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## Algebra (all content)

### Unit 16: Lesson 10

Multiplying & dividing complex numbers in polar form

# Dividing complex numbers: polar & exponential form

Sal shows how complex division affects the modulus and the argument of the divisor and the dividend. Created by Sal Khan.

## Video transcript

Voiceover:So this kind of hairy looking expression, we're just dividing one complex number, written in blue, by another complex number. This first complex - actually, both of them are written in polar form, and we also see them plotted over here. This first complex number, seven times, cosine of seven pi over six, plus i times sine of seven pi over six, we see that the angle, if we're thinking in polar form is seven pi over six, so if we start from the positive real axis, we're gonna go seven pi over six. So we're gonna go seven pi over six, all the way to that point right over there. And then from the origin, we're going to step seven out, or seven away from the origin. So one, two, three, four, five, six, seven to get to that point right over there. And then the second number, the angle is seven pi over four. So that takes us all the way around here. It takes us all the way - so I should do it smaller since that has a smaller distance, so let me - so let's say we start over here, so we're going to go all the way over here to that point right over here, and it's distance from the origin is one. You can imagine that there's a one in front of that. And we want to divide the two, and I encourage you to pause this video and try to do this on your own, and then plot the resulting number when you divide this blue complex number by this green one. Well, as you might have realized, if you somehow just try to divide it straight up, it can get quite hairy, and the way to tackle it, is to write it in another form. And what might have jumped out at you is that exponential form would be much, much simpler. And the way that we convert this to exponential form is to recognize that this business right over here is the same thing, this comes straight out of Euler's formula, this is e, to the seven pi over six i. Seven pi over six i. That's this expression right over here. So this entire top complex number can be rewritten as seven e, to the seven pi over six i. And this bottom complex number can be rewritten as one times, we didn't really have to write the one, but this bottom part right over here is going to be the same thing as e, to the seven pi over four i. Seven pi over four i, and this comes straight out of Euler's formula. And when you write it in this way, then we can just use exponent properties to simplify it. We have the same base, and so we can just subtract this exponent from that exponent right over there. So this is going to be equal to seven divided by one, is just seven. So it's going to be seven to the, or seven times e. Let me do this in a brighter color. Seven times e, to the seven pi over six i, minus seven pi over four i. Minus seven pi over four ith power. So what is this going to be equal to? Well if I have seven pi over six of something and I subtract seven pi over four of that thing, how many do I have left over? Well, let's see, I'm essentially - let's just rewrite all of this. This is really just about subtracting fractions at this point. So if I were to rewrite this, let's see, if I were to rewrite it with a denominator of 12, then I'll have a common denominator. It's my least common multiple of six and four. So this one I can rewrite as 14 pi i over 12. Or I can write it as 14 pi i over 12, and then minus - so I multiplied the numerator and denominator by three, minus 21 pi i over 12, as well. Right? I just multiplied numerators and denominators by three right over here. And so this is going to be equal to, I'm going to have, 12 in the denominator, 14 pi i, minus 21 pi i, is going to be negative seven pi i. So, this thing is equal to seven e, to the negative seven pi i, over 12. Now let's see if we can do a decent job of plotting this. So let's see, each of these increments that they've done, in each of these quadrants, we have one, two, three, four, five, six. So they split this quadrant into six equal angles, and this quadrant is pi over two, so each of these are pi over 12. Each of these little subangles are pi over 12. So we're going to have negative seven pi over 12. So we're going to go in the negative direction. We're going to go counter-clockwise, or sorry, we're going to go clockwise. So we're going to go, let me start here, so if I go one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, gets us right over there. Did I do that right? That doesn't - let me make sure I got that - negative seven, so each of these are pi over - let me make sure. So this whole thing is pi, so we have one - let me make sure, this whole thing is pi. We have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12. So each of these are a twelfth of that, so it's pi over 12, that's right. Then we want to go negative seven of them. Negative one, negative two, negative three, negative four, negative five, negative six, negative seven, oh ya, I just kept going all the way to negative 12 the last time. So this is the angle, this is our angle right over here. And our distance that we go out from the origin is seven, so we go out one, two, three, four, five, six, seven, so we come out right over there. So this complex number divided by that complex number is equal to this complex number, seven times e, to the negative seven pi i over 12. And if we wanted to now write this in polar form, we of course could. We could say that this is the same thing as seven, times cosine of negative seven pi over 12, plus i sine of negative seven pi over 12.