Trigonometry: More Roots of Complex Numbers

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More Roots of Complex Numbers
Let's take another look at taking roots of complex numbers and let's consider the following. Let's find the cube root of -27. So what are the cube roots of -27? There should be three of them. Let's see if we can think about what they're going to be. Well, one of them we can actually figure out by just taking the good old-fashioned cube root that we know, and I think we'd see -3. Let's check. , times another would be -27. So one of the three roots should be -3. But then there are two other roots and those two other roots are probably going to be complex. And now how do we find them? Well, we use that formula we just derived.
So the first thing I have to do is write -27 in polar form. Now how in the world do you do that? Well, you just graph it. For -27 there's no imaginary part; it's just -27. So you can see the length of this vector is -27. That's pretty easy. And what's the angle? Well, the argument or the angle is just . So in fact it's easy to write -27 in polar form. It just equals 27, because that's the length; the length of this is 27, . I take the length of this, which we measure as 27 units, and then the goes in there.
Well, now I just use the formula, because what does the formula say? So the th root--the cube roots are three, so the cube roots are going to be what? Well, I take the cube root of 27, and the cube root of 27 is 3. So that's the cube root of 27. And then I've got cosine of what? I take , which is this case is , and I add and I divide by the root I'm taking, which in this case is 3. And I do the same thing with sine, . And what does do? Well, is going to range from 0, 1 and 2. Those are my three roots. One of those hopefully should be the -3 that we already knew and then the other two should be the new roots that we haven't found yet.
So let's plug in and see what happens. If I plug in , the first thing I see is 3 times cosine of what? Well, if I plug in zero, then I just get . So I see , which is 60°, plus . And what number is that? What's ? Well, that's . That's a standard one that we know. Plus times , which is . And so the number I see is . So that's one complex number, one complex root.
Let's see if we can find the other complex root. Let's let . If , then I have 3 cosine of--if I let , then I have . Well, that's . is just . So I just see , and that's 3 times what? Well, , , so this is just -1. So in fact all I see is -3. Hey, -3, there's that one root that we already knew! You see how this formula bubbles that one out? So we actually found that there's the one real root. Cool. Now there must be another complex root. Let's see if we can find that.
Three times cosine--that must be when . If I put a 2 in here, then I see a 4. , over 3, plus . And where is ? Well, if you actually think about that, what angle is that? You could figure out what angle that is. That's going to be 300° if you work it out. Three hundred degrees. So what you're looking at is an angle--if I could find my angle stuff--that looks like this, so it's 300°. So this angle here--if the whole thing is 360, this must be 60°. So basically we're looking at a reference angle of 60°. But what sign do we have to look at? All Students Take Calculus. So the cosine will be positive but then the sine will be negative. So we're looking at 60°, so we're looking at 3 times--and so cosine is going to be positive, so it's going to be , but the sine is negative there, so .
So there are the three roots. We actually found it using the formula. One root is 3 times the quantity . The other one is very similar, notice. In fact, notice it's the complex conjugate. If you remember the complex conjugate, it's the exact same thing but this sign has been switched. That, by the way, is not a fluke. That will always happen. These things will come in pairs like that when you're taking the roots of a regular real number. And then we have that real root, -3. So there is the answer.
And I want to close by just thinking about this thing visually. Where are these points in the complex plane? Well, if you graph them and plot them, you see something really interesting. So if you compute the length or the moduli or the absolute value of these three numbers--and I'll let you try to do that on your own--you'll see that they all have length 3. So all these points actually live on a circle of radius 3. This is radius 3 and these points are going to be somewhere on there. Now, the -3, that's pretty easy to see. The -3 is just right over here at -3. So there's one root. Now, where is this root? Well, that's at an angle, remember, of 60°, so I go up 60° here, and I go right here. There's another root. And this is actually -60°, so I go down that exact amount. And if you're thinking about it, this is 60° and this is 60°, and this whole thing is 120°. But if this is 120°, we can figure out this angle. If this is 60°, this is 120°. So in fact these points are equally spaced around this circle. They're equally spaced around this circle, and that is exactly what happens. Whenever you take roots of a number, they're all going to lie on a circle, the circle whose radius is the root of the number you're taking--so in this case, if you take the root of, in this case 27, you're going to see 3. They're all going to live on a circle of radius 3, and they're going to be equally spaced on that circle. If there are three roots, they're all going to be 120° apart. If there are four roots, they're going to be one here, one here, one here, and one here. They're all going to be the same distance apart. They may be a little skewed--they may be like this--but they'll always be the same distance apart from each other. So it will divide the circle up evenly into those pieces. That's always how roots will fall.
Up next we'll take a look at some very special roots. Those are called "roots of unity." That's when in fact this number is just -1. You want to take all the roots of -1. So for example, is just . And you'll notice even there, and , it cuts the circle in two equal pieces. What about if I take higher roots? It will cut the circle into more and more pieces, and in fact you can see how that plays out. So it's a neat geometric observation that if you take roots of a number, they live on the same circle and they spread themselves out evenly along that circle. Really cool. We'll see more of that up next. Talk to you there.