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### Course: Precalculus>Unit 3

Lesson 7: Graphically multiplying complex numbers

# 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?

• 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?
• 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)``

Then:

``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.
• 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)
• 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.