Correspondence between Rational Numbers and Simple Finite Continued Fractions

Theorem
Let $\Q$ be the set of rrational 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

are reverse bijections.

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

are reverse bijections.

Odd length
Let $S_0$ 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

are reverse bijections.

Proof
Note that indeed Value of Simple Finite Continued Fraction is Rational Number.

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

Also see

 * Correspondence between Irrational Numbers and Simple Infinite Continued Fractions