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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

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3 Answers

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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 $1$.

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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).

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

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