Floor of Negative equals Negative of Ceiling

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

Proof
From Integer equals Ceiling iff between Number and One More we have:
 * $x \le \ceiling x < x + 1$

and so, by multiplying by -1:
 * $-x \ge -\ceiling x > -x - 1$

From Integer equals Floor iff between Number and One Less we have:
 * $\floor x = n \iff x - 1 < n \le x$

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

Also see

 * Ceiling of Negative equals Negative of Floor