Definition:Ceiling Function

Definition

Let $x \in \R$.

Then $\left \lceil {x} \right \rceil$ is defined as:

$\left \lceil {\cdot} \right \rceil: \R \to \Z: \left \lceil {x} \right \rceil = \inf \left({\left\{{m \in \Z: m \ge x}\right\}}\right)$

That is, $\left \lceil {x} \right \rceil$ is the smallest integer greater than or equal to $x$.

It immediately follows that:

• $\left \lceil {x} \right \rceil$ is an integer
• $\left \lceil {x} \right \rceil - 1 < x \le \left \lceil {x} \right \rceil < x + 1$
• $\forall n \in \Z: \left \lceil {x + n} \right \rceil = \left \lceil {x} \right \rceil + n$

This is called the ceiling 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 floor function.

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.4$