Definition:Floor Function

From ProofWiki
Jump to: navigation, search

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 $\left[{x}\right]$.

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 $\operatorname {fl} \paren x$ 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.


Also see

  • 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