Properties of Ceiling Function

Theorem

This page gathers together some basic propeties of the ceiling function.

Number is between Ceiling and One Less

$\ceiling x - 1 < x \le \ceiling x$

Ceiling is between Number and One More

$x \le \ceiling x < x + 1$

where $\ceiling x$ denotes the ceiling of $x$.

Ceiling of Number plus Integer

$\forall n \in \Z: \ceiling x + n = \ceiling {x + n}$

Ceiling minus Real Number

$\forall x \in \R: \ceiling x - x \in \hointr 0 1$

Real Number is between Ceiling Functions

$\forall x \in \R: \ceiling {x - 1} \le x < \ceiling x$

Real Number is Ceiling minus Difference

Let $n$ be a integer.

The following statements are equivalent:

$(1): \quad$ There exists $t \in \hointr 0 1$ such that $x = n - t$

$(2): \quad n = \ceiling x$

Ceiling Function is Idempotent

$\ceiling {\ceiling x} = \ceiling x$

Range of Values of Ceiling Function

Number greater than Integer iff Ceiling greater than Integer

$\ceiling x > n \iff x > n$

Number not greater than Integer iff Ceiling not greater than Integer

$\ceiling x \le n \iff x \le n$

Integer equals Ceiling iff between Number and One More

$\ceiling x = n \iff x \le n < x + 1$

Integer equals Ceiling iff Number between Integer and One Less

$\ceiling x = n \iff n - 1 < x \le n$

Also see

• Properties of Floor Function