# Definition:Floor Function

## Definition

Let $x$ be a real number.

Informally, the **floor function of $x$** is the greatest integer less than or equal to $x$.

### Definition 1

The **floor function of $x$** is defined as the supremum of the set of integers no greater than $x$:

- $\floor x := \sup \set {m \in \Z: m \le x}$

where $\le$ is the usual ordering on the real numbers.

### Definition 2

The **floor function of $x$**, denoted $\floor x$, is defined as the greatest element of the set of integers:

- $\set {m \in \Z: m \le x}$

where $\le$ is the usual ordering on the real numbers.

### Definition 3

The **floor function of $x$** is the unique integer $\floor x$ such that:

- $\floor x \le x < \floor x + 1$

## Notation

Before around $1970$, the usual symbol for the **floor function** was $\sqbrk x$.

The notation $\floor x$ for the **floor function** is a relatively recent development.

Compare the notation for the corresponding ceiling function, $\ceiling x$, which in the context of discrete mathematics is used almost as much.

Some sources use $\map {\mathrm {fl} } x$ for the **floor function** of $x$. However, this notation is clumsy, and will not be used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Examples

### Floor of $0 \cdotp 99999$

- $\floor {0 \cdotp 99999} = 0$

### Floor of $1 \cdotp 1$

- $\floor {1 \cdotp 1} = 1$

### Floor of $-1 \cdotp 1$

- $\floor {-1 \cdotp 1} = -2$

### Floor of $\sqrt 2$

- $\floor {\sqrt 2} = 1$

## Also known as

The **floor function** of a real number $x$ is usually just referred to as the **floor of $x$**.

The **floor function** is sometimes called the **entier function**, from the French for **integer**.

The **floor of $x$** is also often referred to as the **integer part** or **integral part** of $x$, particularly in older treatments of number theory.

Some sources give it as the **greatest integer function**.

## Also see

- Equivalence of Definitions of Floor Function
- Properties of Floor Function
- Definition:Ceiling Function
- Definition:Fractional Part
- Definition:Nearest Integer Function

- Results about
**the floor function**can be found here.

## Historical Note

The notation $\floor x$ for the floor function was introduced in the $1960$s by Kenneth Eugene Iverson and made popular by Donald Ervin Knuth.

## Technical Note

The $\LaTeX$ code for \(\floor {x}\) is `\floor {x}`

.

When the argument is a single character, it is usual to omit the braces:

`\floor x`

## Sources

- 1961: David V. Widder:
*Advanced Calculus*(2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: Exercise $2$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20 \ (1)$: Introduction - 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**greatest integer function** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**greatest integer function** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**greatest integer function (floor)**