Irrational Number has Periodic Continued Fraction iff Quadratic

Theorem

Let $x$ be an irrational number.

Then $x$ is a quadratic irrational if and only if its continued fraction expansion is periodic.

Proof

Follows from:

• Limit of Simple Infinite Periodic Continued Fraction is Quadratic Irrational
• Quadratic Irrational has Periodic Continued Fraction Expansion

$\blacksquare$