# Correspondence between Irrational Numbers and Simple Infinite Continued Fractions

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## Theorem

Let $\R \setminus \Q$ be the set of irrational numbers.

Let $S$ be the set of all simple infinite continued fractions in $\R$.

The mappings:

- $\R \setminus \Q \to S$ that sends an irrational number to its continued fraction expansion
- $S \to \R \setminus \Q$ that sends a simple infinite continued fractions to its value

## Proof

Note that indeed a Simple Infinite Continued Fraction Converges to Irrational Number.

The result follows from:

- Continued Fraction Expansion of Irrational Number Converges to Number Itself
- Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself

$\blacksquare$