---
tag: 0R9B0B5458
kind: statement
role: proposition
name: Characterization of the floor
msc: [26A09]
uses: [0YVRNEDSVG]
sources:
  - url: https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
    locator: Equivalences
---

## Statement

Let $x$ be a real number and let $n$ be an integer.
Then $n = \lfloor x \rfloor$ if and only if $n \le x < n + 1$.
In particular, $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$.

## Remarks

Thus the [[floor|0YVRNEDSVG]] of $x$ is the unique integer $n$ with $n \le x < n + 1$.
The fractional part of $x$ is $x - \lfloor x \rfloor$. It lies in the interval $[0, 1)$.
An equivalent form: $x - 1 < \lfloor x \rfloor \le x$.

## Explanations

### L1 human

The intervals $[n, n + 1)$ for integers $n$ cover the number line without overlap.
Each real number lies in exactly one of them, and the floor of the number is the left end of that interval.