Basic concepts of simple continued fractions are introduced and some important theorems explored. The effect of an infinite continued fraction's elements forming a convergent series is looked at via an example of geometric series. The Gauss-Kuzmin-Wirsing operator, operating on functions on the interval [0, 1], is studied numerically. Its invariant density is explored and the rate at which an initial density transforms into the invariant density shown to be O(e-Sn) for iteration n. The transformation associated with the operator is applied numerically to a single random point in [0, 1] and interpretations of the results given.