Комплексни бројеви 2

Текст задатака објашњених у видео лекцији.

Пр.1   Одреди реални и имагинарни део комплексног броја.

          $z = 5 + 3i$; $z = 2 - i$; $z = \sqrt 3  + i$; $z =  - 4i$; $z =  - \sqrt 7 $

Пр.2   Одреди модуо датог комплексног броја и коњуговано комплексни

           број.

           $z = 3 + 4i$         $\left| z \right| = ?$  $\overline z  = ?$

Пр.3   $z = 2 - \sqrt 5i $       $\left| z \right| = ?$  $\overline z  = ?$

Пр.4   $z =  - 4i + 1$       $\left| z \right| = ?$  $\overline z  = ?$

Пр. 1

\[\begin{gathered}
z{\text{ }} = {\text{ }}5{\text{ }} + {\text{ }}3i \hfill \\
\operatorname{Re} \left( z \right) = 5 \hfill \\
\operatorname{Im} \left( z \right) = 3 \hfill \\
\hfill \\
z{\text{ }} = {\text{ }}2{\text{ }} - {\text{ }}i \hfill \\
\operatorname{Re} \left( z \right) = 2 \hfill \\
\operatorname{Im} \left( z \right) = - 1 \hfill \\
\hfill \\
z = \sqrt 3 + i \hfill \\
\operatorname{Re} \left( z \right) = \sqrt 3 \hfill \\
\operatorname{Im} \left( z \right) = 1 \hfill \\
\hfill \\
z{\text{ }} = {\text{ }} - {\text{ }}4i \hfill \\
\operatorname{Re} \left( z \right) = 0 \hfill \\
\operatorname{Im} \left( z \right) = - 4 \hfill \\
\hfill \\
z = - \sqrt 7 \hfill \\
\operatorname{Re} \left( z \right) = - \sqrt 7 \hfill \\
\operatorname{Im} \left( z \right) = 0 \hfill \\
\end{gathered} \]

Пр. 2

\[\begin{gathered}
z{\text{ }} = {\text{ }}3{\text{ }} + {\text{ }}4i \hfill \\
\left| z \right| = \sqrt {{3^2} + {4^2}} \hfill \\
\left| z \right| = \sqrt {9 + 16} \hfill \\
\left| z \right| = \sqrt {25} \hfill \\
\left| z \right| = 5 \hfill \\
\bar z = 3 - 4i \hfill \\
\end{gathered} \]

Пр. 3

\[\begin{gathered}
z = 2 - \sqrt 5 i \hfill \\
\left| z \right| = \sqrt {{2^2} + {{\left( { - \sqrt 5 } \right)}^2}} \hfill \\
\left| z \right| = \sqrt {4 + 5} \hfill \\
\left| z \right| = \sqrt 9 \hfill \\
\left| z \right| = 3 \hfill \\
\bar z = 2 + \sqrt 5 i \hfill \\
\end{gathered} \]

Пр. 4

\[\begin{gathered}
z{\text{ }} = {\text{ }} - {\text{ }}4i{\text{ }} + {\text{ }}1 \hfill \\
\left| z \right| = \sqrt {{1^2} + {{\left( { - 4} \right)}^2}} \hfill \\
\left| z \right| = \sqrt {1 + 16} \hfill \\
\left| z \right| = \sqrt {17} \hfill \\
\bar z = 4i + 1 \hfill \\
\end{gathered} \]