Definition:Floor Function

Definition

Let $x \in \R$. Then $\left \lfloor {x} \right \rfloor$ is defined as:

$\left \lfloor {\cdot} \right \rfloor: \R \to \Z: \left \lfloor {x} \right \rfloor = \sup \left({\left\{{m \in \Z: m \le x}\right\}}\right)$

That is, $\left \lfloor {x} \right \rfloor$ is the greatest integer less than or equal to $x$.

Alternatively it can be defined as:

$z = \left \lfloor {x} \right \rfloor \iff z \in \Z \land x \in \left\{{y \in \R: z \le y < z+1}\right\}$

It immediately follows that:

• $\left \lfloor {x} \right \rfloor$ is an integer;
• $x - 1 < \left \lfloor {x} \right \rfloor \le x < \left \lfloor {x} \right \rfloor + 1$;
• $\forall n \in \Z: \left \lfloor {x} \right \rfloor + n = \left \lfloor {x + n} \right \rfloor$.

This is called the floor function.

Notation

The notation given here was introduced in the 1960s by Kenneth Iverson and popularised by Donald E. Knuth.

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

The older and more usual symbol for this is $\left[{x}\right]$, and it is often referred to as the integer part of $x$, but there are already enough uses to which square brackets are put.

This function is sometimes called the entier function, from the French for integer.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): Exercise $2.3$
• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.4$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$: Example $4.8$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 10, \ \S 10.4 \ \text{(i)}$