Today I learned how to do ceiling division in Python just with //.

Floor division //

I recently published a tweet telling people about the floor division operator in Python, //:

Are you familiar with the `//` operator in Python 🐍?

`//` is the “floor division” operation, which is equivalent to dividing and then rounding down.

`q = n // m` is always an integer, and the value of `q` is equivalent to `q = floor(n / m)`.

How many years fit in 10_000 days? pic.twitter.com/ecRQoz3qkM

— Rodrigo 🐍📝 (@mathsppblog) September 14, 2021

This operator is equivalent to doing regular division and then flooring down:

>>> # How many years in 10_000 days?
>>> from math import floor; floor(10_000 / 365)
27
>>> 10_000 // 365
27

Then, someone asked if Python also had a built-in for ceiling division, that is, an operator that divided the operands and then rounded up.

While there is no direct built-in operator for that, someone replied saying that we can use a couple of minus signs and floor division to do that.

Ceiling division with a and b would be equivalent to ceil(a / b). And they showed that we can do it with -(-a // b):

>>> from math import ceil
>>> a, b = 10, 3
>>> ceil(a / b)
4
>>> -(-a // b)
4

Why does this work?

floor rounds down and ceil rounds up. By using -a in the division, it's as if you flip a upside down, so “its ceiling is now on the floor”, so you can use -a // b. Then, you just need to put everything back in place, using a final negation: -(-a // b).

At first, I thought this would fail for some combination of positive/negative values for a and b, but it most certainly doesn't.

For one, the explanation works regardless of the sign of a and/or b. Secondly, one can always test it:

>>> for a, b in [(10, 3), (10, -3), (-10, 3), (-10, -3)]:
...     assert ceil(a / b) == -(-a // b)
...
>>>

Here's the original tweet that taught me this:

You can do ceiling division with '//' and some unary '-' signs, since '//' truncates to the next *lowest* number. If the divisor is negative, that means it goes to the next "most-negative" number, which, when negated, is actually "truncating up". pic.twitter.com/uAQmJJbYhw

— Paul McGuire - pyparsing guy (@ptmcguire) September 15, 2021

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