Definition:Floor Function/Definition 2

Definition

Let $x \in \R$ be a real number.

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.

Also see

• Set of Integers Bounded Above by Real Number has Greatest Element
• Greatest Element is Unique
• Equivalence of Definitions of Floor Function

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

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $3$
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations: Example $3$ (footnote $4$)
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Example $4.8$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 10$: The well-ordering principle
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.2$: Divisibility and factorization in $\mathbf Z$: Exercise $4$
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integer part
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integer part