Definition:Continued Fraction/Expansion of Real Number

Irrational Number
Let $x$ be an irrational number.

The continued fraction for $x$ is the continued fraction resulting from the Continued Fraction Algorithm.

It is a simple infinite continued fraction.

Rational Number
Let $x$ be a rational number.

The continued fraction for $x$ is found using the Euclidean Algorithm.

Also see

 * Existence and Uniqueness of Simple Infinite Continued Fraction