Unveiling the Mandelbrot setIssue 40
A brief introduction to complex numbers
Complex numbers are based on the number i which is defined to be the square root of -1, so i times i equals -1. This number isn't a real number, in other words it does not appear on the usual number line. For this reason it is called an imaginary number, a slightly contentious name. Now any complex number is of the form a + ib, where a and b are ordinary real numbers. The numbers 1 + i2 or 5 - i8 are both complex numbers.
Complex numbers are added (as you would expect) like this:
and multiplied (again as you would expect) like this:
You can apply a function x2 + c even when the seed x0 and the constant c are complex numbers: if x = a + ib and c = s + it then
which is a new complex number.
Unless a complex number a + ib has b = 0, we cannot find it on the ordinary number line. We can, however, visualise it as a point on the plane: to the number a + ib simply associate the point with co-ordinates (a,b). You can see that the real numbers are contained in the complex numbers: a real number x seen as a complex number is simply x + i0 and corresponds to the point with co-ordinates (x,0).
To summarise, every complex number represents a point on the plane and vice versa. We can visualise the orbit of any seed, including 0, as a sequence of points on the plane.