Multiplying complex numbers graphically example: -1-i
About this video
Multiplying Z by a complex number rotates Z by that number's argument and scales Z's modulus by that number's modulus. For -1 - i, the argument is 225 degrees and the modulus is sqrt(2) ≈ 1.4, so a point 3 units from the origin is rotated 225 degrees and moved to about 4.2 units, which is choice B.
You'll be able to
- find the argument and modulus of a complex number such as -1 - i.
- explain multiplication by a complex number as a rotation plus a scaling.
- pick the point that shows a product on the complex plane.
Key words
- argument
- The angle a complex number makes with the positive real axis, measured counterclockwise.
- modulus
- The distance of a complex number from the origin.
- 45-45-90 triangle
- A right triangle with two equal legs; its acute angles are both 45 degrees.
Transcript
- [Instructor] We are told, suppose we multiply a complex number Z by -1-i. So this is Z right over here. Which point represents the product of Z and -1-I? Pause this video and see if you can figure that out. All right, now let's work through this together. So the way I think about this is, when you multiply by a complex number, you are going to rotate by the argument of that complex number.
And you're going to scale the modulus of Z by the modulus of this complex number. Now, let me just think about that a little bit. So I'm gonna draw another complex plane here. And so this is my real axis, this is my imaginary axis, right over here. And -1-I, so that's -1 and then minus 1i.
So it would go right over there. It would be that right over here. And so let's think about two things. Let's think about what its argument is, and let's think about what it's modulus is. So its argument is going to be this angle right over here. And you might already recognize that if this has a length of one, if this has a length of one, or another way of thinking about, this has a length of one, this is a 45, 45, 90 triangle.
So this is 45 degrees but then of course you have this 180 before that. So that's going to be 180 plus 45, is a 225 degree argument. So the argument here is going to be equal to 225 degrees. So when you multiply by this, you are going to rotate by 225 degrees.
So let's see this is going to be rotating by 180 degrees and then another 45. So if you just rotate it by that, you would end up right over here. Now we also are going to scale the modulus. And you can see two choices that scale that modulus. And so we know it's going to be choice A or choice B because choices C or D you'd have to rotate more to get over there.
And so to think about that, we have to just think about the modulus of -1-i, this point right over here and then just scale up this modulus by that same amount. Well, the modulus is just the distance from zero in the complex plane. So it's going to be this distance right over here.
And you could use the Pythagorean theorem to know that this squared, if you call this C, C squared is equal to one squared plus one squared or C squared is equal to two or C is equal to the square root of two. So that's the modulus right over here. Modulus is equal to square root of two which is approximately, it's a little bit more than 1.4.
So let's just call it approximately 1.4. So not only going to rotate by 225 degrees, we're going to scale the modulus, the distance from the origin by 1.4. So it looks like it's three units from the origin right over here. If you multiply that by 1.4, three times 1.4 is about four, or it is exactly 4.2.
So 4.2 of these units is one, two, three, four, a little bit further, you get right over here to choice B and we're done.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.