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### Course: Precalculus (Eureka Math/EngageNY)>Unit 1

Lesson 5: Topic B: Lessons 14-17: Multiplying and dividing complex numbers in polar form

# Visualizing complex number multiplication

Learn how complex number multiplication behaves when you look at its graphical effect on the complex plane.

## What complex multiplication looks like

By now we know how to multiply two complex numbers, both in rectangular and polar form. In particular, the polar form tells us that we multiply magnitudes and add angles:
$\begin{array}{rl}& \phantom{=}r\left(\mathrm{cos}\left(\alpha \right)+i\mathrm{sin}\left(\alpha \right)\right)\cdot s\left(\mathrm{cos}\left(\beta \right)+i\mathrm{sin}\left(\beta \right)\right)\\ \\ & =rs\left[\mathrm{cos}\left(\alpha +\beta \right)+i\mathrm{sin}\left(\alpha +\beta \right)\right]\end{array}$
One great strength of thinking about complex multiplication in terms of the polar representation of numbers is that it lends itself to visualizing what's going on.
What happens if we multiply every point on the complex plane by some complex number $z$? If $z$ has polar form $r\left(\mathrm{cos}\left(\theta \right)+i\mathrm{sin}\left(\theta \right)\right)$, the rule outlined above tells us that every point on the plane will be scaled by a factor $r$, and rotated by an angle of $\theta$.

### Examples

For $z=\sqrt{3}+i=2\left(\mathrm{cos}\left({30}^{\circ }\right)+i\mathrm{sin}\left({30}^{\circ }\right)\right)$, multiplying $z$ would scale everything by a factor of $2$ while rotating by ${30}^{\circ }$, like this:

For $z=\frac{1}{3}-\frac{i}{3}$, the absolute value of $z$ is
$\sqrt{{\left(\frac{1}{3}\right)}^{2}+{\left(\frac{1}{3}\right)}^{2}}=\frac{\sqrt{2}}{3}$
and its angle is $-{45}^{\circ }$, so multiplying by $z$ would scale everything by a factor of $\frac{\sqrt{2}}{3}\approx 0.471$, which will mean shrinking, while rotating $-{45}^{\circ }$ about the origin, which is a clockwise rotation.

For $z=-2$, which has absolute value $2$ and angle ${180}^{\circ }$, multiplication rotates by a half turn about the origin while stretching by a factor of $2$.

Another way to think about these transformations, and complex multiplication in general, is to put a mark down on the number $1$, and a mark down on the number $z$, and to notice that multiplying by $z$ drags the point for $1$ to the point where $z$ started off, since $z\cdot 1=z$. Of course, it must do this in a way which fixes the origin, since $z\cdot 0=0$.
Isn't it interesting how facts as simple as $z\cdot 1=z$ and $z\cdot 0=0$ can be so helpful in visualizing complex multiplication!

## A visual understanding of complex conjugates

Let's look at what happens when we multiply the plane by some complex number $z$, then multiply the result by its conjugate $\overline{z}$:
If the angle of $z$ is $\theta$, the angle of the complex conjugate $\overline{z}$ is $-\theta$, so the successive multiplications have no total rotation. We can see this by the fact that the spot that started on $1$ ultimately lands on the positive real number line.
What about the magnitude? Both numbers have the same absolute value, $|z|=|\overline{z}|$, so the total effect of multiplying by $z$ then $\overline{z}$ is to stretch everything by a factor of $|z|\cdot |\overline{z}|=|z{|}^{2}$.
Of course, this fact is simple enough to see with the formulas, since $\left(a+bi\right)\left(a-bi\right)={a}^{2}+{b}^{2}=|a+bi{|}^{2}$, but it can be enlightening to see it in action!

## What complex division looks like

What happens if we divide every number on the complex plane by $z$? If $z$ has angle $\theta$ and absolute value $r$, then division does the opposite of multiplication: It rotates everything by $-\theta$ and scales by a factor of $\frac{1}{r}$ (which means shrinking by a factor of $r$).

### Example 1: Division by $\sqrt{3}+i$‍

The angle of $\sqrt{3}+i$ is ${30}^{\circ }$, and its absolute value is $2$, so everything rotates by $-{30}^{\circ }$, which is clockwise, and scales by a factor of $\frac{1}{2}$ (which means shrinking by a factor of $2$).

### Example 2: Division by $\frac{1}{3}-\frac{i}{3}$‍

The angle of $\frac{1}{3}-\frac{i}{3}$ is $-{45}^{\circ }$, and its absolute value is
$\sqrt{{\left(\frac{1}{3}\right)}^{2}+{\left(\frac{1}{3}\right)}^{2}}=\frac{\sqrt{2}}{3}$
So now everything rotates by $+{45}^{\circ }$, and is scaled by a factor of $\frac{3}{\sqrt{2}}\approx 2.121$.

You may have noticed that these divisions can also be seen as taking the dot that sits on top of $z$ and placing it over $1$.

