Now that we have seen two forms for specifying complex numbers, let us summarize the ideas for converting from one form to the other.

## From rectangular form to polar form

Let $$z = a+bi$$ be a complex number. We want to write $$z$$ as $$r\operatorname{cis} \theta$$ with $$r = |z|= \sqrt{a^2+b^2}$$ and $$0 \leq \theta \lt 2\pi$$.

We take care of a few special cases first. If $$z = 0$$, then $$r = 0$$. Hence, $$\theta$$ can be any value. However, we will adopt the convention that $$\theta = 0$$ in this case unless otherwise stated.

Suppose that $$a = 0$$. In this case, the modulus is just $$r = |b|$$ and $$\theta =\left\{\begin{array}{ll} \frac{\pi}{2} & \text{if } b \gt 0 \\ \frac{3\pi}{2} & \text{otherwise.} \end{array}\right.$$

Suppose that $$a \neq 0$$. Let $$\alpha = \displaystyle\tan^{-1} \frac{|b|}{|a|}$$. The following table shows the value of the argument $$\theta$$ depending on the signs of $$a$$ and $$b$$:
Case $$\theta$$
$$a \gt 0, b \geq 0$$ $$\alpha$$
$$a \lt 0, b \geq 0$$ $$\pi - \alpha$$
$$a \lt 0, b \lt 0$$ $$\pi + \alpha$$
$$a \gt 0, b \lt 0$$ $$2\pi -\alpha$$

### Examples

1. To convert $$13 + 4i$$ to polar form, note that both its real part and imaginary part are positive. Hence, its polar form is $$r \operatorname{cis} \theta$$ where $$r = \sqrt{ 13^2 + 4^2} = \sqrt{ 285 }$$ and $$\theta = \tan^{-1} \frac{4}{13}$$.

2. To convert $$-1 - 17i$$ to polar form, note that both its real part and its imaginary part are negative. Hence, its polar form is $$r \operatorname{cis} \theta$$ where $$r = \sqrt{ (-1)^2 + (-17)^2} = \sqrt{ 290 }$$ and $$\theta = \pi + \tan^{-1} \frac{|-17|}{|-1|} = \pi + \tan^{-1} 17$$.

## From polar form to rectangular form

Let $$z = r \operatorname{cis} \theta$$ be a complex number with $$r \geq 0$$ and $$0 \leq \theta \lt 2\pi$$.

To convert $$z$$ to rectangular form, recall that $$\operatorname{cis} \theta$$ is an abbreviation for $$\cos \theta + i \sin \theta$$. Thus, $z = r (\cos \theta + i\sin \theta) = (r \cos \theta) + (r \sin \theta) i.$

In short, $$\mathbf{Re} (z) = r \cos \theta$$ and $$\mathbf{Im} (z) = r \sin \theta.$$

### Examples

1. The rectangular form of $$3 \operatorname{cis} \frac{3\pi}{2}$$ is $$3 \cos \frac{3\pi}{2} + 3\sin \frac{3\pi}{2} i = 0 + (-3)i = -3i$$.

2. The rectangular form of $$\sqrt{2} \operatorname{cis} \frac{3}{7}$$ is $$\sqrt{2} \cos \frac{3}{7} + \sqrt{2}\sin \frac{3}{7} i \approx 1.2863 + 0.5877 i$$ with real and imaginary parts rounded to 4 decimal places.

## Exercises

1. Convert each of the following to rectangular form with real and imaginary parts rounded to 4 decimal places.

1. $$\operatorname{cis} \frac{3\pi}{2}$$

2. $$\sqrt{3} \operatorname{cis} \frac{5\pi}{6}$$

3. $$2.5 \operatorname{cis} \frac{13}{5}$$

2. Convert each of the following to polar form.

1. $$12 + 7i$$

2. $$-7i$$

3. $$7 - 12i$$

4. $$13 - i$$

5. $$-23 - 4i$$