# 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

- $x - 1 < \left\lfloor{x}\right\rfloor \le x$

where $\left\lfloor{x}\right\rfloor$ is 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

- $\left \lfloor{x}\right \rfloor < n \iff x < n$

### Number not less than Integer iff Floor not less than Integer

- $x \ge n \iff \left \lfloor{x}\right \rfloor \ge n$

### 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

- $\left \lfloor{x}\right \rfloor = n \iff n \le x < n + 1$