data/problems/57.yml
---
:id: 57
:name: Square root convergents
:url: https://projecteuler.net/problem=57
:content: "It is possible to show that the square root of two can be expressed as
an infinite continued fraction.\n\n√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...\n\nBy
expanding this for the first four iterations, we get:\n\n1 + 1/2 = 3/2 = 1.5 \n1
+ 1/(2 + 1/2) = 7/5 = 1.4 \n1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666... \n1 +
1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...\n\nThe next three expansions are
99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example
where the number of digits in the numerator exceeds the number of digits in the
denominator.\n\nIn the first one-thousand expansions, how many fractions contain
a numerator with more digits than denominator?\n\n"