Range of Values of Floor Function

Theorem

Let $x \in \R$ be a real number.

Let $\floor x$ denote the floor of $x$.

Let $n \in \Z$ be an integer.

Then the following results apply:

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

$\floor x = n \iff n \le x < n + 1$

Also see

• Range of Values of Ceiling Function