Floor of Non-Integer

Theorem

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

Let $x \notin \Z$.

Then:

$\left\lfloor{x}\right\rfloor < x$

where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.

Proof

From Floor is between Number and One Less:

$\left\lfloor{x}\right\rfloor \le x$

From Real Number is Integer iff equals Floor:

$x = \left \lfloor {x} \right \rfloor \iff x \in \Z$

But we have $x \notin \Z$.

So:

$\left \lfloor {x} \right \rfloor \ne x$

and so:

$\left \lfloor {x} \right \rfloor < x$

$\blacksquare$

Also see

• Ceiling of Non-Integer