If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Multiplying complex numbers graphically example: -3i

We can multiply complex numbers graphically on the complex plane by rotating and scaling. Multiplying a complex number z by -3i rotates and scales z. Created by Sal Khan.

Want to join the conversation?

  • blobby green style avatar for user festavarian2
    In the follow-up video to this, Sal states that "you scale the modulus of Z by the modulus of the multiplying complex number". Here, the modulus of -3i is 3 (correct?), so why does he multiply by minus 3?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user TotallyNotAFurryÒwÓ
      It's possible that Sal is using a different convention for representing complex numbers in polar form.

      One common convention is to write a complex number in the form r(cosθ + i sinθ), where r is the modulus (or magnitude) and θ is the argument (or phase angle) of the complex number.

      However, another convention is to write a complex number in the form r cis(θ), where "cis" is shorthand for "cos + i sin". This notation is sometimes used because it simplifies certain algebraic operations involving complex numbers.

      Using this notation, if we have a complex number z = a + bi in rectangular form, its polar form can be expressed as:

      r cis(θ) = |z| cis(θ) = (a^2 + b^2)^(1/2) cis(atan(b/a))

      Multiplying a complex number in polar form by another complex number in polar form involves multiplying their moduli and adding their arguments. So, if we have:

      z = r cis(θ) and w = s cis(φ)


      zw = rs cis(θ + φ)

      In the video, Sal is using the second convention and writes the complex number -3i as 3 cis(-90°). When he multiplies z by -3i, he is multiplying the moduli (|z| = 5) and adding the arguments (θ = 45° and φ = -90°), giving the product:

      zw = 5 x 3 cis(45° - 90°) = -15 cis(-45°)

      So the answer is expressed in polar form as -15 cis(-45°), which can be converted to rectangular form as -15 cos(45°) - 15i sin(45°) = -10.6 - 10.6i.
      (2 votes)
  • blobby green style avatar for user Nop
    One thing I don't understand is when you multiply by -1, why did it flip to 135degree instead of 225 degree (both of them are negative on Real (x))? Why did it must be 135degree?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user TotallyNotAFurryÒwÓ
      When you multiply by -1, you are essentially reflecting the complex number across the real axis. This reflection will cause the angle (or argument) of the complex number to change by 180 degrees.

      In this case, the original angle was -45 degrees (as it was in the fourth quadrant), and when you reflect it across the real axis, the new angle will be 180 degrees - 45 degrees = 135 degrees (as it is in the second quadrant).

      So the reason the new angle is 135 degrees instead of 225 degrees is that the reflection caused the angle to change by 180 degrees, not 360 degrees.
      (2 votes)
  • piceratops ultimate style avatar for user ANB
    The exact coordinates would be -(3sqrt2)/2 - i(3sqrt2)/2
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

- [Instructor] Suppose we multiply a complex number Z by negative three i, and there shows Z right over here Plot the point that represents the product of Z and negative three i. I so pause this video and see if you can work through that. All right, now let's do it step-by-step. First I wanna think about what would, where would three Z be? Well, three Z would have the same angle as Z, but it's absolute value or it's modulus would be three times larger. So you'd be going in this direction but it'd be three times further. So that's one times it's modulus. That's two times it's modulus. That's three times it's modulus or it's three times it's absolute value. So three Z would be right over here. Now what about negative three Z? Well, if you multiply it by a negative, it's just going to flip it around. You can think about it as flipping it at 180 degrees but it's going to have the same modulus. So instead of being right over here, at three in this direction, it's going to be one, two, three in this direction, right over here. So that is negative three Z. And now perhaps most interestingly, what happens when you multiply it by i? So if we have negative three i times Z, now which is exactly what they want us to figure out. Well, let's think about what happens if you had one and if you multiply that by i. So one times i becomes one i. So it goes over there. What if you then took one i and multiplied it by i? Well, then you have negative one. What if you took negative one and you multiplied it by i? Well, then now you have negative one i. So notice every time we multiply by i, we are rotating by 90 degrees. So over here, if we take negative three Z and multiply it by i, you're just going to rotate 90 degrees and you're going to get right over there. So this is negative three i times Z, which is exactly what we were looking for.