Continued fraction expansion of a real number

Definition

Let $x$ be a real number. Put $x_0 = x$ and $a_0 = \lfloor x_0 \rfloor$. For $n \ge 0$, if $x_n \neq a_n$, put $x_{n+1} = \dfrac{1}{x_n - a_n}$ and $a_{n+1} = \lfloor x_{n+1} \rfloor$. If $x_n = a_n$, the process stops. The continued fraction expansion of $x$ is the simple continued fraction $[a_0; a_1, a_2, \dots]$, finite or infinite.

Remarks

Here $\lfloor y \rfloor$ is the largest integer that is at most $y$. Since $0 < x_n - a_n < 1$, each $a_{n+1}$ with $n \ge 0$ is a positive integer. For example, $\frac{43}{19} = [2; 3, 1, 4]$, and $\sqrt{2} = [1; 2, 2, 2, \dots]$. The process stops if and only if $x$ is rational, by the termination theorem. For a rational $x$, the steps are the steps of the Euclidean algorithm on its numerator and denominator.

Explanations

L1 human

Take the integer part of $x$, then flip the remaining fractional part, and repeat. The integer parts that you collect are the terms of the continued fraction of $x$.