Properties of Ceiling Function
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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
- $\ceiling x = n \iff n - 1 < x \le n$