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Section2.2Polar Form of a Complex Number

A point \((x,y)\) in the plane can be represented in polar form \((r,\theta)\) according to the relationships in Figure 2.2.1.

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Figure2.2.1Polar coordinates of a point in the plane

Using these relationships, we can rewrite \begin{align*} x+yi \amp = r\cos(\theta) + r\sin(\theta) i\\ \amp = r(\cos(\theta) + i \sin(\theta)). \end{align*}

Definition2.2.2

For any real number \(\theta\text{,}\) we define \begin{equation*} e^{i\theta} = \cos(\theta) + i\sin(\theta). \end{equation*}

For instance, \(e^{i\pi/2} = \cos(\pi/2) + i\sin(\pi/2) = 0 + i\cdot 1 = i.\)

Similarly, \(e^{i0} = \cos(0) + i\sin(0) = 1\text{,}\) and it's a quick check to see that \(e^{i\pi} = -1\text{,}\) which leads to a simple equation involving the most famous numbers in mathematics (except 8, see [29]), truly an all-star equation: \(e^{i \pi} + 1 = 0\text{.}\)

If \(z = x+yi\) and \((x,y)\) has polar form \((r,\theta)\) then \(z = re^{i\theta}\) is called the polar form of \(z\text{.}\) The non-negative scalar \(|r|\) is the modulus of \(z\text{,}\) and the angle \(\theta\) is called the argument of \(z\), denoted \(\arg(z\)).

Example2.2.3Exploring the polar form

On the left side of Figure 2.2.4, we plot the points \(z = 2e^{i\pi/4}, w = 3e^{i\pi/2}, v = -2e^{i\pi/6}, u = 3e^{-i\pi}.\)

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Figure2.2.4The polar form of a complex number.

To convert \(z = -3 + 4i\) to polar form, refer to the right side of Figure 2.2.4. We note that \(r = \sqrt{9 + 16} = 5\text{,}\) and \(\tan(\alpha) = 4/3\text{,}\) so \(\theta = \pi - \tan^{-1}(4/3)\approx 2.21\) radians. Thus, \begin{equation*} -3+4i = 5e^{i(\pi-\tan^{-1}(4/3))} \approx 5e^{2.21i}. \end{equation*}

Thus, the product of two complex numbers is obtained by multiplying their magnitudes and adding their arguments, and \begin{equation*} \arg(zw) = \arg(z) + \arg(w), \end{equation*} where the equation is taken modulo \(2\pi\text{.}\) That is, depending on our choices for the arguments, we may have \(\arg(vw) = \arg(v)+ \arg(w) + 2\pi k\) for some integer \(k\text{.}\)

Example2.2.6Expressing \(z\) in polar form with \(r\geq 0\)

When representing a complex number \(z\) in polar form as \(z = re^{i\theta}\text{,}\) we may assume that \(r\) is non-negative. If \(r \lt 0\text{,}\) then \begin{align*} re^{i\theta} \amp = - |r|e^{i\theta}\\ \amp = (e^{i\pi})\cdot |r| e^{i\theta} ~~\text{since}~ -1 = e^{i\pi}\\ \amp = |r|e^{i(\theta+\pi)}, ~~\text{by}~{\knowl{./knowl/th_polarmult.html}{\text{Theorem 2.2.5}}}. \end{align*}

Thus, by adding \(\pi\) to the angle if necessary, we may always assume that \(z = re^{i\theta}\) where \(r\) is non-negative.

SubsectionExercises

1

Convert the following points to polar form and plot them: \(3 + i\text{,}\) \(-1 - 2i\text{,}\) \(3 - 4i\text{,}\) \(7,002,001\text{,}\) and \(-4i\text{.}\)

2

Express the following points in Cartesian form, and plot them: \(z = 2e^{i\pi/3}, w = -2e^{i\pi/4}\text{,}\) \(u = 4e^{i5\pi/3},\) and \(z\cdot u\text{.}\)

3

Modify the all-star equation to involve 8. In particular, write an expression involving \(e, i, \pi, 1,\) and 8, that equals 0. You may use no other numbers, and certainly not 3.

4

If \(z = re^{i\theta}\text{,}\) prove that \(\overline{z} = re^{-i\theta}\text{.}\)