Division in Haskell

While learning about division in Haskell, I realized that the concept is not as trivial as I initially wanted it to be. There are subtle differences between the functions that can easily trip someone who’s not aware of them. quot performs integer division that rounds towards zero. div is like quot, but it rounds below zero—negative infinity. rem returns the remainder of a division. mod, on the other hand, performs modular arithmetic.

In the GHCi interaction below, quotRem returns a tuple of the application of quot and rem to its arguments, while divMod returns a tuple of the application div and mod to its arguments.

Prelude> quotRem 13 5
(2,3)
Prelude> quotRem (-13) 5
(-2,-3)
Prelude> quotRem 13 (-5)
(-2,3)
Prelude> quotRem 5 13
(0,5)
Prelude> quotRem (-5) 13
(0,-5)
Prelude> quotRem 5 (-13)
(0,5)
Prelude> divMod 13 5
(2,3)
Prelude> divMod (-13) 5
(-3,2)
Prelude> divMod 13 (-5)
(-3,-2)
Prelude> divMod 5 13
(0,5)
Prelude> divMod (-5) 13
(-1,8)
Prelude> divMod 5 (-13)
(-1,-8)

Giving special attention to negative numbers, here are some observations about it:

• quotRem and divMod behave the same, if all the arguments are positive.
• quot returns 0, if the dividend is less than the divisor.
• rem follows the sign of the dividend.
• rem returns the dividend, if the dividend is less than the divisor.
• div rounds off the divisor to the negative infinity, if either the dividend or divisor is negative.
• mod follows the sign of the divisor.
• div returns 0, if the dividend is less than the divisor, and both arguments are positive.
• mod returns the dividend, if the dividend is less than the divisor, and both arguments are positive.
• div returns -1 if the dividend is negative, and its absolute value is less than the divisor.
• quot and div returns 0 if the dividend is 0, and the divisor is not zero.
• mod returns the difference of the absolute values of the dividend and the divisor, following the sign of the divisor, if either the dividend or the divisor is negative, and if the absolute of the dividend is less than the absolute value of the divisor.
• If either the dividend or divisor of mod is a negative, and that the absolute value of the dividend is larger than the abvolute value of the divisor, mod returns a value such that when this value is added to the result of the div of the dividend and divisor, multiplied by the divisor, it returns the dividend of mod. That is:

  x == (x `mod` y) + (x `div` y) * y
  

  So if x = (-13) and y = 5, then

  (-13) == ((-13) `mod` 5) + ((-13) `div` 5) * 5
  

  evaluates to True.