In
mathematics, the
floor function is the
function defined as follows: for a
real number x, floor(
x) is the largest
integer less than or equal to
x. For example, floor(2.3) = 2, floor(-2) = -2 and floor(-2.3) = -3. The floor function is also denoted by <math> [ x ] </math> or <math>\lfloor x \rfloor</math>.
We always have
- <math> \lfloor x\rfloor \le x < \lfloor x + 1 \rfloor</math>
with equality on the left if and only if
x is an integer. For any integer
k and any real number
x, we have
- <math> \lfloor k+x \rfloor = k + \lfloor x\rfloor</math>
The ordinary rounding of the number
x to the nearest integer can be expressed as floor(
x + 0.5).
The floor function is not continuous, but it is upper semi-continuous.
A closely related mathematical function is the ceiling function,
which is defined as follows: for any given real number x, ceiling(x)
is the smallest integer no less than x. For example, ceiling(2.3) = 3,
ceiling(2) = 2 and ceiling(-2.3) = -2. The ceiling function is also denoted
by <math>\lceil x \rceil</math>. It is easy to show the following:
- <math>\lceil x \rceil = - \lfloor - x \rfloor</math>
and the following:
- <math>x \leq \lceil x \rceil < x + 1</math>
For any integer
k, we also have the following equality:
- <math>\lfloor k / 2 \rfloor + \lceil k / 2 \rceil = k</math>.
If m and n are coprime positive integers, then
- <math>\sum_{i=1}^{n-1} \lfloor im / n \rfloor = (m - 1) (n - 1) / 2</math>