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 or .

We always have

with equality on the left if and only if x is an integer. For any integer k and any real number x, we have

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 . It is easy to show the following:

and the following:

For any integer k, we also have the following equality:

.

If m and n are coprime positive integers, then

Beatty's theorem shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.