Visualizing complex number multiplication

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:

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 $r(\cos (\theta )+i\sin (\theta ))$‍, 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 $\theta$‍.

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

For $z={\displaystyle \frac{1}{3}}-{\displaystyle \frac{i}{3}}$‍, the absolute value of $z$‍ is

and its angle is $-{45}^{\circ }$‍, so multiplying by $z$‍ would scale everything by a factor of ${\displaystyle \frac{\sqrt{2}}{3}}\approx 0.471$‍, 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!

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 $(a+bi)(a-bi)=a^2+b^2=|a+bi{|}^2$‍, but it can be enlightening to see it in action!

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 ${\displaystyle \frac{1}{r}}$‍ (which means shrinking by a factor of $r$‍).

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 ${\displaystyle \frac{1}{2}}$‍ (which means shrinking by a factor of $2$‍).

The angle of ${\displaystyle \frac{1}{3}}-{\displaystyle \frac{i}{3}}$‍ is $-{45}^{\circ }$‍, and its absolute value is

So now everything rotates by $+{45}^{\circ }$‍, and is scaled by a factor of ${\displaystyle \frac{3}{\sqrt{2}}}\approx 2.121$‍.

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 $1$‍.

To compute ${\displaystyle \frac{z}{w}}$‍, where let's say $z=a+bi$‍ and $w=c+di$‍, we learned to multiply both numerator and denominator by the complex conjugate of $w$‍, $\bar{w}=c-di$‍.

In other words, dividing by $w$‍ is the same as multiplying by ${\displaystyle \frac{\bar{w}}{|w{|}^2}}$‍. 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 ${\displaystyle \frac{1}{r}}$‍. Since $\bar{w}$‍, the conjugate, has the opposite angle from $w$‍, multiplying by $\bar{w}$‍ will rotate by $-\theta$‍, like we want. However, multiplying by $\bar{w}$‍ 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.