What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original.
While there has been plenty commentary on Egyptian fractions, I’d like to suggest my favorite alternative, the Pierce and Engel fractions. Engel are defined by x= 1/a_1 + 1/(a_1a_2) + 1/(a_1a_2a_3) + … where u_1=x, a_k = ceil(1/u_k), u_{k+1}=u_ka_k-1. And a lovely obscure number representation was introduced by Marshall Hall Jr in the 1940s. Define a bounded continued fraction as one where the partial quotients can only be so big. For example, F(4) is the set of all continued fractions with partial quotients from 1 to 4. Hall proved that every real can be represented as the sum of two bounded continued fractions with bound 4, which he wrote as R=F(4)+F(4).