Question

From the haskell report:

The quot, rem, div, and mod class methods satisfy these laws if y is non-zero:

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

quot is integer division truncated toward zero, while the result of div is truncated toward negative infinity.

For example:

Prelude> (-12) `quot` 5 -2 Prelude> (-12) `div` 5 -3 

What are some examples of where the difference between how the result is truncated matters?

Answer 1

Many languages have a ‘mod’ or ‘%’ operator that gives the remainder after division with truncation towards 0; for example C, C++, and Java, and probably C#, would say:

(-11)/5 = -2 (-11)%5 = -1 5*((-11)/5) + (-11)%5 = 5*(-2) + (-1) = -11. 

Haskell’s quot and rem are intended to imitate this behaviour. I can imagine compatibility with the output of some C program might be desirable in some contrived situation.

Haskell’s div and mod, and subsequently Python’s / and %, follow the convention of mathematicians (at least number-theorists) in always truncating down division (not towards 0 — towards negative infinity) so that the remainder is always nonnegative. Thus in Python,

(-11)/5 = -3 (-11)%5 = 4 5*((-11)/5) + (-11)%5 = 5*(-3) + 4 = -11. 

Haskell’s div and mod follow this behaviour.