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
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: