We want to add, subtract, multiply, and divide continued fractions
We'll see a slightly simpler case first
Say we have a continued fraction, x
We want to calculate 
, which we will abbreviate as 
We will think of z as a machine which can perform two kinds of actions:
If it thinks it knows a term of the result, it can output that term
If it needs more information about x, it can request a term from x
Asking z for its next term might cause it to demand one or more terms from x
Eventually, it will emit the value we asked for