# Polar form and argument properties

## Polar form of a complex number

You can express every complex number in terms of its modulus and argument. Taking a complex number $$z = x+yi$$, we let $$r$$ be the modulus of $$z$$ and $$\theta$$ be an argument of $$z$$.

Using trigonometry, we find that $$x = r \cos \theta$$ and $$y = r \sin \theta$$. Thus we can write $$z = r \cos \theta + ir \sin \theta = r( \cos{\theta} + i \sin{\theta}).$$

This can be abbreviated as $r \text{cis}{\theta}$, and is referred to as the polar form of a complex number.

Example. The complex number $$1+i$$ has modulus $$\sqrt{2}$$ and principal argument $$\frac{\pi}{4}$$, so its polar form is $$1+i = \sqrt{2} \left( \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) \right).$$

## Argument properties

With this in mind, we have a couple of properties relating to arguments \begin{align*} \arg(zw) &= \arg(z) + \arg(w) \\ \arg\left( \frac{z}{w}\right) &= \arg(z)-\arg(w). \end{align*}

Let us prove the first one.

Proof. Let $$z = r_1 (\cos{\theta_1} + i \sin{\theta_1})$$ and $$w = r_2 (\cos{\theta_2} + i \sin{\theta_2})$$. So we have $$\arg(z) + \arg(w) = \theta_1 + \theta_2$$. We want to show that $$\arg(zw) = \theta_1 + \theta_2$$. We have \begin{align*} zw &= r_1 r_2 (\cos{\theta_1} + i \sin{\theta_1})(\cos{\theta_2} + i \sin{\theta_2}) \\ &= r_1 r_2 ( \cos{\theta_1} \cos{\theta_2} – \sin{\theta_1}\sin{\theta_2} + i(\sin{\theta_1}\cos{\theta_2} + \cos{\theta_1} \sin{\theta_2})) \\ &= r_1 r_2 ( \cos(\theta_1+\theta_2)+i \sin(\theta_1+\theta_2)), \end{align*}
making use of the compound angle formula in the last line. We can see that the argument of $$zw$$ is indeed $$\theta_1+\theta_2$$.

## Problems

1. Find the polar form of the following complex numbers.
1. $$\sqrt{3}+i$$
2. $$\sqrt{3}-i$$
3. $$-\sqrt{3}+i$$
4. $$-\sqrt{3}-i$$
5. $$-2-2\sqrt{3}i$$
6. $$-4+4i$$
2. Prove that for complex $$z,w$$, we have $$\arg\left( \frac{z}{w}\right) = \arg(z)-\arg(w)$$.