Multiplying by j is rotation

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 1.

Now, what I want to do is a sequence of multiplications by j. This is a really important property of this imaginary unit. We do powers of j, and I'm going to plot them. While we're doing that, I want to plot them on the imaginary or on the complex plane. So, this is the real axis, and this is the imaginary axis.

I'm going to take powers of j. So, first one is j to the zero, and anything to the zero is one. So, j to the zero is one. If we plot that on the imaginary axis, here's one, and there's no real part, so we're just right on the real axis.

Okay, let's do j to the one. That's j times itself one time, and 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.

Okay, let's keep going. Let's do j squared. And what does that equal to? Well, I wrote it down up here: j squared is equal to minus 1. So, j squared is equal to minus 1. Where's that in the complex plane? That's over here at minus 1 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. We have those two right here: j squared is minus one, and j to the one is j, so that equals minus j. 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 do 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 1 times minus 1. What does that equal to? That equals 1. 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 does 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: 1, j, minus 1, minus j, 1, j, it's going to be minus 1 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 multiplied by j, when I multiplied one by j, it rotated it 90 degrees, and that was the first step right here. When I multiplied 1 times j, I got j.

Now, when we went to j squared, we ended up at -1. So multiplying by j again caused the vector to go down here like that, and that was another 90 degrees. 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 is the nature of it; it's 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. It's the reason that 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, 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. What is j squared equal to? We wrote it down right here: j squared is minus 1.

So, this equals j over minus 1 or equals minus j. So, whenever we see j in a fraction, j to the minus one basically 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. 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.