Numere complexe

      $\mathbb{C}$={ z = x + iy | x,y$\in$$\mathbb{R}$}, unde ${{i}^{2}}$= -1, este multimea numerelor complexe.

  Daca z = x + iy, unde x,y$\in$$\mathbb{R}$, numerele reale x si y se numesc partea reala si respective partea imaginara a numarului complex z; notam x = Rez , y = Imz.

   Elementele multimii i${{\mathbb{R}}^{*}}$= {iy | y$\in$$\mathbb{R}$\{0}} se numesc numere pur imaginare.

      Modulul unui numar complex z = x+ iy este numarul real |z| =$\sqrt{{{x}^{^{2}}}+{{y}^{2}}}$.    

Proprietati: 1. |z|$\ge$0, $\forall$z$\in$$\mathbb{C}$; |z|=0 $\Leftrightarrow$z = 0.

                      2. |${{z}_{1}}$$\centerdot$${{z}_{2}}$| = |${{z}_{1}}$| $\centerdot$|${{z}_{2}}$|, $\forall$${{z}_{1}}$,${{z}_{2}}$$\in$$\mathbb{C}$.

                      3. |${{z}_{1}}$+${{z}_{2}}$| $\le$ |${{z}_{1}}$|+|${{z}_{2}}$|, $\forall$${{z}_{1}}$,${{z}_{2}}$$\in$$\mathbb{C}$.

    Conjugatul unui numar complex z = x+iy este numarul complex $\overline{z}$= x – iy.

Proprietati: 1. $\overline{{{z}_{_{1}}}+{{z}_{2}}}$ = $\overline{{{z}_{1}}}$+$\overline{{{z}_{2}}}$, $\forall$${{z}_{1}}$,${{z}_{2}}$$\in$$\mathbb{C}$;

                   2. $\overline{{{z}_{1}}\centerdot {{z}_{2}}}$ = $\overline{{{z}_{1}}}\centerdot \overline{{{z}_{2}}}$, $\forall$${{z}_{1}}$,${{z}_{2}}$$\in$$\mathbb{C}$;

                   3. |z| = |$\overline{z}$|, $\forall$${{z}_{1}}$,${{z}_{2}}$$\in$$\mathbb{C}$;

                     4. z$\centerdot$$\overline{z}$ = |z${{|}^{2}}$, $\forall$z$\in$$\mathbb{C}$.

Observatii: 1. z$\in$$\mathbb{R}$ daca si numai daca $\overline{z}$ = z ;

                    2. z$\in$i${{\mathbb{R}}^{*}}$daca si numai daca $\overline{z}$ = -z .