Complex Numbers

A Complex Number is a combination of:

a Real Number

Real Numbers are just numbers like:

1 12.38 -0.8625 3/4 √2 1998

Nearly any number you can think of is a Real Number

and an Imaginary Number

Imaginary Numbers are special because:

When squared, they give a negative result.

Normally this doesn't happen, because:

But just imagine there is such a number, because we will need it!

The "unit" imaginary number (like 1 for Real Numbers) is i, which is the square root of -1

equals the square root of -1

(Read Imaginary Numbers to find out more.)

A Combination

So we have this definition:

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

Complex Number

Examples:

1 + i 39 + 3i 0.8 - 2.2i -2 + πi √2 + i/2

 

Can a Number be a Combination of Two Numbers?

Can we make up a number from two other numbers? Sure we can!

We do it with fractions all the time. The fraction 3/8 is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts".

Well, a Complex Number is just two numbers added together (a Real and an Imaginary Number).

Either Part Can Be Zero

So, a Complex Number has a real part and an imaginary part.

But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

Complex Number Real Part Imaginary Part
3 + 2i 3 2
5 5 0
-6i 0 -6

Complicated?

Complex does not mean complicated.

It means the two types of numbers, real and imaginary, together form a complex, just like a building complex (buildings joined together).

Adding

To add two complex numbers we add each element separately:

(a+bi) + (c+di) = (a+c) + (b+d)i

Example: (3 + 2i) + (1 + 7i) = (4 + 9i)

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  

And this:

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 we have the (ac - bd) + (ad + bc)i  pattern.

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

Let us try i2

Just for fun, let's use the method to calculate i2

Example: i2

i can also be written with a real and imaginary part as 0 + i

i2 = (0 + i)2   = (0 + i)(0 + i)  
    = (0×0 - 1×1) + (0×1 + 1×0)i  
    = -1 + 0i  
    = -1  

And that agrees nicely with the definition that i2 = -1

So it all works nicely!

Learn more at Complex Number Multiplication.

 

 

Complex Plane

We can also put complex numbers on a Complex Plane.

  • The Real part goes left-right
  • The Imaginary part goes up-down

Conjugates

A conjugate is where we change the sign in the middle like this:

Complex Conjugate

A conjugate is often written with a bar over it:

Example:

  5 - 3i   =   5 + 3i

Dividing

The conjugate is used to help division.

The trick is to multiply both top and bottom by the conjugate of the bottom.

Example: Do this Division:

  2 + 3i
4 - 5i

Multiply top and bottom by the conjugate of 4 - 5i :

  2 + 3i × 4 + 5i = 8 + 10i + 12i + 15i2
4 - 5i 4 + 5i 16 + 20i - 20i - 25i2

Now remember that i2 = -1, so:

  = 8 + 10i + 12i - 15
16 + 20i - 20i + 25

Add Like Terms (and notice how on the bottom 20i - 20i cancels out!):

  = -7 + 22i
41

We should then put the answer back into a + bi form:

  = -7 + 22 i
41 41

DONE!

Yes, there is a bit of calculation to do. But it can be done.

Multiplying By the Conjugate

We can save a little bit of time, though.

In that example, what happened on the bottom was interesting:

(4 - 5i)(4 + 5i) = 16 + 20i - 20i - 25i2

The middle terms cancel out!
And since i2=-1 we ended up with this:

(4 - 5i)(4 + 5i) = 42 + 52

Which is really quite a simple result

In fact we can write a general rule like this:

(a + bi)(a - bi) = a2 + b2

So that can save us time when do division, like this:

Example: What is

  2 + 3i
4 - 5i

Multiply top and bottom by the conjugate of 4 - 5i :

  2 + 3i × 4 + 5i = 8 + 10i + 12i + 15i2 = -7 + 22i
4 - 5i 4 + 5i 16 + 25 41

And then back into a + bi form:

  = -7 + 22 i
41 41

DONE!

 

Mandelbrot Set

Mandelbrot Set

The beautiful Mandelbrot Set (pictured here) is based on Complex Numbers.

It is a plot of what happens when we take the simple equation z2+c (both complex numbers) and feed the result back into z time and time again.

The color shows how fast z2+c grows, and black means it stays within a certain range.

Here is an image made by zooming into the Mandelbrot set

Mandelbrot Set Zoomed In
And here is the center of the previous one zoomed in even further:

 

 
Challenging Questions: 1 2