Unveiling the Mandelbrot set

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September 2006

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:

(a + ib) + (c + id) = (a + c) + i(b + d),

and multiplied (again as you would expect) like this:
(a + ib)(c + id) = ac + iad + ibc + i2bd = ac - bd + i(ad + bc).

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

x2 + c = (a + ib)2 + (s + it) = a2 - b2 + i(2ab) + s +it = (a2 - b2 +s) + i(2ab + t),

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.

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