Complex rotation

So now we've seen rotation by multiplying J by J over and over again, and we see that that's rotation. Now let's do it for the general idea of any complex number.

So if I have a complex number, we'll call it Z, and we'll say it's made of two parts: a real part called A and an imaginary part called B. So now what I want to do is what happens if we multiply Z by J one time.

J * Z and that equals J times A plus J B. And let's just multiply it through: J * A plus J * J * B. A and B have now switched places. So we're going to put J A on this side, J A on this side, and what do we have here? J * J is minus one, so we have minus B plus J A.

So now we have expressions for Z and JZ, and I want to go plot these on a complex plane and see what they look like.

Here's the real axis; here's the imaginary axis. Let's first plot, let's plot Z. Let's say Z has a large real value, and that would be A. And let's say that B is a smaller value; we'll put B here. That means that Z is at a location in the complex plane right there.

We can plot the dotted lines; that's Z in the complex plane. So now let's put JZ on the same plot. JZ has a real component of minus B, so that would be right about here. Here's minus B, and it has an imaginary component of plus A, so let's swing A. A goes all the way up to about here, and so that's the location of JZ.

And let me draw the hypotenuse of that. This is the vector representing JZ right there.

So now we have a bunch of triangles on the page, and what I want to demonstrate is that this angle right here is 90°. So one way to do that, let's see if we can do that. Let's say this angle here is Theta; that's the angle right there.

Now this triangle here, this triangle that we sketched in, just imagine in your head that we're going to rotate that angle up until the A leg of that triangle is resting right here on the imaginary axis. So this triangle rotates up to become this triangle here. Since we moved that triangle, we know that this angle here, that's also Theta.

It's the same triangle, just rotated up. And what does that make this angle here? This angle here, that equal 90° minus Theta. So if I combine this Theta angle with this angle here, what do I get? Theta plus 90° minus Theta, and we get 90°.

So we just showed that this angle right here is a 90° angle. That demonstrates that any complex number, number Z, if I multiply it by J, that results in a positive rotation of 90°.

So let's do this rotation again; only this time, instead of using the rectangular coordinate system, let's use the exponential representation. So in the exponential notation, we say in general Z equals some radius times e to the J Theta, where this is the angle Theta and R is the length of this hypotenuse here to get out to Z.

So what is, in this notation, what is JZ? And that equals J * R e to the J Theta. So now I'm going to do a little trick where I'm going to represent J in exponential notation.

So if I color in dark here, this is J. The vector J is right there, and it has a magnitude of one and points straight up on the imaginary axis. So I can represent J like this: I can say J is e to the J 90°. That's equivalent to this J here, and it's multiplied by R e to the J Theta.

And now the last step is we just combine these two exponents together, and we get JZ equal R * e to the J (Theta + 90°). So in exponential notation, we get this vector; here we go, an additional 90° rotation, and we go out the same distance we had originally, R.

So now we've shown that we can rotate any complex number by 90°. If we multiply it by J, we're going to get to apply this kind of transformation to working out the current and voltage relationships in inductors and capacitors, and that'll happen in a couple of videos from now.