Imaginary and Complex Numbers

In order to graph fractals on a number plane, you first need to understand imaginary numbers. The square root of 25 is 5 (because 5*5=25). The square root of 9 is 3, and the square root of 1 is 1. However, what is the square root of -25? It can't be -5 because (-5)*(-5)=25, while 5*5=25 also. Therefore, mathematicians invented the imaginary number i, equal to -1. We can now say that -25=5i, because (5-1)²=25.

We can mix together imaginary numbers and real numbers (real numbers are numbers on the number line like 4, 25, 1, 0, 1/2, 3/4, etc.) to create expressions like 3i+4. Numbers like this are called complex numbers. Complex numbers can be added, subtracted, multiplied, and divided quite easily. To add complex numbers, the real and imaginary are added together separately, like this:

To subtract, we follow the same method, but switch the operation:

3i+7

-5i-2

-2i=9

To multiply, a method called FOIL is used. FOIL stands for First Outer Inner Last. We'll FOIL the following example:

To quickly explain how we derived the second line in the example above, we first multiply the two first terms in each quantity, 3i and 5i. 5*3=15, but i*i=-1, so the result is -15. To do the outers, 3i*(-2)=-6i. The inner terms are 7 and 5i; those two multiplied together are 35i. Finally, the product of the last terms, 7 and -2 is -14. To get the last line, the like terms are simply collected (added together).