Characterization of the floor

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 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.