# Problem 57: Square root convergents
It is possible to show that the square root of two can be expressed as
an infinite continued fraction. √ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) =
1.414213... By expanding this for the first four iterations, we get: 1 +
1/2 = 3/2 = 1.5 1 + 1/(2 + 1/2) = 7/5 = 1.4 1 + 1/(2 + 1/(2 + 1/2)) =
17/12 = 1.41666... 1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...
The 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.
In the first one-thousand expansions, how many fractions contain a
numerator with more digits than denominator?