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### Course: Precalculus > Unit 3

Lesson 8: Multiplying and dividing complex numbers in polar form- Multiplying complex numbers in polar form
- Dividing complex numbers in polar form
- Multiply & divide complex numbers in polar form
- Taking and visualizing powers of a complex number
- Complex number equations: x³=1
- Visualizing complex number powers
- Powers of complex numbers
- Complex number polar form review

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# Taking and visualizing powers of a complex number

Given a complex number in rectangular form, we can convert it to polar form to show how to visualize powers of the complex number by scaling and rotating it by its own modulus and argument. Created by Sal Khan.

## Want to join the conversation?

- That's the De Moivre's theorem isnt it(6 votes)
- Yes, that's correct! De Moivre's theorem states that for any complex number z and any positive integer n, (cos θ + i sin θ)^n = cos nθ + i sin nθ. It is a useful tool for raising complex numbers to powers and expressing them in polar form.(5 votes)

- Can we apply polar form to real numbers as well?(3 votes)
- Yes, the angle would be 0 for positive numbers and 180 for negative numbers. Example: -5=5(cos(180)+isin(180)). The imaginary component will disappear because sin(180) and sin(0) are both 0.(3 votes)

- So to have a formula, for a complex number z where z = r*((cos(θ))+(sin(θ)i)) then z^x = (r^x)*((cos(θ*x))+(sin(θ*x)i))

Could anyone with more experience with this topic confirm? That's my understanding of this video.(3 votes) - At3:27why isn't it
*i^2*instead of just i. I thought we were squaring the entire expression.(2 votes)- Sal is using a form of the Pythagorean theorem, square root of a^2 + b^2 = c, and supplementing the two sides of his "triangle" to find the "hypotenuse", or modulus, of z.(2 votes)

## Video transcript

- We're told to consider the
complex number z is equal to -1 plus i times the square root of 3. 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 -1 squared plus square root of 3 squared, which is going to be equal to 1 plus 3. So principal 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 -1 in the real direction. So that might be -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 2. 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, sorry, it's hard to see that 30 degree. The reason why that helps us is if 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 2. 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 'cause 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 4. 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 gonna 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 4th. Well, I'm just gonna
take this modulus here since I'm going to multiply
z to the third times z, I'm gonna 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 is the same
thing as zero degrees. So this I could say is zero
degrees. This is zero 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 1/2 of that. So it has length 8. And then the long side is
gonna be square root of 3 times the short side. So it's going to be 8 square roots of 3. So if we wanted to write z to the fourth in rectangular form, it would be the real part is -8. Plus i times 8 square
roots of 3, and we're done.