i2 = −1 i3 = −i i4 = 1 i5 = i
z1 ± z2 = (x1±x2) + i(y1±y2)
z1 • z2 = (x1•x2−y1•y2) + i(x1 • y2 + x2 • y1)
$$\frac{z_{1}}{z_{2}}\mathbf{=}\frac{\mathbf{(}x_{1} \bullet x_{2} + y_{1} \bullet y_{2})}{\mathbf{x}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathbf{y}_{\mathbf{2}}^{\mathbf{2}}}\mathbf{+ i}\frac{\mathbf{(}x_{2} \bullet y_{1} - x_{1} \bullet y_{2})}{\mathbf{x}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathbf{y}_{\mathbf{2}}^{\mathbf{2}}}$$
$$\sqrt{\mathbf{x + iy}}\mathbf{= \pm}\left( \sqrt{\frac{\mathbf{x +}\sqrt{\mathbf{x}^{\mathbf{2}}\mathbf{+}\mathbf{y}^{\mathbf{2}}}}{\mathbf{2}}}\mathbf{+ isgny}\sqrt{\frac{\mathbf{- x +}\sqrt{\mathbf{x}^{\mathbf{2}}\mathbf{+}\mathbf{y}^{\mathbf{2}}}}{\mathbf{2}}} \right)$$
$$\mathbf{z}_{\mathbf{k}}\mathbf{=}\sqrt[\mathbf{n}]{\mathbf{r}}\left( \mathbf{\cos}\frac{\mathbf{\varphi + 2}\mathbf{\text{kπ}}}{\mathbf{n}}\mathbf{+ isin}\frac{\mathbf{\varphi + 2}\mathbf{\text{kπ}}}{\mathbf{n}} \right)$$