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:
- 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
$\blacksquare$