Interesting Application of the Greedy Algorithm for Egyptian Fractions

Here we discuss a new system to represent numbers, for instance constants such as Pi, e, or log 2, using rational fractions. Each iteration doubles the precision (the number of correct decimals computed) making it converging much faster than current systems such as continued fractions, to represent any positive real number. The algorithm discussed here is equivalent to the greedy algorithm to compute Egyptian fractions, except that we use it here mostly for irrational numbers. You start with a seed p(1) 1 (though you can work with other seed values) and loop over k 1, 2, and so on. As k tends to infinity, x(k) tends to x.