Cartesian and Polar Forms of Complex Numbers

Q1: What is a complex number?
A1: A complex number $z$ is a number with:
1. A real component $a$
2. An imaginary component $b$
where $z=a+jb$ where $j=\sqrt{-1}$

This is the Cartesian form of the complex number.

Q2: How else may a complex number be expressed?
A2: Alternatively, $z$ may be expressed in polar form with:
1. A magnitude $\left|z\right|$
2. An angle $\theta$
where $z=\left|z\right|e^j\theta$

Q3: Why should I bother learning Cartesian and polar form of complex numbers?
A3: Cartesian form is convenient for complex addition/subtraction while polar form is convenient for complex multiplication/division.

Q4: How do I switch from Cartesian to polar form?
A4: Use the relations:
1. $\left|z\right|=\sqrt{a^2+b^2}$
2. $\tan{\theta}=\frac {b}{a}$

Q5: How do I switch from polar to Cartesian form?
A5: Use the relations:
1. $a=\left|z\right|\cos{\theta}$
2. $b=\left|z\right|\sin{\theta}$