カテゴリー: 剰余演算

\[ \mod\left(\mod\left(\alpha,n\beta\right),\beta\right)=\mod\left(\alpha,\beta\right) \]

\[ \mod\left(\alpha,\beta\right)=\Re\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)-\Im\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+i\left\{ \Re\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+\Im\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)\right\} \]

\[ \mod\left(1,\overline{\alpha}\right)=\overline{\mod\left(1,\alpha\right)}+i\overline{\alpha}\left|\sgn\mod\left(\Im\alpha,\left|\alpha\right|^{2}\right)\right| \]

\[ \mod\left(\alpha,1\right)=\mod\left(\Re\alpha,1\right)+i\mod\left(\Im\alpha,1\right) \]

\[ \Arg\alpha=\mod\left(\Arg\left(\alpha\right),-2\pi,\pi\right) \]

\[ \frac{1}{\delta}\mod\left(\alpha,\beta,\gamma\right)=\mod\left(\frac{\alpha}{\delta},\frac{\beta}{\delta},\frac{\gamma}{\delta}\right) \]

\[ \alpha=\beta\left\lfloor \frac{\alpha-\gamma}{\beta}\right\rfloor +\mod\left(\alpha,\beta,\gamma\right) \]

\[ \mod\left(\alpha,\beta,\gamma\right):=\mod\left(\alpha-\gamma,\beta\right)+\gamma \]

\[ \mod\left(-x,a,b\right)=-\mod\left(x+2b,a,b\right)+a\left|\sgn\left\{ \mod\left(x+2b,a,b\right)-b\right\} \right|+2b \]

\[ \mod\left(x,a,b\right)+\mod\left(y,a,b\right)=\mod\left(x+y,a,b\right)+a\mzp_{0,1}\left(b\sgn\left(a\right),b\sgn\left(a\right)+\left|a\right|;\sgn\left(a\right)\left(\mod\left(x,a,b\right)+\mod\left(y,a,b\right)\right)\right) \]