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A Complex Number is a combination of a Real Number and an Imaginary Number:

A Real Number is the type of number we use every day.

Examples: 12.38, ½, 0, −2000

An Imaginary Number, when squared gives a negative result:

The 'unit' imaginary number when squared equals −1

i² = −1

Examples: 5i, −3.6i, i/2, 500i

Examples of Complex Numbers:

Either part can be zero:

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):

Like this:

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

Here is another example:

Example: (1 + i)²

But There is a Quicker Way!

Use this rule:

(a+bi)(c+di) = (ac−bd) + (ad+bc)i

Example:

Why Does That Rule Work?

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

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

Multiplying By i

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?

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:

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

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

How do we do the conversions?

Example: the number 3 + 4i

We can do a Cartesian to Polar conversion:

• r = √(x² + y²) = √(3² + 4²) = √25 = 5
• θ = tan^-1 (y/x) = tan^-1 (4/3) = 0.927 (to 3 decimals)

We can also take Polar coordinates and convert them to Cartesian coordinates:

• x = r × cos( θ ) = 5 × cos( 0.927 ) = 5 × 0.6002... = 3 (close enough)
• y = r × sin( θ ) = 5 × sin( 0.927 ) = 5 × 0.7998... = 4 (close enough)

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

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

x + iy = r cis θ

cis is just shorthand for cos θ + i sin θ

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

And here is the result on the Complex Plane:

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

Example: (continued)

Convert 1+i to Polar:

• r = √(1² + 1²) = √2
• θ = tan^-1 (1/1) = 0.785 (to 3 decimals)

Convert 3+i to Polar:

• r = √(3² + 1²) = √10
• θ = tan^-1 (1/3) = 0.322 (to 3 decimals)

Convert 2+4i to Polar:

• r = √(2² + 4²) = √20
• θ = tan^-1 (4/2) = 1.107 (to 3 decimals)

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:

• √2 x √10 = √20
• 0.785 + 0.322 = 1.107

So:

The magnitudes get multiplied.
And the angles get added.

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

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

Squaring

To square a complex number, multiply it by itself:

• multiply the magnitudes: magnitude × magnitude = magnitude²
• add the angles: angle + angle = 2 , so we double them.

Result: square the magnitudes, double the angle.

Example: Let us square 1 + 2i:

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

Also:

• The magnitude of (1+2i) = √(1² + 2²) = √5
• The magnitude of (−3+4i) = √(3² + 4²) = √25 = 5

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's Formula

And the mathematician Abraham de Moivre found it works for any integer exponent n:

[ r(cos θ + i sin θ) ]ⁿ = rⁿ(cos nθ + i sin nθ)

(the magnitude becomes rⁿ the angle becomes nθ.)

Or in the shorter 'cis' notation:

(r cis θ)ⁿ = rⁿ cis nθ

Example: What is (1+i)⁶

Convert 1+i to Polar:

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• r = √(1² + 1²) = √2
• θ = tan^-1 (1/1) =π/4

In 'cis' notation: 1+i = √2 cis π/4

Now, with an exponent of 6, r becomes r⁶, θ becomes 6θ:

(√2 cis π/4)⁶ = (√2)⁶ cis 6π/4 = 8 cis 3π/2

The magnitude is now 8, and the angle is 3π/2 (=270°)

Which is also 0−8i (see diagram)

Summary

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• Use 'FOIL' to multiply complex numbers,
• Or use the formula:
• Or use polar form and then multiply the magnitudes and add the angles.
• De Moivre's Formula can be used for integer exponents:
  [ r(cos θ + i sin θ) ]ⁿ = rⁿ(cos nθ + i sin nθ)
• Polar form r cos θ + i r sin θ is often shortened to r cis θ

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