整数を含む床関数と天井関数
\[
\left\lfloor \frac{n}{2}\right\rfloor +\left\lceil \frac{n}{2}\right\rceil =n
\]
三角関数・双曲線関数の実部と虚部
\[
\sin z=\sin\left(\Re z\right)\cosh\left(\Im z\right)+i\cos\left(\Re z\right)\sinh\left(\Im z\right)
\]
剰余演算の実部と虚部
\[
\mod\left(\alpha,\beta\right)=\Re\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)-\Im\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+i\left\{ \Re\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+\Im\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)\right\}
\]
複素数と複素共役の実数での剰余演算
\[
\mod\left(\alpha,1\right)=\mod\left(\Re\alpha,1\right)+i\mod\left(\Im\alpha,1\right)
\]