Multiplying complex numbers graphically example: -1-i | Precalculus | Khan Academy

We are told suppose we multiply a complex number z by negative one minus i. So, this is z right over here. Which point represents the product of z and negative one minus 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 going to draw another complex plane here, and so this is my real axis. This is my imaginary axis right over here.

And negative 1 minus i, so that's negative 1, and then minus 1 i, 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 its 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 rotated 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 negative 1 minus i, this point right over here, and then just scale up these modulus by that same amount.

Well, the modulus is just the distance from 0 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 1 squared plus 1 squared, or c squared is equal to 2, or c is equal to the square root of 2.

So, that's the modulus right over here. Modulus is equal to the square root of 2, which is approximately 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, if it is, 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.