Proposition: Properties of Floors and Ceilings

The floor and ceiling functions have the following properties:

  1. $\lfloor x \rfloor\le x$ and $\lceil x \rceil \ge x$ for all real numbers $x\in\mathbb R.$
  2. $\lfloor x \rfloor = x$ and $\lceil x \rceil = x$ if and only if $x\in\mathbb Z$ is an integer.
  3. If $x\not\in\mathbb Z$, then $\lceil x \rceil-\lfloor x \rfloor=1,$ otherwise $\lceil x \rceil-\lfloor x \rfloor=0.$
  4. For every integer $n\in\mathbb Z$, $\lfloor x+n \rfloor = \lfloor x \rfloor+n.$
  5. In general, $\lfloor nx\rfloor\neq n\lfloor x\rfloor.$
  6. Redundant floor and ceilings brackets: $(a)$ $x < n\Leftrightarrow \lfloor x\rfloor < n,$ $(b)$ $n < x\Leftrightarrow n < \lceil x\rceil,$ $(c)$ $x \le n\Leftrightarrow \lceil x\rceil\le n,$ $(d)$ $n \le x\Leftrightarrow n\le \lfloor x\rfloor.$
  7. Redundant floor and ceilings brackets: $(a)$ $x < n\Leftrightarrow \lfloor x\rfloor < n,$ $(b)$ $n < x\Leftrightarrow n < \lceil x\rceil,$ $(c)$ $x \le n\Leftrightarrow \lceil x\rceil\le n,$ $(d)$ $n \le x\Leftrightarrow n\le \lfloor x\rfloor.$

Proofs: 1

Theorems: 1


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References

Bibliography

  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
  2. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927