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# Complex rotation

Multiplying a real or complex number by the imaginary unit j corresponds to a rotation by +90 degrees. This is the key feature of j that makes it such a useful number. Created by Willy McAllister.

## Want to join the conversation?

• At , he plot z with an axis of (b,a) then in the video He plots jz with an axis of (a, -b). Which completely switches the coordinates names of the same line. Thoroughly confusing. How can he switch the names of the (x,y) plots on the same graph?
• The orange coordinate system does not change. The horizontal axis is called Re (for "real") and the vertical axis is called Im (for imaginary). The variables a and b represent distances, not axis names. There are two different complex numbers plotted on the coordinate system, z and jz. We defined z to be z = a + jb. Then we asked "If (a + jb) is the value of z, what is the value of j time z?" An expression for jz is worked out on the left side of the screen.
It turned out, jz = -b + ja. This is shown on the plot, too.

The cool/tricky thing that happens when you multiply something by j is the real and imaginary parts switch positions. That is, when we multiplied z by j, "a" became the imaginary part, and "b" became the real part. This is why complex multiplication has a fantastic rotation property.
• At ; I could not understand the conversion of j into e to the j 90 degrees? Can you please explain me the same?
• When we say exponent is unit-less, then how can we represent j by e^j90 deg at ?
• 90 degrees is an alternate representation of the radian angle pi/2. A radian is a unit-less number, it is the ratio of two lengths (the distance around the arc of a circle divided by the radius).
• So can you use Electrical Engineering in Mechanical Engineering?
(1 vote)
• A lot of the math we use for RC and RLC circuits applies to mechanical systems as well. The math describing a swinging pendulum is exactly the same as the RLC circuit. Anything having to do with mechanical vibration uses the same math EE's use for signal processing (frequency analysis).
(1 vote)
• Hi,
I was trying to find rotation in the complex plane for precalculus and saw this. This is pretty helpful but sometimes confusing (because of different letterings for engineering? idk). It is interesting how this is in electrical engineering. Can you make a video like this in precalc? It would be soo helpful :). Also, how would I do this it i need to do this multiple times and scale it? Thanks.
(1 vote)
• Lots of details left out of this discussion made it thoroughly confusing. Is j a unit vector on the imaginary axis? How does he get from j to e^j90degrees?
A warmup explanation of what re^jTheta represents would help a lot.
(1 vote)
• j (the imaginary unit) is sqrt(-1). If you plot j on a Complex plane it is a vector pointing straight up on the vertical axis, with a magnitude of 1.

There are multiple equivalent ways to represent a complex number for j...

Re/Im notation: z = 0 + j

Radius/Angle notation: z = r / 90°

Trig notation: z = cos 90° + j sin 90°

Euler notation: z = r e^(j 90°)

(assume r=1 in the last three notations)

This last "Euler" notation is first mentioned at minutes in the previous Complex Numbers video, and explained further in the next video, "Euler's Formula". The videos are slightly out of order for this line of explanation. Sorry about that.
(1 vote)

## Video transcript

- [Voiceover] 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 times z. And that equals, j times a plus jb. And let's just multiply it through. Equals j times a plus j times j times b. A and b have now switched places. So, we're gonna put ja on this side, ja on this side. And what do we have here? J times j is minus one, so we have minus b plus ja. So, now we have expressions for z and jz. And I wanna plot these on a complex plane and see what they look like. And here's the real axis, here's the imaginary axis. And 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. And 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 this same plot. Jz has a real component of minus b, so that would be right about here. Here's minus b and it has a 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 wanna demonstrate is that this angle right here is 90 degrees. 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 gonna 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 equals 90 degrees minus theta. So, if I combine this theta angle with this angle here, what do I get? Theta plus 90 degrees minus theta and we get 90 degrees. So, we just showed that this angle right here is a 90 degree angle. That demonstrates that any complex number z, if I multiply it by j, that results in a positive rotation of 90 degrees. 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. Or 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 times r e to the j theta. So, now I'm gonna do a little trick, where I'm gonna 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 it points straight up on the imaginary axis. So, I can represent j, like this. I can say j is e to the j 90 degrees. 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 equals r times e to the j theta plus 90 degrees. So, in exponential notation, we get this vector here. We go an additional 90 degree rotation and we go out the same distance we had originally, r. So, now we've shown that we we can rotate any complex number by 90 degrees if we multiply it by j. We're gonna 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 videos from now.