Integer division by negative number [duplicate]

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Short answer: Language designers get to choose if their language will round towards zero, negative infinity, or positive infinity when doing integer division. Different languages have made different choices.

Long answer: The language authors of Python and Ruby both decided that rounding towards negative infinity makes more sense than rounding towards zero (like C does). The creator of python wrote a blog post about his reasoning here. I’ve excerpted much of it below.

I was asked (again) today to explain why integer division in Python returns the floor of the result instead of truncating towards zero like C.

For positive numbers, there’s no surprise:

>>> 5//2
2

But if one of the operands is negative, the result is floored, i.e., rounded away from zero (towards negative infinity):

>>> -5//2
-3
>>> 5//-2
-3

This disturbs some people, but there is a good mathematical reason.
The integer division operation (//) and its sibling, the modulo
operation (%), go together and satisfy a nice mathematical
relationship (all variables are integers):

a/b = q with remainder r

such that

b*q + r = a and 0 <= r < b
(assuming a and b are >= 0).

If you want the relationship to extend for negative a (keeping b
positive), you have two choices: if you truncate q towards zero, r
will become negative, so that the invariant changes to 0 <= abs(r) <
otherwise, you can floor q towards negative infinity, and the
invariant remains 0 <= r < b. [update: fixed this para]

In mathematical number theory, mathematicians always prefer the latter
choice (see e.g. Wikipedia). For Python, I made the same choice
because there are some interesting applications of the modulo
operation where the sign of a is uninteresting. Consider taking a
POSIX timestamp (seconds since the start of 1970) and turning it into
the time of day. Since there are 24*3600 = 86400 seconds in a day,
this calculation is simply t % 86400. But if we were to express times
before 1970 using negative numbers, the “truncate towards zero” rule
would give a meaningless result! Using the floor rule it all works out
fine.

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