z⁻²⁰¹¹ gdzie $z = \frac{1 + i}{\sqrt{}2}$

z = a + bi $\left| z \right| = \sqrt{}a^{2} + b^{2}$ $\left| z \right| = \sqrt{{(\frac{1}{\sqrt{}2})}^{2} + {(\frac{i}{\sqrt{}2})}^{2}} = \sqrt{\frac{1}{2} + \frac{1}{2}} = 1$

$cos \propto \ = \ \frac{a^{2}}{\left| z \right|}$ $sin \propto \ = \frac{b^{2}}{|z|}$

$cos \propto \ = \ \frac{\sqrt{}2}{2}$ $sin \propto \ = \ \frac{\sqrt{}2}{2}$

$\propto \ = \ \frac{\pi}{4}$ z = |z|(cos ∝ +sin ∝ ) $z = 1\left( \cos\frac{\pi}{4} + sin\frac{\pi}{4} \right)$

zⁿ = |z|ⁿ(cosn∝+sinn∝) $z^{- 2011} = {|z|}^{- 2011}\left( \cos{\lbrack - 2011\frac{\pi}{4}\rbrack} + \operatorname{sin\lbrack}{- 2011}\frac{\pi}{4}\rbrack \right)$

$\cos{\frac{- 2011\pi}{4} = \cos\frac{- 3\pi}{4}}$ $\sin\frac{- 2011\pi}{4} = \sin\frac{- 3\pi}{4}$

$$z^{- 2011} = 1^{- 2011}\left( - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \right) = - \sqrt{}2$$

z² − 6z + 13 = 0 =36 − 52 = −16 $\sqrt{} = \sqrt{- 16} = i\sqrt{16} = 4i$

$z_{1} = \frac{- 6 - 4i}{2} = - 3 - 2i$ z₂ = −3 + 2i