# Correspondence between Rational Numbers and Simple Finite Continued Fractions

## Theorem

Let $\Q$ be the set of rational numbers.

Let $S$ be the set of all simple finite continued fractions in $\Q$, whose last partial quotient is not $1$.

The mappings:

- $\Q \to S$ that sends an rational number to its continued fraction expansion
- $S \to \Q$ that sends a simple finite continued fractions to its value

### Even length

Let $S_0$ be the set of all simple finite continued fractions in $\Q$ of even length.

The mappings:

- $\Q \to S$ that sends an rational number to:
- its continued fraction expansion $(a_0, \ldots, a_n)$ if $n$ is even
- $(a_0, \ldots, a_n-1, 1)$ if $n$ is odd

- $S \to \Q$ that sends a simple finite continued fractions to its value

### Odd length

Let $S_1$ be the set of all simple finite continued fractions in $\Q$ of odd length.

The mappings:

- $\Q \to S$ that sends an rational number to:
- its continued fraction expansion $(a_0, \ldots, a_n)$ if $n$ is odd
- $(a_0, \ldots, a_n-1, 1)$ if $n$ is even

- $S \to \Q$ that sends a simple finite continued fractions to its value

## Proof

Note that indeed Finite Simple Continued Fraction has Rational Value.

The result follows from:

- Value of Continued Fraction Expansion of Rational Number equals Number Itself
- Continued Fraction Expansion of Value of Simple Finite Continued Fraction equals Expansion Itself

$\blacksquare$