床関数と天井関数の性質
床関数と天井関数の性質
\(\left\lceil z\right\rceil \)は天井関数
(1)
\[ \left\lceil x\right\rceil -1\leq\left\lfloor x\right\rfloor \leq x\leq\left\lceil x\right\rceil \leq\left\lfloor x\right\rfloor +1 \](2)
\[ \left\lceil z\right\rceil =-\left\lfloor -z\right\rfloor \](3)
\[ \left\lfloor z\right\rfloor =-\left\lceil -z\right\rceil \](4)
\[ \left\lfloor \left\lfloor z\right\rfloor \right\rfloor =\left\lfloor z\right\rfloor \](5)
\[ \left\lceil \left\lceil z\right\rceil \right\rceil =\left\lceil z\right\rceil \]-
\(\left\lfloor z\right\rfloor \)は床関数\(\left\lceil z\right\rceil \)は天井関数
(1)
\begin{align*} \left\lceil x\right\rceil -1 & =x-\mod\left(x,-1\right)-1\\ & =x+\mod\left(-x,1\right)-1\\ & =x+\left|\sgn\mod\left(x,1\right)\right|-\mod\left(x,1\right)-1\\ & =x-\mod\left(x,1\right)-\delta_{0,\mod\left(x,1\right)}\\ & \leq x-\mod\left(x,1\right)\\ & =\left\lfloor x\right\rfloor \\ & \leq x\\ & \leq x-\mod\left(x,-1\right)\\ & =\left\lceil x\right\rceil \\ & =x+\mod\left(-x,1\right)\\ & =x-\mod\left(x,1\right)+\left|\sgn\mod\left(x,1\right)\right|\\ & \leq x-\mod\left(x,1\right)+1\\ & =\left\lfloor x\right\rfloor +1 \end{align*}(2)
\begin{align*} -\left\lfloor -z\right\rfloor & =-\left(-z-\mod\left(-z,1\right)\right)\\ & =z+\mod\left(-z,1\right)\\ & =z-\mod\left(z,-1\right)\\ & =\left\lceil z\right\rceil \end{align*}(3)
\begin{align*} -\left\lceil -z\right\rceil & =-\left(-z-\mod\left(-z,-1\right)\right)\\ & =z+\mod\left(-z,-1\right)\\ & =z-\mod\left(z,1\right)\\ & =\left\lfloor z\right\rfloor \end{align*}(4)
\begin{align*} \left\lfloor \left\lfloor z\right\rfloor \right\rfloor & =\left\lfloor \left\lfloor z\right\rfloor +\mod\left(z,1\right)\right\rfloor \\ & =\left\lfloor z\right\rfloor \end{align*}(5)
\begin{align*} \left\lceil \left\lceil z\right\rceil \right\rceil & =\left\lceil \left\lceil z\right\rceil +\mod\left(z,-1\right)\right\rceil \\ & =\left\lceil z\right\rceil \end{align*}ページ情報
タイトル | 床関数と天井関数の性質 |
URL | https://www.nomuramath.com/i1ds7rwg/ |
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天井関数と床関数の和と差
\[
\left\lceil z\right\rceil -\left\lfloor z\right\rfloor =\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right)
\]
床関数と天井関数の別表記
\[
\left\lfloor z\right\rfloor =z-\mod\left(z,1\right)
\]
整数を含む床関数と天井関数
\[
\left\lfloor \frac{n}{2}\right\rfloor +\left\lceil \frac{n}{2}\right\rceil =n
\]
床関数と天井関数の定義
\[
\left\lfloor x\right\rfloor =\max\left\{ n\in\mathbb{Z};n\leq x\right\}
\]