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# Dividing complex numbers: polar & exponential form

## Video transcript

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 7 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 going to go seven PI over six so we're going to 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 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 a smaller distance so let me so let's say we start over here so maybe go all the way over here to that point right over here and its 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 could be rewritten as seven e to the seven I / 6i and this bottom complex number can be rewritten as as one times we don't really have to write the 1 but this bottom part right over here is going to be the same thing as e to the 7 PI over 4 I 7 PI over 4 I this comes straight out of Oilers formula when you write it in this way then we could 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 well 7 divided by 1 is just 7 so it's going to be 7 to the or 7 times e let me just in a brighter color 7 times e to the 7 PI over 6 I - minus 7 PI over 4 I minus 7 PI over 4 I power so what is this going to be equal to well if I have 7 PI over 6 of something and I subtract 7 PI over 4 of that thing how many do I have left over well let's see I'm essentially let's just rewrite let me 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 write it with a denominator 12 then I'll have a common denominator it's my least common multiple of 6 and 4 so this one I can rewrite as 14 PI over 12 or I could write it is 14 pi like let me just write this 14 PI over 12 and then minus so I multiply the numerator and denominator by 3 - 21 pi PI I PI I over 12 as well right I just want to find numerator denominator by 3 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 7 negative 7 PI I so this thing is equal to 7 e to the negative 7 PI I over 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 let's see in each of these quadrants we have 1 2 3 4 5 6 so they split this quadrant into six equal angles and this quadrant is PI over 2 so each of these is our PI over 12 each of these little sub angles are PI over 12 so we're going to have negative 7 PI over 12 so we're going to go in the negative direction we're going to go counterclockwise or start we're going to go clockwise so we're going to go let me start here and so if I go 1 2 3 4 5 6 7 8 9 10 11 12 gets us right over there did I do that right that that doesn't let me let me make sure I got that let me show 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 ok let me make sure this whole thing is PI with 1 2 3 4 5 6 7 8 9 10 11 12 so each of these are 12 that's just PI over 12 that's right and we want to go negative seven of them negative 1 negative 2 negative 3 negative 4 negative 5 negative 6 negative 7 I just kept going all the way to negative 12 the last time so this is the angle this is our this is our angle right over here and our distance that we go out from the origin is 7 so we go out 1 2 3 4 5 6 7 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 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