Ceiling of Negative equals Negative of Floor

Theorem

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

Let $\floor x$ be the floor of $x$, and $\ceiling x$ be the ceiling of $x$.

Then:

$\ceiling {-x} = -\floor x$

Proof

From Integer equals Floor iff between Number and One Less we have:

$x - 1 < \floor x \le x$

and so, by multiplying both sides by -1:

$-x + 1 > -\floor x \ge -x$

From Integer equals Ceiling iff between Number and One More we have:

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

Hence:

$-x \le -\floor x < -x + 1 \implies \ceiling {-x} = -\floor x$

$\blacksquare$

Also see

• Floor of Negative equals Negative of Ceiling