# Floor and Ceiling Functions

## Definitions

Let $$x$$ be a real number. The floor function of $$x,$$ denoted by $$\lfloor {x} \rfloor$$ or $$\text{floor}\left( x \right),$$ is defined to be the greatest integer that is less than or equal to $$x.$$

The ceiling function of $$x,$$ denoted by $$\lceil {x} \rceil$$ or $$\text{ceil}\left( x \right),$$ is defined to be the least integer that is greater than or equal to $$x.$$

For example,

$\lfloor{\pi}\rfloor = 3,\;\;\lceil{\pi}\rceil = 4,\;\;\lfloor{5}\rfloor = 5,\;\;\lceil{5}\rceil = 5.$
$\lfloor{- e}\rfloor = -3,\;\;\lceil{-e}\rceil = -2,\;\;\lfloor{-1}\rfloor = -1,\;\;\lceil{-1}\rceil = -1.$

It follows from the definitions that the floor and ceiling functions have type $$\mathbb{R} \to \mathbb{Z}.$$ Formally, for any $$x \in \mathbb{R},$$ they can be defined as

$\begin{array}{*{20}{l}} \text{floor:} & {\lfloor {x} \rfloor = \max \left\{ {n \in \mathbb{Z}:n \le x} \right\}}\\[1em] \text{ceiling:} & {\lceil {x} \rceil = \min \left\{ {n \in \mathbb{Z}:n \le x} \right\}} \end{array}$

## Graphs of the Floor and Ceiling Functions

The floor and ceiling functions look like a staircase and have a jump discontinuity at each integer point.

## Properties of the Floor and Ceiling Functions

There are many interesting and useful properties involving the floor and ceiling functions, some of which are listed below. The number $$n$$ is assumed to be an integer.

1. $$\left\lfloor x \right\rfloor = n \;\text{ iff }\; n \le x \lt n + 1$$
2. $$\left\lceil x \right\rceil = n \;\text{ iff }\; n - 1 \lt x \le n$$
3. $$\left\lfloor x \right\rfloor = n \;\text{ iff }\; x - 1 \lt n \le x$$
4. $$\left\lceil x \right\rceil = n \;\text{ iff }\; x \le n \lt x + 1$$
5. $$\left\lfloor { - x} \right\rfloor = - \left\lceil x \right\rceil$$
6. $$\left\lceil { - x} \right\rceil = - \left\lfloor x \right\rfloor$$
7. $$\left\lfloor x \right\rfloor + \left\lfloor { - x} \right\rfloor$$ $$= \left\{ {\begin{array}{*{20}{l}} 0 &{\text{ if } x \in \mathbb{Z}}\\ { - 1} &{\text{ if } x \notin \mathbb{Z}} \end{array}} \right.$$
8. $$\left\lceil x \right\rceil + \left\lceil { - x} \right\rceil$$ $$= \left\{ {\begin{array}{*{20}{l}} 0 &{\text{ if } x \in \mathbb{Z}}\\ 1 &{\text{ if } x \notin \mathbb{Z}} \end{array}} \right.$$
9. $$\left\lfloor {x + n} \right\rfloor = \left\lfloor x \right\rfloor + n$$
10. $$\left\lceil {x + n} \right\rceil = \left\lceil x \right\rceil + n$$

## Fractional Part Function

The fractional part of a number $$x \in \mathbb{R}$$ is the difference between $$x$$ and the floor of $$x:$$

$\left\{ x \right\} = x - \left\lfloor x \right\rfloor .$

For example,

$\left\{ 2 \right\} = 2 - \left\lfloor 2 \right\rfloor = 2 - 2 = 0,$
$\left\{ {3.51} \right\} = 3.51 - \left\lfloor {3.51} \right\rfloor = 3.51 - 3 = 0.51,$
$\left\{ {\frac{7}{3}} \right\} = \frac{7}{3} - \left\lfloor {\frac{7}{3}} \right\rfloor = \frac{7}{3} - 2 = \frac{1}{3},$
$\left\{ { - 5.98} \right\} = - 5.98 - \left\lfloor { - 5.98} \right\rfloor = - 5.98 - \left( { - 6} \right) = - 5.98 + 6 = 0.02$

The graph of the fractional part function looks like a sawtooth wave, with a period of $$1.$$

The range of fractional part function is the half-open interval $$\left[ {0,1} \right).$$

Some other properties of the fractional part are

1. $$\left\{ x \right\} = 0 \;\text{ iff }\; x \in \mathbb{Z}$$
2. $$\left\{ {x + n} \right\} = \left\{ x \right\}, n \in \mathbb{Z}$$
3. $$\left\{ x \right\} + \left\{ { - x} \right\}$$ $$= \left\{ {\begin{array}{*{20}{l}} 0 &{\text{if } x \in \mathbb{Z}}\\ 1 &{\text{if } x \notin \mathbb{Z}} \end{array}} \right.$$