# A mathematical way for defining the Floor and Ceiling functions

Given:

• $Floor(x)=\lfloor x \rfloor$
• $Ceiling(x)=\lceil x \rceil$

Where $x$ is a real number.

Is there any other (mathematical) way for defining $Floor(x)$ and $Ceiling(x)$?

Restrictions:

• Do not use the Floor function in order to define the Ceiling function.
• Do not use the Ceiling function in order to define the Floor function.
• Do not use the Round function in order to define either one of them.

Please excuse the possible duplicate, as I haven't been able to do find this question anywhere...

Thanks

## 3 Answers

Use $\bmod$ (not necessarily $\diamond$ mods) $$\lfloor x \rfloor = x - \bmod(x,1)$$ To get $\bmod(x,1)$, use $\frac12\Biggr(\frac{\log\left(\exp\left(2\pi i(x- \frac12)\right)\right)}{\pi i}+1\Biggr)$, since $\exp\left(2\pi i(x- \frac12)\right)$ has a period of $1$.

• And how do you define $mod$? Apr 2, 2014 at 15:29
• @barakmanos what about $\log(\exp(i2\pi x)/(i2\pi))$? Apr 2, 2014 at 15:38
• I don't know. Can you please elaborate on this in your answer? It actually looks very interesting, though I'd like to be sure that it gives the correct answer (and whether it is $Floor$ or $Ceiling$). Thanks Apr 2, 2014 at 15:45
• So the result is a complex number... Do I take the $Real$ part then? Apr 2, 2014 at 15:55
• Here is the graph: wolframalpha.com/share/… Apr 3, 2014 at 9:32

Floor and ceiling functions are usually defined as

$$\lfloor x \rfloor=\max\, \{m\in\mathbb{Z}\mid m\le x\}$$ and $$\lceil x \rceil=\min\,\{n\in\mathbb{Z}\mid n\ge x\}$$ for $x\in\mathbb R$ (see Floor and ceiling functions for more details).

$$\lfloor x \rfloor = \max\{n \in \mathbb Z: n \le x\}$$