# Writing Complex Numbers

## The Prompt

Express the complex numbers $z_1$ and $z_2$, where $z_1=i$ in polar form.

Write $e^{-i3\pi}$ in rectangular form.

Represent both of these numbers in the complex plane.

## Context

Students have often had little exposure to Euler's Identity, particularly in a quantum mechanical context. This activity gives students the opportunity to convert complex values from polar to rectangular form. Students also have the chance to use the general form of Euler's Identity and practice using the identity for explicit examples. The answers can also be drawn in the complex plane to show how the complex number is located using each form.
This SWBQ could be included in any lecture pertaining to complex numbers and their varied representations. Furthermore, this prompt leads well into any activities that feature inner products with complex components.

## Wrap Up

Students may represent the two complex numbers with arrows of different lengths. This is a good time to remind students that the given complex numbers were both unit norm, so both arrows must be the same length. Some students never advance to representing the given complex numbers graphically. Stressing the reliability of Euler's Identity for moving between complex representations may help the students that are struggling to transition between representations of complex numbers.

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