Properties of Floor Function
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Theorem
This page gathers together some basic propeties of the floor function.
Floor is between Number and One Less
- $x - 1 < \floor x \le x$
where $\floor x$ denotes the floor of $x$.
Floor of Number plus Integer
- $\forall n \in \Z: \floor x + n = \floor {x + n}$
Real Number minus Floor
- $x - \floor x \in \hointr 0 1$
Real Number is between Floor Functions
- $\forall x \in \R: \floor x \le x < \floor {x + 1}$
Real Number is Floor plus Difference
- There exists an integer $n \in \Z$ such that for some $t \in \hointr 0 1$:
- $x = n + t$
- $n = \floor x$
Floor Function is Idempotent
- $\floor {\floor x} = \floor x$
Range of Values of Floor Function
Number less than Integer iff Floor less than Integer
- $\floor x < n \iff x < n$
Number not less than Integer iff Floor not less than Integer
- $x \ge n \iff \floor x \ge n$
Integer equals Floor iff between Number and One Less
- $\floor x = n \iff x - 1 < n \le x$
Integer equals Floor iff Number between Integer and One More
- $\floor x = n \iff n \le x < n + 1$