Complex Numbers
A Complex Number is a combination of a Real Number and an Imaginary Number:
Real Numbers are numbers like:
1  12.38  0.8625  3/4  √2  1998 
Nearly any number you can think of is a Real Number
Imaginary Numbers are special because:
When squared, they give a negative result.
Normally this doesn't happen, because:
 when we square a positive number we get a positive result, and
 when we square a negative number we also get a positive result (because a negative times a negative gives a positive)
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
A Combination
So we have this definition:
A Complex Number is a combination of a Real Number and an Imaginary 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):


(a+bi)(c+di) = ac + adi + bci + bdi^{2} 
Like this:
Example: (3 + 2i)(1 + 7i)
(3 + 2i)(1 + 7i)  = 3×1 + 3×7i + 2i×1+ 2i×7i  
= 3 + 21i + 2i + 14i^{2}  
= 3 + 21i + 2i  14  (because i^{2} = 1)  
= 11 + 23i 
And this:
Example: (1 + i)^{2}
(1 + i)^{2} = (1 + i)(1 + i)  = 1×1 + 1×i + 1×i + i^{2}  
= 1 + 2i  1  (because i^{2} = 1)  
= 0 + 2i 
But There is a Quicker Way!
Use this rule:
(a+bi)(c+di) = (acbd) + (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 + bdi^{2}  FOIL method  
=  ac + adi + bci  bd  (because i^{2}=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 i^{2}
Just for fun, let's use the method to calculate i^{2}
Example: i^{2}
i can also be written with a real and imaginary part as 0 + i
i^{2} = (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 i^{2 }= 1
So it all works nicely!
Learn more at Complex Number Multiplication.
Complex PlaneWe can also put complex numbers on a Complex Plane.

Conjugates
A conjugate is where we change the sign in the middle like this:
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 + 15i^{2}  
4  5i  4 + 5i  16 + 20i  20i  25i^{2} 
Now remember that i^{2} = 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  25i^{2}
The middle terms cancel out!
And since i^{2}=1 we ended up with this:
(4  5i)(4 + 5i) = 4^{2} + 5^{2}
Which is really quite a simple result
In fact we can write a general rule like this:
(a + bi)(a  bi) = a^{2} + b^{2}
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 + 15i^{2}  =  7 + 22i  
4  5i  4 + 5i  16 + 25  41 
And then back into a + bi form:
=  7  +  22  i  
41  41 
DONE!
The 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 z^{2}+c (both complex numbers) and feed the result back into z time and time again. The color shows how fast z^{2}+c grows, and black means it stays within a certain range. 

Here is an image made by zooming into the Mandelbrot set 

And here is the center of the previous one zoomed in even further: 