GCT Polar Form of a Complex Number

🔗

Section 2.2 Polar 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.

🔗

Figure 2.2.1. Polar coordinates of a point in the plane

🔗

Using these relationships, we can rewrite

$\begin{align*} x+yi \amp = r\cos(\theta) + r\sin(\theta) i\\ \amp = r(\cos(\theta) + i \sin(\theta))\text{.} \end{align*}$

🔗

This leads us to make the following definition. For any real number

$\theta\text{,}$ we define

$\begin{equation*} e^{i\theta} = \cos(\theta) + i\sin(\theta)\text{.} \end{equation*}$

🔗

For instance,

$e^{i\pi/2} = \cos(\pi/2) + i\sin(\pi/2) = 0 + i\cdot 1 = i\text{.}$

🔗

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), truly an all-star equation:

$\begin{equation*} e^{i \pi} + 1 = 0\text{.} \end{equation*}$

🔗

$z = x+yi$ 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

$\arg(z$ ).

🔗

Example 2.2.2. Exploring the polar form.

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

To convert $z = -3 + 4i$ to polar form, refer to the right side of the diagram. 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,

$\begin{equation*} -3+4i = 5e^{i(\pi-\tan^{-1}(4/3))} \approx 5e^{2.21i}\text{.} \end{equation*}$

🔗

Theorem 2.2.3.

The product of two complex numbers in polar form is given by

$\begin{equation*} \displaystyle re^{i\theta}\cdot se^{i\beta} = (rs)e^{i(\theta+\beta)}\text{.} \end{equation*}$

🔗

Proof.

We use the definition of the complex exponential and some trigonometric identities.

\begin{align*} re^{i\theta}\cdot se^{i\beta} \amp = r(\cos\theta + i\sin\theta)\cdot s(\cos\beta+i\sin\beta)\\ \amp = (rs)(\cos\theta + i\sin\theta)\cdot (\cos\beta+i\sin\beta)\\ \amp = rs[\cos\theta\cos\beta - \sin\theta\sin\beta + (\cos\theta\sin\beta+\sin\theta\cos\beta)i]\\ \amp = rs[\cos(\theta+\beta) + \sin(\theta+\beta)i]\\ \amp =rs[e^{i(\theta+\beta)}]\text{.} \end{align*}

🔗

Thus, the product of two complex numbers is obtained by multiplying their magnitudes and adding their arguments, and

$\begin{equation*} \arg(zw) = \arg(z) + \arg(w)\text{,} \end{equation*}$

🔗

where the equation is taken modulo

$2\pi\text{.}$ That is, depending on our choices for the arguments, we have

$\arg(vw) = \arg(v)+ \arg(w) + 2\pi k$ for some integer

$k\text{.}$

🔗

Example 2.2.4. 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

$\begin{align*} re^{i\theta} \amp = - |r|e^{i\theta}\\ \amp = (e^{i\pi})\cdot |r| e^{i\theta} ~~\text{since}~ -1 = e^{i\pi}\\ \amp = |r|e^{i(\theta+\pi)}, ~~\text{by}~{\knowl{./knowl/th_polarmult.html}{\text{Theorem 2.2.3}}}\text{.} \end{align*}$

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

🔗

Exercises Exercises

🔗

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: $z = 2e^{i\pi/3}\text{,}$ $w = -2e^{i\pi/4}\text{,}$ $u = 4e^{i5\pi/3},$ 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 $\overline{z} = re^{-i\theta}\text{.}$