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## Precalculus

### Course: Precalculus > Unit 3

Lesson 5: Modulus (absolute value) and argument (angle) of complex numbers- Absolute value of complex numbers
- Complex numbers with the same modulus (absolute value)
- Modulus (absolute value) of complex numbers
- Absolute value & angle of complex numbers
- Angle of complex numbers
- Complex numbers from absolute value & angle
- Complex number absolute value & angle review

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# Complex numbers with the same modulus (absolute value)

Sal shows how to determine which members in a set of complex numbers have the same modulus (or absolute value). He also shows how to visualize all of the complex numbers with a given modulus as a circle centered at the origin on the complex plane, since all points on such a circle are the same distance from the origin. Created by Sal Khan.

## Want to join the conversation?

- What are complex numbers?

What is a complex conjugate?

At about2:24-2:29, Sal says, "When you have your complex conjugate, it has the same modulus." What does he mean by that?(1 vote)- A complex number is any number of the form x + yi where x,y are real numbers and i is the imaginary number √(-1). (All real and imaginary numbers are also complex numbers.) Let N be the complex number x + yi. The complex conjugate of N is equal to x - yi. The modulus of N is √(x^2 + y^2). The modulus N and its conjugate are equal.(2 votes)

- Hi!

So, one of the questions said to find the modulus of 4 - i. My answer was sqrt of 15 because I did (4)^2 + (-i)^2 = 16 + -1 = 15. And then I put it under a radical. However, the correct answer was 17. Why is that? Was I just supposed to do (4 - i)^2 and then sqrt it?(1 vote)- When you take the modulus of a complex number a+bi, you don't square i along with b. You just square b. So instead of it being (4)^2+(-i)^2, it should just be (4)^2+(-1)^2, which is 17.(1 vote)

## Video transcript

- [Instructor] We are asked,
which of these complex numbers has a modulus of 13? And just as a bit of a hint, when we're talking about the
modulus of a complex number, we're really just talking
about its absolute value. Or if we were to plot
it in the complex plane, which is what we have right over here, what is its distance from the origin? So really you need to find
which of these complex numbers has a distance of 13 from the
origin in the complex plane. Pause this video and see
if you can figure that out. All right, now let's work
through this together. Now one might jump out at you immediately that's going to have a
distance of 13 from the origin. If this is the origin right over here, we see that if we go exactly 13 units down we have this point right
over here, negative 13i. So immediately right
out of the gate, I say, "Okay, that complex number
has a modulus of 13," but is that the only one? Well, we can actually visualize
all of the complex numbers that have a modulus of
13 by drawing a circle with the radius 13 centered at the origin. So let's do that. And we can see that it contains the first complex number
that we looked for, but it also seems to have included in it this one right over
here, and we can verify that the modulus right over
here is going to be 13. We can just use the Pythagorean theorem. So this distance right over here is 12. And this distance right over here is 5. And so we just need to figure out the hypotenuse right over here. And so we know that the hypotenuse is going to be the
square root of 5 squared plus 12 squared, which is equal to the
square root of 25 plus 144, which is equal to the square root of 169, which indeed does equal 13. So I like that choice as well. And we can see visually that
none of these other points that they already plotted
sit on that circle. So they don't have a modulus of 13. If we wanted to come up with
some other interesting points, we could instead of having
negative 5 plus 12i, we could have negative 5 minus 12i. It would get us right over there. And that would have a modulus of 13. And notice, when you have
your complex conjugate, it has the same modulus. Or you could go the other way around. Instead of negative 5 plus
12i, you could have 5 plus 12i. That also would have a modulates of 13. Or you could have 5 minus 12i. That also would have a modulus of 13. Now there's an infinite number of points, any of these points on the circle, that will have a modulus of 13.