| 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, |
| 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.