# 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 denominators 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 $\sqbrk {a_0, \ldots, a_n}$ if $n$ is even
$\sqbrk {a_0, \ldots, a_n - 1, 1}$ 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 $\sqbrk {a_0, \ldots, a_n}$ if $n$ is odd
$\sqbrk {a_0, \ldots, a_n - 1, 1}$ if $n$ is even
$S \to \Q$ that sends a simple finite continued fractions to its value

## Proof

Note that indeed Simple Finite 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

$\blacksquare$