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Contents Index Dark Mode Prev Up Next \(\def\versionnumber{1.54}
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Worksheet 1.6 Optional Lab
1.
Convert the following complex numbers into their polar representation, i.e., give the absolute value and the argument of the number.
\begin{align*}
34 \amp = \fillinmath{XXX}\amp i \amp = \fillinmath{XXX}\amp -\pi \amp = \fillinmath{XXX}\amp 2+2i \amp = \fillinmath{XXX}\amp - \frac 1 2 \left( \sqrt 3 + i \right) \amp =\fillinmath{XXX}
\end{align*}
After you have finished computing these numbers, check your answers with the open-source software platform
Geogebra. You might search for
geogebra complex function grapher .
2.
Convert the following complex numbers given in polar representation into their rectangular representation.
\begin{align*}
2 \, e^{ i 0 } \amp = \fillinmath{XXX} \amp 3 \, e^{ \frac{ \pi i} 2 } \amp =\fillinmath{XXX} \amp \tfrac{1}{2} \, e^{ i \pi } \amp = \fillinmath{XXX}\amp e^{ - \frac{ 3 \pi i }{ 2 } } \amp =\fillinmath{XXX} \amp 2 \, e^{ \frac{ 3 \pi i }{ 2 } } \amp =\fillinmath{XXX}
\end{align*}
After you have finished computing these numbers, check your answers with Geogebra.
3. Pick your favorite five numbers
\(z_1\text{,}\) \(z_2\text{,}\) \(z_3\text{,}\) \(z_4\text{,}\) and
\(z_5\) from the ones that you’ve played around with and put them in the table below, in both rectangular and polar form. Apply the functions listed to your numbers. Think about which representation is more helpful in each instance.
rectangular
polar
\(z+1\)
\(z+2-i\)
\(2z\)
\(-z\)
\(\frac z 2\)
\(iz\)
\(\overline z\)
\(z^2\)
\(\Re(z)\)
\(\Im(z)\)
\(i \Im(z)\)
\(|z|\)
\(\frac 1 z\)
4. Play with other examples until you get a feel for these functions.
