Iff Criterion For An Irrational Number to be Periodic

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)
\begin{align} \quad \theta = \frac{a + b\sqrt{d}}{c} \end{align}

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.