Floor function

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>