Complex numbers with the same modulus (absolute value)

• [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 modulus 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.