# 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?

The first is the correct: you round “down” (i.e. the greatest integer LESS THAN $-0.8$).

In contrast, the ceiling function rounds “up” to the least integer GREATER THAN $-0.8 = 0$.

In general, we must have that

And so it follows that

K.Stm’s suggestion is a nice, intuitive way to recall the relation between the floor and the ceiling of a real number $x$, especially when $x\lt 0$. Using the “number line” idea and plotting $-0.8$ with the two closest integers that “sandwich” $-0.8$ gives us:

$\qquad\qquad$ We see that the floor of $x= -0.8$ is the first integer immediately to the left of $-0.8,\;$ and the ceiling of $x= -0.8$ is the first integer immediately to the right of $-0.8$, and this strategy can be used, whatever the value of a real number $x$.