(i\) was some complex number, and thus we could write it in the form \(a+bi\), and thus we got an equation, and solved it for \(a\) and \(b\):
\[\begin{align*} \sqrt{i} &= a+bi \\ &\quad\vdots\\ &\text{lots of algebra later...}\\ &\quad\vdots \\ &=\frac{\pm1}{\sqrt{2}} + \frac{\pm1}{\sqrt{2}}i \end{align*}\]
This method seemed to work, but the result is weird. We still don’t have a good understanding of what the square root of a complex number is. I mean, by definition, a square root of any number is just the number that, when you multiply it by itself, you get the original number back. And that’s what we found here. But it’s still so weird. Where are the fractions coming from? The \(\sqrt{2}\)? We know what it is, but we don’t know why it is.
Contrast this with taking the square roots of real numbers (positive real numbers). We have a pretty good idea about how that works. For example, we don’t know exactly what \(\sqrt{17}\) is, but it’s definitely something less than \(17\). Square roots make things smaller (at least when we’re talking about real numbers). Or, I guess they actually make them bigger if it’s less than \(1\). So I guess we should say that when we’re taking the square root of a real number, that makes it closer to \(1\).
\[1 < \sqrt{17} < 17\]
And we know more than this. We know \(\sqrt{17}\) is a little bigger than \(\sqrt{16}\), which is \(4\), so \(\sqrt{17}\) must be a little bigger than \(4\). And it has to be less than \(\sqrt{25}\), which is \(5\):
\[4<\sqrt{17}<5\]
So we’ve got some basic intuition and ideas for how square roots work with positive real numbers. But with complex numbers?!?! No clue. If we were together in class last semester, you got to watch that wonderful short documentary on the proof of Fermat’s Last Theorem, and Andrew Wiles describes doing mathematics as feeling like:
Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.
Right now, we’re stumbling around in the dark. We’ve bumped into one piece of furniture—the square root of \(i\). Let’s bump into some more furniture!
Can you figure out what \(\sqrt[3]{i}\) is, using the same method as we did to find the square root of \(i\)? (Quick question beforehand: we found that \(i\) has two square roots. This should make sense: \(4\) also has two square roots, \(+2\) and \(-2\). How many cube roots do you think that \(i\) has? Analogously, how many cube roots does \(8\) have?)
Likewise, can you find the fourth roots of \(i\)?
Graph all this stuff, too.