How to resolve the algorithm Arithmetic/Rational step by step in the Haskell programming language
How to resolve the algorithm Arithmetic/Rational step by step in the Haskell programming language
Table of Contents
Problem Statement
Create a reasonably complete implementation of rational arithmetic in the particular language using the idioms of the language.
Define a new type called frac with binary operator "//" of two integers that returns a structure made up of the numerator and the denominator (as per a rational number). Further define the appropriate rational unary operators abs and '-', with the binary operators for addition '+', subtraction '-', multiplication '×', division '/', integer division '÷', modulo division, the comparison operators (e.g. '<', '≤', '>', & '≥') and equality operators (e.g. '=' & '≠'). Define standard coercion operators for casting int to frac etc. If space allows, define standard increment and decrement operators (e.g. '+:=' & '-:=' etc.). Finally test the operators: Use the new type frac to find all perfect numbers less than 219 by summing the reciprocal of the factors.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Arithmetic/Rational step by step in the Haskell programming language
This Haskell code snippet calculates and prints the first N perfect numbers. Perfect numbers are positive integers that are equal to the sum of their proper divisors, excluding the number itself.
The main function:
- Defines a constant
n
to specify how many perfect numbers to print (in this case, 4). - Uses the
mapM_
function to print each perfect number found usingtake
. take
takes the firstn
elements from the list generated by the list comprehension [candidate
| ... ].- The list comprehension generates candidate perfect numbers by iterating through a range of integers [2 .. 2 ^ 19] and filtering for numbers that satisfy the condition getSum candidate == 1, which checks if the sum of their divisors (excluding the number itself) is 1.
The getSum function:
- Calculates the sum of the proper divisors of a candidate perfect number.
- It adds 1 % candidate, which represents the number itself, and then sums over all factors (2 up to the square root of the candidate) that evenly divide the candidate.
- For each factor, it adds 1 % factor and 1 % (candidate
div
factor), which represent the factor and its corresponding pair in the factorization of the candidate.
Source code in the haskell programming language
import Data.Ratio ((%))
-- Prints the first N perfect numbers.
main = do
let n = 4
mapM_ print $
take
n
[ candidate
| candidate <- [2 .. 2 ^ 19]
, getSum candidate == 1 ]
where
getSum candidate =
1 % candidate +
sum
[ 1 % factor + 1 % (candidate `div` factor)
| factor <- [2 .. floor (sqrt (fromIntegral candidate))]
, candidate `mod` factor == 0 ]
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