How do the floor and ceiling functions work on negative numbers? It's clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take $\,-0.8\,$ to $\,-1,\,$ then do you take the floor of $\,-0.8,\,$ or the ceiling?
That is, which of the following are true?
$$\lfloor-0.8\rfloor=-1$$
$$\text{or}$$
$$\lceil-0.8\rceil=-1$$

When $x\in \mathbb{R}\setminus \mathbb{Z}$ ($x$ is not an integer):
$\lfloor-x\rfloor=  -\lfloor x\rfloor - 1.$
When $x \in \mathbb{R}$:
$\lceil-x\rceil= -\lfloor x\rfloor.$
