The regular continued fraction (CFs) expansion of a real number is finite if and only if the number is rational. By the Euler-Lagrange theorem, the expansion is eventually-periodic if and only if the number is a root of a non-degenerate quadratic equation with integer coefficients. We provide a new proof of these results that applies to a wide range of CF algorithms over the real numbers, complex numbers, and much more exotic spaces. The talk will be largely self-contained and will focus on CF algorithms for real numbers. This is joint work with Joseph Vandehey.