# Non-Euclidean geometry and Indra's pearls

### 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.

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)*.

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).*

Division of complex numbers is a little harder. Suppose that *z= x + iy* and that *w = u + iv* are two complex numbers. Then we calculate *z/w* by getting rid of the imaginary part in the denominator:

*(x + iy)/(u + iv) = ((x + iy)(u - iv))/((u + iv)(u - iv)) = (xu + yv + i(yu +xv))/(u*Since the denomninator of this expression is a real number, we now have a new complex number

^{2}+ v^{2}).*(xu + yv)/(u*

^{2}+ v^{2}) + i(yu +xv)/(u^{2}+ v^{2}).Now suppose that you have a MÃ¶bius transformation of the form

*z -> (az+b)/(cz+d),*

where *a*, *b*, *c* and *d* are some fixed complex numbers. Using the rules for addition, multiplication and division above you can calculate its value, which is a new complex number, or, going back to our visual interpretation, a new point on the plane.