Definition:Ceiling Function

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 1960's by Kenneth Iverson and popularised by Knuth.

Compare the notation for the floor function.