Complex Numbers

complex example 7 + 3i
A Complex Number

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

 

right arrowReal Numbers are numbers like:

1 12.38 −0.8625 3/4 √2 1998

Nearly any number you can think of is a Real Number!

 

right arrowImaginary Numbers when squared give a negative result.

Normally this doesn't happen, because:

But just imagine such numbers exist, because we will need them.

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

equals the square root of -1

Because when we square i we get −1

i2 = −1

Examples of Imaginary Numbers:

3i 1.04i −2.8i 3i/4 (√2)i 1998i

And we keep that little "i" there to remind us we need to multiply by √−1

Complex Numbers

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?

pie 3/8

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?

building complex

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).

A Visual Explanation

You know how the number line goes left-right?

Well let's have the imaginary numbers go up-down:

complex plane

And we get the Complex Plane

And a complex number can now be shown as a point:

complex plane 3+4i
The complex number 3 + 4i

Adding

To add two complex numbers we add each part separately:

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

Example: add the complex numbers 3 + 2i and 1 + 7i

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

Let's try one visually:

Example: add the complex numbers 3 + 5i and 4 − 3i

complex plane vector addition

(3 + 5i) + (4 − 3i)
= 3 + 4 + (5 − 3)i
= 7 + 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  

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 wonderfully!

Learn more at Complex Number Multiplication.

Conjugates

We will need to lnow about conjugates in a minute!

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 complex 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

There is a faster way though.

In the previous 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: Let's try this again

  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 + 25
           
          =  
−7 + 22i
41
       

And then back into a + bi form:

  =   −7 + 22 i
  41 41

DONE!

 

The 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: Mandelbrot Set Zoomed In More

 

 
Challenging Questions: 1 2