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Multiplying by j is rotation

Successive powers of the imaginary unit j correspond to a rotation.  Created by Willy McAllister.

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Video transcript

- [Voiceover] Okay, there's one more feature of complex numbers that I want to share with you and we'll do that down here. So, our definition of j is j squared equals minus one, and now what I want to do is a sequence of multiplications by j. This is a really important property of this imaginary unit. We're gonna do powers of j and I'm going to plot them, while we're doing that I'm gonna plot them on the imaginary, or on the complex plane. So, this is the real axis and this is the imaginary axis. So, I'm gonna take powers of j, so first one j to the zero, and anything to the zero is one, so j to the zero is one and if we plot that on the imaginary axis, here's one, and there's no real part. So, it's just right on the real axis. Okay, let's do j to the one, that's j times itself one time, so that is equal to j. If I plot that number, that's up here, that's up on the imaginary axis right there, there's j. Alright, this has no real part, it's all imaginary. Let's keep going. Let's do j squared and what is that equal to? Well, I wrote it down up here, j squared is equal to minus one. So, j squared is equal to minus one, where's that in the complex plane? That's over here at minus one, on the real axis, no imaginary part. Let's do the next one. Let's do j cubed. What is that equal to? J to the third power is equal to j squared times j and we have those two right here, j squared is minus one and j to the one is j, so that equals minus j, and where do we plot that? We plot that down on the imaginary axis in the complex plane, right here, minus j. Okay, now we've got four answers. Let's go one or two more. Okay, j to the fourth is equal to what? It's equal to j squared times j squared. Let's look that up. It's minus one times minus one, and what does that equal to? That equals one. So, let's go plot that. Well, we've already plotted it, so that's this answer right here. Alright, so we already have that, let's do one more. J to the fifth, what is that equal to? That equals j to the fourth times j, and j to the fourth is right here, it's one times j equals j, let's go plot that one. Well, we've already plotted it, it's already right here. Okay, so, you can see there's a pattern here, one j minus one minus j. One j, it's gonna by minus one and minus j, it keeps repeating. Here's what's interesting about this, okay. If we draw this as vectors, if I draw these imaginary numbers as vectors, when I multiply by j, when I multiplied one by j, it rotated it 90 degrees and then that was the first step right here when I multiplied one times j, I got j. Now, when we went to j squared, we ended up at minus one. So, multiplying by j again, caused the vector to go down here, like that and that was another 90 degrees and if I take it again, the next one went this way and the final one went this way. So, this is the property of j, this is the key property of the imaginary unit, multiplying by the imaginary unit, the nature of it is this 90 degree rotation. There's this idea of a number that causes other numbers to rotate and that's the feature of j that makes it super important and it's the reason we use imaginary numbers in electrical engineering. So, the key idea here is that j rotates. That's the point. That's what we love about j. So, the last thing I want to mention is the negative powers of j. What happens if we have j to the minus one? Let's figure out what that means. So, that, of course, is one over j and if I multiply this by j over j and anything over itself is one, so I haven't changed the value of this and that equals j on top over j times j or j squared and what is j squared equal to? We wrote it down right here, j squared is minus one, so this equals j over minus 1 or equals minus j. So, whenever we see a j in a fraction, j to the minus one, basically it introduces a minus sign and the j comes up to the top out of the fraction. So, j to the minus one equals minus j and we'll use that occasionally to help us in our math. So, this was a really quick review of complex numbers and if any of this was new to you, I really encourage you to go back and watch Sal's videos on complex numbers and if this is something you've seen before, I hope it knocked off some of the rust and we're ready to go and use these numbers.