剰余の剰余
\[
\mod\left(\mod\left(\alpha,n\beta\right),\beta\right)=\mod\left(\alpha,\beta\right)
\]
整数を含む床関数と天井関数
\[
\left\lfloor \frac{n}{2}\right\rfloor +\left\lceil \frac{n}{2}\right\rceil =n
\]
天井関数と床関数の和と差
\[
\left\lceil z\right\rceil -\left\lfloor z\right\rfloor =\left|\sgn\mod\left(\Re z,1\right)\right|+i\left|\sgn\mod\left(\Im z,1\right)\right|
\]
剰余演算の実部と虚部
\[
\mod\left(\alpha,\beta\right)=\Re\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)-\Im\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+i\left\{ \Re\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+\Im\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)\right\}
\]
実数の複素数と複素共役の剰余演算
\[
\mod\left(1,\overline{\alpha}\right)=\overline{\mod\left(1,\alpha\right)}+i\overline{\alpha}\left|\sgn\mod\left(\Im\alpha,\left|\alpha\right|^{2}\right)\right|
\]
複素数と複素共役の実数での剰余演算
\[
\mod\left(\alpha,1\right)=\mod\left(\Re\alpha,1\right)+i\mod\left(\Im\alpha,1\right)
\]
床関数と天井関数の性質
\[
\left\lceil z\right\rceil =-\left\lfloor -z\right\rfloor
\]
床関数と天井関数の別表記
\[
\left\lfloor z\right\rfloor =z-\mod\left(z,1\right)
\]
床関数と天井関数の定義
\[
\left\lfloor x\right\rfloor =\max\left\{ n\in\mathbb{Z};n\leq x\right\}
\]
複素共役の偏角と対数
\[
\Arg\overline{z}=-\Arg z+2\pi\delta_{\pi,\Arg z}
\]