There are all sorts of ways of writing numbers. We can use arithmetics with different bases, fractions, decimals, logarithms, powers, or simply words. Each is more convenient for one purpose or another and each will be familiar to anyone who has done some mathematics at school. But, surprisingly, one of the most striking and powerful representations of numbers is completely ignored in the mathematics that is taught in schools and it rarely makes an appearance in university courses, unless you take a special option in number theory. Yet continued fractions are one of the most revealing representations of numbers. Numbers whose decimal expansions look unremarkable and featureless are revealed to have extraordinary symmetries and patterns embedded deep within them when unfolded into a continued fraction. Continued fractions also provide us with a way of constructing rational approximations to irrational numbers and discovering the most irrational numbers.