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 $2$
- 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(i)}$ Periodic Functions
- 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): greatest integer function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): greatest integer function
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): greatest integer function (floor)