Complex Iteration

Think of the complex number z as a pair (x, y) of real numbers, and think of the complex number c as a pair of real numbers (a, b).

In these terms, z -> z² + c becomes

x -> x² - y² + a

and

y -> 2*x*y + b

Why is this? Recall the rules of complex arithmetic. Any complex number z can be written as z = x + i*y.

We call x the real part of z

and y the imaginary part of z.

To add two complex numbers, add the real parts and add the imaginary parts:

(v + i*w) + (x + i*y) = (v + x) + i*(w + y).

The product of two complex numbers (v + i*w)*(x + i*y) can be obtained by multiplying binomials (FOIL, for example) and recalling i² = -1. Grouping together the real and the imaginary parts of the product, we obtain

(v + i*w)*(x + i*y) = (v*x - w*y) + i*(v*y + w*x)

Note the special case

(x + i*y)² = (x² - y²) + i*2*x*y

The relation between the iteration formula for z and those for x and y now should be clear.

Return to Complex Iteration.