# 2.2: Polar Form of a Complex Number

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A point $$(x,y)$$ in the plane can be represented in polar form $$(r,\theta)$$ according to the relationships in Figure $$\PageIndex{1}$$. Figure $$\PageIndex{1}$$: Polar coordinates of a point in the plane. (Copyright; author via source)

Using these relationships, we can rewrite

\begin{align*} x+yi &= r\cos(\theta) + r\sin(\theta) i\\ &= 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

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

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

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:

$e^{i \pi} + 1 = 0\text{.}$

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 $$\arg(z$$).

##### Example $$\PageIndex{1}$$: 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 $$\tan(\alpha) =\dfrac{4}{3} \text{,}$$ so $$\theta = \pi - \tan^{-1}(\dfrac{4}{3})\approx 2.21$$ radians. Thus,

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

##### Theorem $$\PageIndex{1}$$

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

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

Proof

We use the definition of the complex exponential and some trigonometric identities. $\begin{array} d re^{i\theta}\cdot se^{i\beta} &= r(\cos\theta + i\sin\theta)\cdot s(\cos\beta+i\sin\beta)\\ &= (rs)(\cos\theta + i\sin\theta)\cdot (\cos\beta+i\sin\beta)\\ &= rs[\cos\theta\cos\beta - \sin\theta\sin\beta + (\cos\theta\sin\beta+\sin\theta\cos\beta)i]\\ &= rs[\cos(\theta+\beta) + \sin(\theta+\beta)i]\\ &=rs[e^{i(\theta+\beta)}]\text{.} \end{array}$

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

$\arg(zw) = \arg(z) + \arg(w)\text{,}$

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 $$\PageIndex{2}$$: 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 $\begin{array}d re^{i\theta} &= - |r|e^{i\theta}\\ &= (e^{i\pi})\cdot |r| e^{i\theta} \;\; \text{since} \; -1 = e^{i\pi}\\ &= |r|e^{i(\theta+\pi)}, \;\; \text{by Theorem 2.2.1} \end{array}$ Thus, by adding $$\pi$$ to the angle if necessary, we may always assume that $$z = re^{i\theta}$$ where $$r$$ is non-negative.

## Exercises

##### Exercise $$\PageIndex{1}$$

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

##### Exercise $$\PageIndex{2}$$

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 $$z\cdot u\text{.}$$

##### Exercise $$\PageIndex{3}$$

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 $$3$$.

##### Exercise $$\PageIndex{4}$$

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

This page titled 2.2: Polar Form of a Complex Number is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.