# 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?

$$\lfloor-0.8\rfloor=-1$$

$$\text{or}$$ $$\lceil-0.8\rceil=-1$$

• Look at the number line - Floor: Go to the next integer left of where you are. - Ceiling: Go to the next integer right of where you are. Commented Mar 28, 2013 at 15:26
• Lower limit: Lower than or equal to it. Upper bound: More than or equal to it. So, First one is right.! Commented Mar 28, 2013 at 15:26
• @K.Stm. +1 beat me to it :) Commented Mar 28, 2013 at 15:27
• To almost any mathematician, it is obvious: $\lfloor -1.8\rfloor=-2$. I remember however bumping into a computer implementation that gave $\lfloor -1.8\rfloor=-1$. Commented Mar 28, 2013 at 15:44

The first is the correct: you round "down" (i.e. the greatest integer LESS THAN OR EQUAL TO $$-0.8$$).

In contrast, the ceiling function rounds "up" to the least integer GREATER THAN OR EQUAL TO $$-0.8 = 0$$.

\begin{align} \lfloor{-0.8}\rfloor & = -1\quad & \text{since}\;\; \color{blue}{\bf -1} \le -0.8 \le 0 \\ \\ \lceil {-0.8} \rceil & = 0\quad &\text{since} \;\; -1 \le -0.8 \le \color{blue}{\bf 0} \end{align}

In general, we must have that $$\lfloor x \rfloor \leq x\leq \lceil x \rceil\quad \forall x \in \mathbb R$$

And so it follows that $$-1 = \lfloor -0.8 \rfloor \leq -0.8 \leq \lceil -0.8 \rceil = 0$$

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 $$x$$.

• With a vertical number line you even have "above" for ceiling and "below" for floor... Commented Aug 27, 2017 at 11:31
• This is really by convention of definition not intuition. Pivoting towards or away from zero also has its intuition and places where it is useful. floor(x)={-floor(-x) if x<0, floor(x) otherwise. Commented Oct 14, 2021 at 15:21

Keep in mind that $\lfloor x \rfloor \le x \le \lceil x \rceil$, so it's the first one.

More precisely, for $x \in \Bbb{R}$, $$\lfloor x \rfloor = \max \{ z \in \Bbb{Z} : z \le x \}$$ while $$\lceil x \rceil = \min \{ z \in \Bbb{Z} : x \le z \}.$$

• Again this does not have to be so. By convention or common definition. Other definitions define it as rounding towards or away from 0 Commented Oct 14, 2021 at 15:27
• @GregoryMorse I see your point, but in that case you aren't really interested in the true floor or ceil of the value itself. Instead, you are then interested in the floor or ceil of the deviation of the value from zero, i.e. it's absolute value. So, you can define such an alternative as the floor or ceil of the absolute value of the given number, with the sign tacked on afterwards. Commented Jan 21 at 0:02

The first one is true: $\lfloor x\rfloor\le x\le\lceil x\rceil$, no matter if $x$ is negative or not.

• According to your definition. It could also be rounding towards or away from zero. Commented Oct 14, 2021 at 15:26

When $$x\in \mathbb{R}\setminus \mathbb{Z}$$ ($$x$$ is not an integer):

$$\lfloor-x\rfloor= -\lfloor x\rfloor - 1.$$

When $$x \in \mathbb{R}$$:

$$\lceil-x\rceil= -\lfloor x\rfloor.$$

• Don't you mean $\lfloor -x \rfloor=-\lceil x\rceil$? For example, what happens when $x$ is an integer, say $3$? Commented Oct 11, 2015 at 18:16

The correct answer is it depends how you define floor and ceil. You could define as shown here the more common way with always rounding downward or upward on the number line.

OR

Floor always rounding towards zero. Ceiling always rounding away from zero. E.g floor(x)=-floor(-x) if x<0, floor(x) otherwise

If gravity were reversed, the ceiling would become the floor. So from a physics standpoint the standard mathematical definition might be inadequate.