Irrational Number is Limit of Unique Simple Infinite Continued Fraction
Jump to navigation
Jump to search
Theorem
Let $x$ be an irrational number.
Then the continued fraction expansion of $x$ is the unique simple infinite continued fraction that converges to $x$.
Proof
Follows from:
- Continued Fraction Expansion of Irrational Number Converges to Number Itself
- Simple Infinite Continued Fraction is Uniquely Determined by Limit.
$\blacksquare$
Also see
- Correspondence between Irrational Numbers and Simple Infinite Continued Fractions, a more precise statement
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continued fraction