### 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 *x ^{2} + c* even when the seed

*x*and the constant

_{0}*c*are complex numbers: if

*x = a + ib*and

*c = s + it*then

*x*

^{2}+ c = (a + ib)^{2}+ (s + it) = a^{2}- b^{2}+ i(2ab) + s +it = (a^{2}- b^{2}+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.