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 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

are inverses of each other.

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

are inverses of each other.

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

are inverses of each other.

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

$\blacksquare$

Also see

• Correspondence between Irrational Numbers and Simple Infinite Continued Fractions