## Relating the visualization of complex division with the formula

To compute $\frac{z}{w}$, where let's say $z=a+bi$ and $w=c+di$, we learned to multiply both numerator and denominator by the complex conjugate of $w$, $\stackrel{―}{w}=c-di$.
$\frac{z}{w}=\frac{a+bi}{c+di}=\frac{a+bi}{c+di}\cdot \frac{c-di}{c-di}=\frac{\left(a+bi\right)\left(c-di\right)}{{c}^{2}+{d}^{2}}=\frac{z\cdot \stackrel{―}{w}}{\phantom{\rule{0.278em}{0ex}}\phantom{\rule{0.167em}{0ex}}|w{|}^{2}}$
In other words, dividing by $w$ is the same as multiplying by $\frac{\stackrel{―}{w}}{\phantom{\rule{0.278em}{0ex}}\phantom{\rule{0.167em}{0ex}}|w{|}^{2}}$. Is there a visual way to understand this?
Suppose $w$ has angle $\theta$ and absolute value $r$, then to divide by $w$, we must rotate by $-\theta$ and scale by $\frac{1}{r}$. Since $\stackrel{―}{w}$, the conjugate, has the opposite angle from $w$, multiplying by $\stackrel{―}{w}$ will rotate by $-\theta$, like we want. However, multiplying by $\stackrel{―}{w}$ scales everything by a factor of $r$, when we need to go the other way, so we divide by ${r}^{2}=|w{|}^{2}$ to correct.
For instance, this is what directly dividing by $1+2i$ looks like:
And here is what it looks like to first multiply by its conjugate, $1-2i$, then to divide by the square of its magnitude $|1+2i{|}^{2}=5$.
The end result of both is the same.

## Want to join the conversation?

• As these short videos look nice I am at a complete loss what I should be able to learn or understand from looking at it?
• They are very helpful. It helps some to work through them multiple times. Even to draw the graphs on paper following the videos. Multiplication transforms Z = 1 +0i to a new complex grid with a new grid, W= 1 + 0i rotated from Z by a + bi.

These videos answer a question I asked in an earlier video on complex numbers.

My question and attempted answer are linked here, in the Dividing Complex Numbers video
.
• I'm even more confused after reading this section...
• I'll try to make an easy to understand summary:

First of all, if you don't want to visualize the multiplication of the two complex numbers, you can simply multiply their "rectangular form" and you will get the correct result, but here we are trying to understand the multiplication from a visual point of view.

You can multiply two complex numbers by following two single steps:
2) multiplying their distance to the origin (magnitude)

Think of it as a sequence of transformations.
1) adding their angle -> rotation
2) multiplying their distance to the origin -> dilation.

Thankfully, we have a notation that directly shows these two transformations that we want to apply (rotation & dilation).
It's called the "polar" form.

Here's the "rectangular" form of any complex number:
Z = a + bi

Here's its "polar" form (far more convenient for our two transformations):
Z = r(cos(θ) + sin(θ)i)

Where "r" is the magnitude (distance from the origin) and "θ" is its angle.

Now, to multiply any complex number (Z = a+bi), you refer to the two transformations:

1) Rotation by "θ" (angle "p" + angle "θ").
2) dilation by "r" (magnitude "m" * magnitude "r").

This makes sense from a graph point of view where you can visually add the angles and apply a dilation.
Now, if you had to do it without using a graph, you could multiply the two polar forms like the following:

Z1*Z2 = r(cos(α)+isin(α))⋅s(cos(β)+isin(β))
Z1*Z2 = rs[cos(α+β)+isin(α+β)]

Hope this helped.
• The document opens with: "By now we know how to multiply two complex numbers, both in rectangular and polar form. In particular, the polar form tells us that we multiply magnitudes and add angles". Where in the sequence was this taught? I totally missed this and do not understand.
• I'm on the same boat as you. I don't think he directly mentioned it in any video.

He did show how to divide two complex nubers in polar form by first converting them to exponential form. You can extend that to multiplication.

Polar form -> Exponential Form:
r(cos(α) + sin(α)i) = r * e^(iα)
s(cos(β) + sin(β)i) = s * e^(iβ)

(r * e^(iα)) * (s * e^(iβ)) =
r * s * e^(αi + βi) =
r * s * e^( (α + β) * i )

From there you can convert the exponential form back into polar form where the modulus(?) = r * s and the argument = (α + β) giving us:

r * s (cos(α + β) + sin(α + β) * i)
• By now we know how to multiply two complex numbers, both in rectangular and polar form. In particular, the polar form tells us that we multiply magnitudes and add angles:

Which videos about multiplication is he referring to?
• r(cos(α)+isin(α))⋅s(cos(β)+isin(β))
​=rs[cos(α+β)+isin(α+β)]
how they calculated it ??
• If you work out the multiplication of cos(a) + i*sin(a) and cos(b) + i*sin(b), you'll get:
(cos(a)cos(b) - sin(a)sin(b)) + i*(sin(a)cos(b) + sin(b)cos(a))
The expression in the first set of parentheses is a formula for cos(a + b), and the expression in the second set of parentheses is sin(a + b).

Division requires an extra step, but you will get cos(a - b) + i*sin(a - b)
• Man I am having some difficulty keeping up with all the terminology etc and thinking in a mathematical sense. I can do all the exercises no problem, but I would really like to be able to think in a logical/mathematical sense and work through practice session using that rather than seeing patterns/repetition. Is there a book you guys would recommend to be able to think more objectively/logically?
• What does total rotation refer to in the sentence "successive multiplications have no total rotation"? It means that instead of going back to the starting point (the two angles do not "cancel each other out"), the point moves along the x-axis?