Transcription
So far, we've learned what a complex number is; we've even learned how to graph it. And we learned how to add, subtract, and multiply it. Where I left off in the last video was, how do we divide two complex numbers? So I said, let's say I have one complex number, z1. And that equals a + b i. And I want to divide that by z2, which is c + d i.
So let me ask you a question. And I touched on this in the last video. And let me do it in a different color over here. We know that (a + b)(a - b) = a² - b². And you can multiply it out, in case you're not sure. Remember, it's just a times a + b times a - a times b, plus a times a, and you'll get this. But you know how to do this, anyway. There's a review of it, if you need to do it.
So, given that, what is c + d? What happens if we do something very similar with a complex number? If we say (c + d i)(c - d i). Well, in this case, a is c. And b is d i, right? So this is just going to be equal to c² - (d i)². This equals c² - d²i². And that equals c² - d²(-1). And i² is -1, right? So this is going to be multiplied by -1, so it cancels out this negative. So you get c² + d². That's interesting. When I multiply a complex number times this other number, which is very similar to it, but it's kind of the imaginary part, goes in the other direction. When I multiply the two, I get a completely real number. All of the i's disappear.
And, in general, this number -- if we call this -- well, in our example this was z2, so if we say that z2 = c + d i, the quantity c - d i is called its conjugate. And that's just good terminology to know. And the sign for conjugate is that line over the top. So the conjugate of z2 is c - d i. Or you could say, the conjugate of c - d i = c + d i. Or you could say it the other way around. The conjugate of c + d i = c - d i. And notice, we're just switching the direction in the imaginary -- along the imaginary axis, when we take the conjugate of something.
With that said, let me erase that and go back to our original problem. Because the conjugate is the tool we're going to use to divide this. So we know when we multiply an imaginary number times its conjugate, we get a real number. And we know, also, if we multiply -- we can multiply anything by 1, and we get the same number. So let's multiply the numerator and denominator of this expression by the conjugate of the denominator. So let me do that. So the conjugate of the denominator is going to be c - d i. So (c - d i)/(c - d i). So this was c + d i, so this is its conjugate.
And so what do we get? So in the numerator, we get a c -- I don't want to run out of space, I always do -- a c, so a times c, - a d i, - a d i -- these i's are looking funny -- this is an i. + b c i; + b c i. And then the last term, we have a + b - b. So it's - b d i². - b d i². All of that. And this is (a + b)(a - b). So it's equal to a² - b². So this is going to be equal to -- and this will become second nature to you after a while, but you might want to just multiply it out. This equals c² + d². And don't take my word for it. Actually, algebraically, multiply this out and just realize you can only add real parts to real parts and imaginary parts to imaginary parts.
So let me simplify that. That equals -- let's see, the real parts. This is real, a c. And this is - b d i². So the i² is -1. So it switches the sign here, so it becomes + b d. And we can get rid of d i. So the real parts are a c + b d. That's that, and that. And then the imaginary parts are + -- this one's positive, so I'll put one first -- b c - a d i, all of that over c² + d². And that still might not look like a complex number to you. But then we can separate them out and we could say well, that equals (a c + b d)/(c² + d²). And that's the real part. + (b c - a d)/(c² + d²) And that times i, and that's the imaginary part. So you can't merge, when you're adding and subtracting, the real part to the imaginary part. But you can most definitely scale an imaginary number by a real number. And that's essentially what we're doing. We're multiplying 1/(c² + d²) times this.
So, division might seem a little complicated when I write it all in variables. But let me give you an example and you will hopefully see that it -- with real numbers, and -- not real numbers, with actual numbers, I should be careful with what I say. Let's say I have 1 + 2 i. And I want to divide that by, I don't know, let's divide it by, I'm going to pick a random number. 2 + 3i. And so what do we do? We multiply it times the conjugate of the denominator. (2 - 3i)/(2 - 3i). Because then we're not changing the number. This is just 1, this simplifies to 1. It equals -- the bottom, we can multiply it out. But hopefully it's second nature to you. It equals 4 + 9, right? Because that's just a² + b². Right? Well, I mean, it's a² - b², but then the i's, when you multiply, and it becomes a negative number. Try it out if you don't believe me.
And then the top, we get 1 times 2 is 2. 1 times -3i is -3i. And you have 2i times 2, which is +4i. And then you have 2i times -3i. So that's -6i². Well, what does i² equal? That equals -1. So -1 times -6. Get rid of the i² and this becomes a positive. So then, what are our real parts? Our real parts are 2 and 6. so 2 + 6 is 8. And what are our imaginary parts? -3i + 4i. So that's just +1i, right? -3 + 4 is positive 1. So it's just +1i. Over 13. Or we could write that as -- if we wanted to write that in the traditional complex form -- is 8/13 + 1/13i.
So when I divided one complex number by another, I got another complex number. And an interesting exercise for you to do is, pick some random complex numbers. Plot them out on the complex plane, and see what happens when you multiply them, when you divide them, when you add them, when you subtract them. And when you scale them. Or when you take the conjugate. And that'll give you a better intuition of what's going on with these numbers. Anyway, I will see you in the next video.