Complex Number Multiplication

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

A Real Number is the type of number you are used to dealing with 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

Remember to keep the "i" there so you know it is an imaginary number.

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

Examples: 3.6 + 4i, -0.02 + 1.2i, 25 - 0.3i, 0 + 2i

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: a × c
  • Outers: a × di
  • Inners: bi × c
  • Lasts: bi × di

(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!

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).
 

Multiplying By i

This is what happens when we multiply it by i:

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

And i2 = -1, so:

3i + 4i2 = -4 + 3i
 

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

Was that just a weird coincidence?

Let's try multiplying by i again:

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

and again:

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

and again:

(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 look more closely at angles now.

Polar Form

Our friend the complex number 3 + 4i  
     

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).

How do you 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

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

And "cos θ + i sin θ" can be shortened to "cis θ":

So 3 + 4i = 5 cis 0.927

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 it is on the Complex Plane:

 

But it is really 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)

 

So the multiplication (in Polar "cis" form) is

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

 

And here is the fun thing:

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

 

The magnitudes got multiplied. But the angles got added.

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

 

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:

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

Result: square the magnitudes, double the angle.

Let us square 1 + 2i:

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

You can see on the diagram that the angle doubles.

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.)

 

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 .)

Or in the shorter "cis" notation:

(r cis θ)n = rn cis nθ

 

Example: What is (1+i)6

Convert 1+i to Polar:

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

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

(√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

 

Summary

Use "FOIL" to multiply complex numbers,

Or the formula:

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Or you can multiply the magnitudes and add the angles.

De Moivre's Formula can be used for exponents:

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