Complex number absolute value & angle review

Review your knowledge of the complex number features: absolute value and angle. Convert between them and the rectangular representation of a number.
Absolute value of $a+bi$
$\mid\!\! z\!\mid=\sqrt{a^2+b^2}$
Angle of $a+bi$
$\theta=\tan^{-1}\left(\dfrac{b}{a}\right)$
Rectangular form from absolute value $r$ and angle $\theta$
$r\cos\theta+r\sin\theta i$

What are the absolute value and angle of complex numbers?

We are used to writing complex numbers in their rectangular form, that gives their $\blueD{\text{real}}$ and $\greenD{\text{imaginary}}$ parts. For example, $\blueD3+\greenD4i$.
We can plot numbers in the complex plane according to their parts:
Considered graphically, there's another way to uniquely describe complex numbers — their $\goldD{\text{absolute value}}$ and $\purpleC{\text{angle}}$:
The $\goldD{\text{absolute value}}$, or $\goldD{\text{modulus}}$, gives the distance of the number from the origin in the complex plane, while its $\purpleC{\text{angle}}$, or $\purpleC{\text{argument}}$, is the angle the number forms with the positive Real axis.
The absolute value of a complex number $z$ is written in the same way as the absolute value of a real number, $|z|$.
Want to learn more about the absolute value and angle of complex numbers? Check out this video.

Practice set 1: Finding absolute value

To find the absolute value of a complex number, we take the square root of the sum of the squares of the parts (this is a direct result of the Pythagorean theorem):
$|\blueD a+\greenD bi|=\sqrt{\blueD a^2+\greenD b^2}$
For example, the absolute value of $\blueD 3+\greenD4i$ is $\sqrt{\blueD3^2+\greenD4^2}=\sqrt{25}=5$.
Problem 1.1
$|3+7i|=$

Want to try more problems like this? Check out this exercise.

Practice set 2: Finding angle

To find the angle of a complex number, we take the inverse tangent of the ratio of its parts:
$\theta=\tan^{-1}\left(\dfrac{\greenD b}{\blueD a}\right)$
This results from using trigonometry in the right triangle formed by the number and the Real axis.

Example 1: Quadrant $\text{I}$

Let's find the angle of $\blueD3+\greenD4i$:
$\tan^{-1}\left(\dfrac{\greenD4}{\blueD3}\right)\approx 53^\circ$

Example 2: Quadrant $\text{II}$

Let's find the angle of $\blueD{-3}+\greenD4i$. First, notice that $\blueD{-3}+\greenD4i$ is in Quadrant $\text{II}$.
$\tan^{-1}\left(\dfrac{\greenD4}{\blueD{-3}}\right)\approx -53^\circ$
$-53^\circ$ is in Quadrant $\text{IV}$, not $\text{II}$. We must add $180^\circ$ to obtain the opposite angle:
$-53^\circ+180^\circ=127^\circ$
Problem 2.1
$z=1+4i$
$\theta=$
$^\circ$
Round your answer, if necessary, to the nearest tenth. Express $\theta$ between $-180^\circ$ and $180^\circ$.
Want to try more problems like this? Check out this exercise.

Practice set 3: Rectangular form from absolute value and angle

To find the real and imaginary parts of a complex number from its absolute value and angle, we multiply the absolute value by the sine or cosine of the angle:
This results from using trigonometry in the right triangle formed by the number and the Real axis.
For example, this is the rectangular form of the complex number whose absolute value is $\goldD 2$ and angle is $\purpleC{30^\circ}$:
$\goldD 2\cos(\purpleC{30^\circ})+\goldD 2\sin(\purpleC{30^\circ})i=\blueD{\sqrt 3}+\greenD1i$
Problem 3.1
$|z_1|=3$ and $\theta_1=20^{\circ}$
$z_1 =$
+
$i$