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Current time:0:00Total duration:5:25

Complex numbers — Harder example

Video transcript

- [Instructor] Which of the following is equivalent to the complex number shown above? And then we got this big, hairy mess here, where we wanna take the rational expression one plus i over one minus i and then add that to one over one plus i. Well, when you add two fractions like this, what you might wanna do is find a common denominator. And the easiest way to find a common denominator is to take the product of both of these things. Let me just rewrite everything. So we wanna take one plus i over one minus i and then add that, and let me switch some colors just so we can keep track of things, and then add that to one plus, or so one over one plus i. And actually, let me give myself some space here. So plus, plus one over one plus i. Well, we're able to add two fractions if we have the same denominator. And we can have the same denominator if we multiply both the numerator and the denominator here by this denominator, by one plus i, and if we multiply both the numerator and the denominator here by this denominator. And this is the way that you've always added fractions with unlike denominators. We're just finding a least common multiple. And that least common multiple, the easiest one, is just multiplying the denominators. So let's do that. So let's multiply the numerator and denominator here, so put some parentheses, times one plus i, and let's multiply the denominator here times one plus i. So notice, one plus i over one plus i, that's just going to be one. So we aren't changing its value, we're just going to find another way of expressing it. And this one over here, we wanna multiply the numerator and the denominator, and the denominator by this denominator, by one minus i, one minus i. So we're gonna multiply times, we're going to take this and multiply one minus i, multiply it by one minus i over one minus i. And once again, this thing right over here is just one. So we're not changing its value, we're just finding a good way to rewrite it so that we're going to have the same denominator. Notice, both of these denominators are now going to be the same thing. They're gonna be one plus i times one minus i. But let's figure out what this is. So this first one right over here, so the numerator, so let's see, we're gonna have one times one. Actually, let me do this in different colors just so we can keep track of everything. So we have this numerator right over here. One times one is one. One times i is i. I times one is i. And then i times i is negative one, is negative one. And then this denominator over here is going to be, so we're gonna have divided by one times one. And you could say that this is a difference of squares right over here. Well, let me just, I'll multiply it out. We'll see that it's a difference of squares. So one times one is one. One times plus i is plus i. Negative i times one is minus i. And then negative i times positive i, well, i times i would be negative one. But then we have this negative, so it's just going to be plus one. And so what does this simplify to? Let's see, you're gonna have one minus one, so those cancel out. And then you're gonna have this i minus i, that's just going to be zero. So this simplifies. In the numerator, you have two times i. And then in the denominator, in the denominator, you have one plus one is equal to two. And I can simplify this, but my goal isn't to simplify this. My goal is to have the same, is to have like denominators, so I'm just gonna leave this like this, just like that right now. And now let's move on to this one. Let's move on to this one. So the numerator, the numerator here is pretty straightforward. One times one minus i, that's just gonna be one minus i. And then the denominator here, one plus i times one minus i, we just figure that out. That simplifies to two. So that simplifies to two. So this has now become two i over two plus one minus i over two. Well, what's that going to be? Well, our denominator's going to be two. You could say we have two i-halves plus one minus i-halves. So how many halves is that going to be? Well, we wanna add, this has no real parts, so we wanna add the imaginary parts. So we're gonna have the one, this one, and then we could add, could take two i and then subtract i from that. Two i minus i is going to be i, so it's going to be one plus i over two. And they don't have exactly that. But if we then, if we just, instead of writing it like this, if we viewed this as, if we viewed this as 1/2 times one plus i, which this is the same thing, or if we viewed this as 1/2, if we distribute the 1/2, 1/2 plus 1/2 i or plus, ah, let me just write it 1/2 plus 1/2 i. And when we write it that way, when we write it that way, we see this choice is exactly that. And we are done.