# Non-Euclidean geometry and Indra's pearls

June 2007

### 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))/(u2 + v2).Since the denomninator of this expression is a real number, we now have a new complex number(xu + yv)/(u2 + v2) + i(yu +xv)/(u2 + v2).

Now suppose that you have a Möbius 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.

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