Polar form

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Section 7.2 Polar form

Another useful representation of a complex number \(z=x+iy\) is polar form. In polar form we represent a complex number by specifying an angle \(\theta\) from the \(x\) axis, and a distance from the origin which, recall, is \(|z|\text{.}\)

Definition 7.2.1.

Let \(z\neq 0\) be a complex number. Let \(r = |z|\) and define the argument of \(z\) be the angle \(\theta\in [0,2\pi)\) between the line from \(0\) to \(z\) and the positive \(x\)-axis. We can then write

\begin{equation*} z= r(\cos\theta+i\sin\theta)\text{.} \end{equation*}

This is the polar form of \(z\text{.}\)

While we specify that the argument \(\theta\) of a complex number \(z\) to be in the range \([0,2\pi)\text{,}\) there are infinitely many numbers \(\phi\) so that \(z= r(\cos\phi+i\sin\phi)\) (where \(r=|z|\)): we can just take \(\phi = \theta+ 2n\pi\) for \(n\in\mathbb{Z}\text{.}\) We have made a choice to only take \(\theta\in [0,2\pi)\text{.}\) This is called the principal argument. Other books and resources sometimes choose \(\theta\in (-\pi,\pi]\text{.}\) Mathematicians are free to “cut” the complex plane where we choose, but be clear about which choice you make and stick to it.