# Iff Criterion For An Irrational Number to be Periodic

Recall from the Periodic and Purely Periodic Continued Fractions page that an infinite continued fraction $\langle a_0; a_1, a_2, ... \rangle$ is said to be periodic if there exists an $n \in \mathbb{N}$ such that $a_{n+r} = a_{r}$ for sufficiently large $r$, and $n$ is called the period of the infinite continued fraction.

When $\theta$ is an irrational number and satisfies a quadratic equation then by the quadratic formula it is of the form:

(1)where $d$ is not a perfect square. We give these special irrationals a name.

Definition: An irrational number is called a Quadratic Irrational if it satisfies a quadratic equation. |

The following Theorem characterizes when an irrational number is a periodic continued fraction.

Theorem 1: Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Then $\theta$ has a periodic continued fraction expansion if and only if $\theta$ is a quadratic irrational. |

We prove the first direction on the Periodic Continued Fractions are Quadratic Irrationals page and then make note of the other dirrection on the Quadratic Irrationals Have Periodic Continued Fractions page.