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Current time:0:00Total duration:6:21

Taking and visualizing powers of a complex number


Video transcript

we're told to consider the complex number z is equal to negative one plus i times the square root of three find z to the fourth in polar and rectangular form so pause this video and see if you can figure that out all right now let's work through this together so first let's just think about what the modulus of z is we know that the modulus is going to be equal to the square root of the real part squared plus the square root of 3 plus the imaginary part squared so it is going to be negative 1 squared plus square root of 3 squared which is going to be equal to 1 plus 3 so principle root of 4 which is equal to 2. now the next interesting question is what is the argument of z and the reason why i'm even going through this is once we put it into polar form it's going to be a lot easier to both visualize what it means to take the various exponents of it and then we can convert back into rectangular form and so let us let me draw another complex plane here imaginary axis that is my real axis and if i were to plot z it would look something like this we have negative 1 in the real direction so that might be negative 1 there and we have square root of 3 in the imaginary direction square root of 3. so our point z is right over here and we know the distance from the origin the modulus we know that this distance right over here is two we know that this distance right over here is square root of 3 and we know that this distance right over here is 1. and so you might immediately recognize this as a 30-60-90 triangle because in a 30-60-90 triangle the short side is half of the hypotenuse and the long side is the square root of 3 times the short side so we know that this is a 60 degree angle we know that this is a 30 degree angle and the reason why that helps us it's hard to see that 30 degree the reason why that helps us is that this is 60 degrees we know that the argument here must be 120 degrees so the arg of z the argument of z is 120 degrees and so just like that we can now think about z in polar form so let me write it right over here we can write that z is equal to its modulus 2 times the cosine of 120 degrees plus i times the sine of 120 degrees and we could also visualize z now over here so its modulus is two so that's halfway to 4 and its argument is 120 degrees so it would put us right over here this is where z is now what would z squared be well when you multiply complex numbers and you've represented them in polar form we know that you would multiply the moduli so it would then be 2 squared so it'd be 4 right over here and then you would add the arguments so you would essentially rotate z by another 120 degrees because you're multiplying it by z so it's going to be cosine of 240 degrees plus i sine of 240 degrees once again 2 times 2 is equal to 4 120 degrees plus another 120 degrees is 240 degrees and so now where would z squared sit well its argument is 240 degrees and its modulus is four so now it is twice as far from the origin and now let's think about what i'll do this in a new color what z to the third power is going to be equal to well that's going to be z squared times z again so we're going to multiply 2 times this modulus so that's going to be equal to 8 times and then we're going to rotate z squared by 120 degrees so cosine of 360 degrees plus i sine of 360 degrees and so that's going to put us at 8 for our modulus and 360 degrees is the same thing as zero degrees so we are right over here so this is z to the third power and i think you know where this is going what is z to the fourth power going to be let me move my screen down a little bit so i have a little more real estate z to the fourth well i'm just going to take this modulus here since i'm going to multiply z to the third times z i'm going to multiply that modulus times 2 to get to 16 and then i'm going to add another 120 degrees well i could write cosine of 480 degrees or 360 degrees the same thing as 0 degrees so this i could say is 0 degrees this is 0 degrees so if i add 120 to that i get cosine of 120 degrees plus i sine of 120 degrees so my argument is back to being at 120 degrees but now my modulus is 16 so there's 4 8 12 16. this outer circle right over here i am right over there with z to the fourth so we're almost done we've just represented z to the fourth in polar form now we just have to think about it in rectangular form now lucky for us we already know what cosine of 120 degrees is and sine of 120 degrees is it is we can construct if we want another 30-60-90 triangle right over here so the hypotenuse here has length 16 the short side is going to be half of that so has length 8 and then the long side is going to be square root of 3 times the short side so it's going to be 8 square roots of three so if we wanted to write z to the fourth in rectangular form it would be the real part is negative eight plus i times eight square roots of three and we're done