Floor and ceiling functions

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 [x] or ⌊x⌋.

We always have

floor(x) ≤ x < floor(x) + 1

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

floor(k+x) = k + floor(x).

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 $\lceil x\rceil$. It is easy to show the following: $\lceil x\rceil =-\lfloor -x\rfloor$ and the following:

$x\leq \lceil x\rceil <x+1$

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

$\lfloor n/2\rfloor +\lceil n/2\rceil =n$.

If m and n are coprime integers, then

∑_1≤i≤n-1 floor(im/n) = (m-1)(n-1)/2.