Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
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- $\R \setminus \Q \to S$ 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
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