Rechenregeln

Addition

$\displaystyle \qquad(a + \mathrm ib) + (x + \mathrm iy) \,=\, (a + x) + \mathrm i(b + y)$

Subtraktion

$\displaystyle \qquad(a + \mathrm ib) - (x + \mathrm iy) \,=\, (a - x) + \mathrm i(b - y)$

Multiplikation

$\displaystyle \qquad(a + \mathrm ib)\cdot(x + \mathrm iy) \,=\, ax - by + \mathrm i(ay + bx)$

Division

$\displaystyle \qquad(a + \mathrm ib):(x + \mathrm iy) \,=\, \frac{ax + by}{x^2 + y^2} + \mathrm i\,\frac{bx - ay}{x^2 + y^2}\quad , \quad x^2+y^2 \, \not= \,0$

Quadratwurzel

$\displaystyle \sqrt{a + \mathrm ib} \,=\, \pm \left(\sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}} + \mathrm i\frac{b}{\sqrt{2(a + \sqrt{a^2 + b^2})}}\right)$