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 .