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

Also see

 * Floor of Negative equals Negative of Ceiling