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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:
=r(cos(α)+isin(α))s(cos(β)+isin(β))=rs[cos(α+β)+isin(α+β)]
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(θ)+isin(θ)), 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 θ.

Examples

For z=3+i=2(cos(30)+isin(30)), multiplying z would scale everything by a factor of 2 while rotating by 30, like this:
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For z=13i3, the absolute value of z is
(13)2+(13)2=23
and its angle is 45, so multiplying by z would scale everything by a factor of 230.471, which will mean shrinking, while rotating 45 about the origin, which is a clockwise rotation.
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For z=2, which has absolute value 2 and angle 180, multiplication rotates by a half turn about the origin while stretching by a factor of 2.
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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 z1=z. Of course, it must do this in a way which fixes the origin, since z0=0.
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Isn't it interesting how facts as simple as z1=z and z0=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 z¯:
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If the angle of z is θ, the angle of the complex conjugate z¯ is θ, 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|=|z¯|, so the total effect of multiplying by z then z¯ is to stretch everything by a factor of |z||z¯|=|z|2.
Of course, this fact is simple enough to see with the formulas, since (a+bi)(abi)=a2+b2=|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 θ and absolute value r, then division does the opposite of multiplication: It rotates everything by θ and scales by a factor of 1r (which means shrinking by a factor of r).

Example 1: Division by 3+i

The angle of 3+i is 30, and its absolute value is 2, so everything rotates by 30, which is clockwise, and scales by a factor of 12 (which means shrinking by a factor of 2).
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Example 2: Division by 13i3

The angle of 13i3 is 45, and its absolute value is
(13)2+(13)2=23
So now everything rotates by +45, and is scaled by a factor of 322.121.
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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 zw, 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, w=cdi.
zw=a+bic+di=a+bic+dicdicdi=(a+bi)(cdi)c2+d2=zw|w|2
In other words, dividing by w is the same as multiplying by w|w|2. Is there a visual way to understand this?
Suppose w has angle θ and absolute value r, then to divide by w, we must rotate by θ and scale by 1r. Since w, the conjugate, has the opposite angle from w, multiplying by w will rotate by θ, like we want. However, multiplying by w scales everything by a factor of r, when we need to go the other way, so we divide by r2=|w|2 to correct.
For instance, this is what directly dividing by 1+2i looks like:
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And here is what it looks like to first multiply by its conjugate, 12i, then to divide by the square of its magnitude |1+2i|2=5.
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The end result of both is the same.