I'm very new to Haskell, and I would like some feedback on my coding style. Is there anything here that could be made more elegant or concise?
This is my solution to Project Euler Problem 57.
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...$$\sqrt 2 = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \ldots}}} = 1.414213\ldots$$
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?
numerator =
3 : 7 : zipWith (+) numerator (map ((*) 2) (tail numerator))
denominator =
2 : 5 : zipWith (+) denominator (map ((*) 2) (tail denominator))
numDigits :: Integer -> Int
numDigits = length . show
moreDigits :: Integer -> Integer -> Bool
moreDigits x y = (numDigits x) > (numDigits y)
countTrue :: [Bool] -> Int
countTrue = length . filter ((&&) True)
pe_057 = countTrue $
take 1000 (zipWith moreDigits numerator denominator)
main :: IO ()
main = do
print pe_057
