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# 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 $[ x ]$ or $\lfloor x \rfloor$.

We always have

$\lfloor x\rfloor \le x < \lfloor x + 1 \rfloor$
with equality on the left if and only if x is an integer. For any integer k and any real number x, we have
$\lfloor k+x \rfloor = k + \lfloor x\rfloor$
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 k, we also have the following equality:
$\lfloor k / 2 \rfloor + \lceil k / 2 \rceil = k$.

If m and n are coprime positive integers, then

$\sum_{i=1}^{n-1} \lfloor im / n \rfloor = (m - 1) (n - 1) / 2$

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