# A mathematical way for defining the $\operatorname{Floor}$ and $\operatorname{Ceiling}$ functions

Given:

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

Where $$x$$ is a real number.

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

Restrictions:

• Do not use the $$\operatorname{Floor}$$ function in order to define the $$\operatorname{Ceiling}$$ function.
• Do not use the $$\operatorname{Ceiling}$$ function in order to define the $$\operatorname{Floor}$$ function.
• Do not use the $$\operatorname{Round}$$ 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...

$$\lfloor x \rfloor=\max\, \{m\in\mathbb{Z}\mid m\le x\}$$ 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).
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 $1$.
$$\lfloor x \rfloor = \max\{n \in \mathbb Z: n \le x\}$$