Algebra with Complex Numbers: Exponential Form

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Section 2.8 Algebra with Complex Numbers: Exponential Form

The paragraphs below discuss how to do basic arithmetic operations for two complex numbers $z_1=r_1e^{i\phi_1}$ and $z_2 = r_2 e^{i\phi_2}\text{.}$

Addition and Subtraction.

There is nothing simple you can do to simplify the sum and difference of two complex numbers written in exponential form, other than to convert them to rectangular form.

COMING SOON: Factoring out a common phase.

Multiplication and Division.

It is easier to do multiplication and division of two complex numbers in exponential form than in rectangular form:

\begin{align} z_1 z_2 \amp = r_1 e^{i\phi_1}\, r_2 e^{i\phi_2}\notag\\ \amp = r_1 r_2 e^{i(\phi_1 + \phi_2)}\tag{2.8.1}\\ \notag\\ \frac{z_1}{ z_2} \amp = r_1 e^{i\phi_1}/ r_2 e^{i\phi_2}\notag\\ \amp = \frac{r_1}{ r_2} e^{i(\phi_1 - \phi_2)}\tag{2.8.2} \end{align}

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