# Is there a ceiling equivalent of // operator in Python?

## Is there a ceiling equivalent of // operator in Python?

No, but you can use upside-down floor division:¹

``````def ceildiv(a, b):
return -(a // -b)
``````

This works because Pythons division operator does floor division (unlike in C, where integer division truncates the fractional part).

Heres a demonstration:

``````>>> from __future__ import division     # for Python 2.x compatibility
>>> import math
>>> def ceildiv(a, b):
...     return -(a // -b)
...
>>> b = 3
>>> for a in range(-7, 8):
...     q1 = math.ceil(a / b)   # a/b is float division
...     q2 = ceildiv(a, b)
...     print(%2d/%d %2d %2d % (a, b, q1, q2))
...
-7/3 -2 -2
-6/3 -2 -2
-5/3 -1 -1
-4/3 -1 -1
-3/3 -1 -1
-2/3  0  0
-1/3  0  0
0/3  0  0
1/3  1  1
2/3  1  1
3/3  1  1
4/3  2  2
5/3  2  2
6/3  2  2
7/3  3  3
``````

## Why this instead of math.ceil?

`math.ceil(a / b)` can quietly produce incorrect results, because it introduces floating-point error. For example:

``````>>> from __future__ import division     # Python 2.x compat
>>> import math
>>> def ceildiv(a, b):
...     return -(a // -b)
...
>>> x = 2**64
>>> y = 2**48
>>> ceildiv(x, y)
65536
>>> ceildiv(x + 1, y)
65537                       # Correct
>>> math.ceil(x / y)
65536
>>> math.ceil((x + 1) / y)
65536                       # Incorrect!
``````

In general, its considered good practice to avoid floating-point arithmetic altogether unless you specifically need it. Floating-point math has several tricky edge cases, which tends to introduce bugs if youre not paying close attention. It can also be computationally expensive on small/low-power devices that do not have a hardware FPU.

¹In a previous version of this answer, ceildiv was implemented as `return -(-a // b)` but it was changed to `return -(a // -b)` after commenters reported that the latter performs slightly better in benchmarks. That makes sense, because the dividend (a) is typically larger than the divisor (b). Since Python uses arbitrary-precision arithmetic to perform these calculations, computing the unary negation `-a` would almost always involve equal-or-more work than computing `-b`.

There is no operator which divides with ceil. You need to `import math` and use `math.ceil`

## Solution 1: Convert floor to ceiling with negation

``````def ceiling_division(n, d):
return -(n // -d)
``````

Reminiscent of the Penn & Teller levitation trick, this turns the world upside down (with negation), uses plain floor division (where the ceiling and floor have been swapped), and then turns the world right-side up (with negation again)

## Solution 2: Let divmod() do the work

``````def ceiling_division(n, d):
q, r = divmod(n, d)
return q + bool(r)
``````

The divmod() function gives `(a // b, a % b)` for integers (this may be less reliable with floats due to round-off error). The step with `bool(r)` adds one to the quotient whenever there is a non-zero remainder.

## Solution 3: Adjust the numerator before the division

``````def ceiling_division(n, d):
return (n + d - 1) // d
``````

Translate the numerator upwards so that floor division rounds down to the intended ceiling. Note, this only works for integers.

## Solution 4: Convert to floats to use math.ceil()

``````def ceiling_division(n, d):
return math.ceil(n / d)
``````

The math.ceil() code is easy to understand, but it converts from ints to floats and back. This isnt very fast and it may have rounding issues. Also, it relies on Python 3 semantics where true division produces a float and where the ceil() function returns an integer.