057_square_root_convergents.hs

-- 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?

import Data.Ratio
import Numeric

import MyLib_haskell

numeratorGTdenum :: Rational -> Bool
numeratorGTdenum n = numLength (numerator n) > numLength (denominator n)

iterativeRoot :: Int -> [Rational]
iterativeRoot n = map (+ (-1)) $ take n $ tail $ iterate (\x -> 2 + 1/x) 2

main :: IO()
main = print . length . filter (== True) . map numeratorGTdenum . iterativeRoot $ 1000