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Starting with $x_0=x>0$, the continued-fraction algorithm setsunless $x_n$ is already an integer. This gives

Every finite continued fraction is rational, since it is built from integers by finitely many additions and reciprocals. Conversely, if $x=p/q$ is rational in lowest terms, thenWhen $r\ne 0$, the next complete quotient is $q/r$. Thus each step is an application of the Euclidean algorithm and replaces the denominator by a smaller nonnegative remainder. The process must terminate. This proves the termination criterion for a simple continued fraction.

For $x=\sqrt{3}$,andFinally,so the complete quotients repeat. Hence the continued fraction of the square root of three is