# Rational Numbers and Simple Finite Continued Fractions are Equivalent

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## Theorem

Simple finite continued fractions and rational numbers are equivalent, in the following precise sense.

### Simple Finite Continued Fraction has Rational Value

Let $n \ge 0$ be a natural number.

Let $\paren {a_0, \ldots, a_n}$ be a simple finite continued fraction of length $n$.

Then its value $\sqbrk {a_0, \ldots, a_n}$ is a rational number.

### Rational Number can be Expressed as Simple Finite Continued Fraction

Let $q \in \Q$ be a rational number.

Then $q$ can be expressed as a simple finite continued fraction.