Correspondence between Rational Numbers and Simple Finite Continued Fractions

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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:
$S \to \Q$ 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:
$S \to \Q$ 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:

$\blacksquare$