Either part can be zero:

0 + 2i
⇒ no real part, imaginary part is 2i, same as 2i
4 + 0i
⇒ real part is 4, no imaginary part, same as 4

Multiplying

To multiply complex numbers:

Each part of the first complex number gets multiplied by
each part of the second complex number

Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" (see Binomial Multiplication for more details):

FOIL: Firsts, Outers, Inners, Lasts

(a+bi)(c+di) = ac + adi + bci + bdi2

Like this:

Example: (3 + 2i)(1 + 7i)

(3 + 2i)(1 + 7i) = 3×1 + 3×7i + 2i×1+ 2i×7i = 3 + 21i + 2i + 14i2 = 3 + 21i + 2i − 14   (because i2 = −1) = −11 + 23i

Here is another example:

Example: (1 + i)2

(1 + i)2= (1 + i)(1 + i) = 1×1 + 1×i + 1×i + i2 = 1 + 2i − 1   (because i2 = −1) = 0 + 2i

But There is a Quicker Way!

Use this rule:

(a+bi)(c+di) = (ac−bd) + (ad+bc)i

Example:

(3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i

Why Does That Rule Work?

It is just the "FOIL" method after a little work:

(a+bi)(c+di) = ac + adi + bci + bdi2   FOIL method
  = ac + adi + bci − bd   (because i2=−1)
  = (ac − bd) + (ad + bc)i   (gathering like terms)

And there you have the (ac − bd) + (ad + bc)i pattern.

This rule is certainly faster, but if you forget it, just remember the FOIL method.

Now let's see what multiplication looks like on the Complex Plane.

The Complex Plane

This is the complex plane

It is a plane for complex numbers!

complex plane

We can plot a Complex number like 3 + 4i

It is placed

  • 3 units along (the real axis),
  • and 4 units up (the imaginary axis)
complex plane 3+4i

Multiplying By i

complex plane vector 3+4i by i = -4+31

Let's multiply it by i:

(3 + 4i) x i = 3i + 4i2

Which simplifies to (because i2 = −1):

−4 + 3i

And here is the cool thing ... it's the same as rotating by a right angle (90° or π/2)

Was that just a weird coincidence?

Let's keep multiplying by i to see what happens:

complex plane vector by i 4 times is full rotation
(−4 + 3i) x i = −4i + 3i2 = −3 − 4i
(−3 − 4i) x i = −3i − 4i2 = 4 − 3i
(4 − 3i) x i = 4i − 3i2 = 3 + 4i

Well, isn't that stunning? Each time it rotates by a right angle, until it ends up where it started.

Let's try it on the number 1:

i cycle on complex plane
1 × i   = i
i × i   = −1
−1 × i   = −i
−i × i   = 1
Back to 1 again!

Each time a right angle rotation.

Choose your own complex number and try that for yourself, it is good practice.

Let's look more closely at angles now.

Polar Form

Our friendly complex number 3 + 4i:
complex plane 3+4i vector
Here it is again, but in polar form (distance and angle):

So the complex number 3 + 4i can also be shown as distance (5) and angle (0.927 radians)
complex plane 3+4i polar

How do we do the conversions?

Example: the number 3 + 4i

We can do a Cartesian to Polar conversion:

We can also go the other way (Polar to Cartesian coordinates):

In fact, a common way to write a complex number in Polar form is

x + iy = r cos θ + i r sin θ
  = r(cos θ + i sin θ)

And "cos θ + i sin θ" is often shortened to "cis θ", so:

x + iy = r cis θ

cis θ is just shorthand for cos θ + i sin θ

So we can write:

3 + 4i = 5 cis 0.927

In some subjects, like electronics, "cis" is used a lot!

Now For Some More Multiplication

Let's try another multiplication:

Example: Multiply 1+i by 3+i

(1+i) (3+i) =1(3+i) + i(3+i) 3 + i + 3i + i2 3 + 4i − 1 2 + 4i

And here is the result on the Complex Plane:

complex plane 1+i, 3+i, 2+4i

But it is more interesting to see those numbers in Polar Form:

Example: (continued)

Convert 1+i to Polar:

Convert 3+i to Polar:

Convert 2+4i to Polar:

Have a look at the r values for a minute. Are they related somehow?
And what about the θ values?

Here is that multiplication in one line (using "cis"):

(√2 cis 0.785) × (√10 cis 0.322) = √20 cis 1.107

This is the interesting thing:

So:

The magnitudes get multiplied
and the angles get added

When multiplying in Polar Form: multiply the magnitudes, add the angles.

complex plane i is right angle

And that is why multiplying by i rotates by a right angle:

i has a magnitude of 1
and forms a right angle on the complex plane

Squaring

To square a complex number, multiply it by itself:

Result: square the magnitudes, double the angle.

complex plane vector 1+2i squared is -3+4i

Example: Let us square 1 + 2i:

(1 + 2i)(1 + 2i) = 1 + 4i + 4i2 = −3 + 4i

On the diagram the angle looks to be (and is!) doubled.

Also:

So the magnitude got squared, too.

In general, a complex number like:

r(cos θ + i sin θ)

When squared becomes:

r2(cos 2θ + i sin 2θ)

The magnitude r gets squared and the angle θ gets doubled.

Or in the shorter "cis" notation:

(r cis θ)2 = r2 cis 2θ

de moivre

De Moivre's Formula

We can make the previous formula more general! With the help of Abraham de Moivre we have a formula for any integer n:

(r cis θ)n = rn cis nθ

The magnitude becomes rn and the angle becomes nθ

Example: What is (1+i)6

First convert 1+i to Polar:

In "cis" notation is now: √2 cis π/4

complex plane 0-8i

Use the formula with an exponent of 6:

(√2 cis π/4)6 = (√2)6 cis 6π/4

Which simplifies to:

8 cis 3π/2

In other words: the magnitude is now 8, and the angle is 3π/2 (=270°)

Which is also 0−8i (see diagram)

Learn more at de Moivre's Formula

Summary

276, 8002, 8003, 8888, 8889, 8891,277, 8004, 8005, 8890, 8892
Complex Numbers de Moivre's Formula Algebra Index
Copyright © 2025 Rod Pierce