Powers of the Imaginary Unit

Q: What are the integer powers of the imaginary unit?
A:

$j^0=1$
$j^1=j$
$j^2=-1$
$j^3=-j$

This basic four-part pattern continues for consecutive integer powers above and below this arbitrarily selected sequence

Complex Conjugates

Given: $z = a + jb = |z|e^{j\theta}$

Q1: What is the complex conjugate of $z$ in Cartesian form?

A1: $z^* = a - jb$

Q2: What is the complex conjugate of $z$ in polar form?

A2: Note that $z^*$ is simply the reflection of $z$ over the real axis of the complex plane. It then follows that:

1.The magnitude of $z^*$ is the same as the magnitude of $z$

2. The angle that $z^*$ makes with respect to the positive real axis is the negative value of the angle that $z$ makes with the positive real axis. 

Hence, $z^*= |z|e^{-j\theta}$

Q3: How can you simplify the expression $zz^*$?
A3: Although the multiplication suggests that you use polar form to simplify the product, Cartesian form yields an interesting result.
$zz^*=(a+jb)(a-jb)$
Notice that the cross-terms (i.e. the O and I products of FOIL) negate each other.
$zz^*=a^2+b^2$
$zz^*={|z|}^2$

Complex Number Exponentiation

Given: $z$ a complex number

Q1: What is $z^n$? Assume $n$ is an integer.

A1: Since exponentiation is repeated multiplication, it's convenient to use the complex polar form of $z$:

$z=\left|z\right|e^{j\theta}$

$z^n={\left|z\right|}^ne^{jn\theta}$

This result is known as De Moivre's formula.

Q2: Suppose you don't have $z$ raised to an integer power. Instead, you have $z^{\frac{1}{n}}$. How do you find the $n$ roots of this expression?
Q3: To answer Q2, let's look at the specific case where $n$ is 3. What are the 3 roots of $z^{\frac{1}{3}}$?
A3: Following the logic from A1,
Root 1: $z^{\frac{1}{3}}={\left|z\right|}^{\frac{1}{3}}e^{\frac{j\theta}{3}}$.
But this is only 1 root. We know we need to find 2 more. Note that $2\pi$ can be added to the angle $\theta$ without changing the result. We can write 2 more cases:
Root 2: $z^{\frac{1}{3}}={\left|z\right|}^{\frac{1}{3}}e^{\frac{j(\theta+2\pi)}{3}}$.
Root 3: $z^{\frac{1}{3}}={\left|z\right|}^{\frac{1}{3}}e^{\frac{j(\theta+4\pi)}{3}}$.
Q4: This gives us 3 roots - just what we want. But why not continue adding 2$\pi$ to the angle again? Couldn't this produce a 4th root?
A4: Suppose you added another 2$\pi$. Then you'd get the fourth case:
Root 4: $z^{\frac{1}{3}}={\left|z\right|}^{\frac{1}{3}}e^{\frac{j(\theta+6\pi)}{3}}$.
But note that this reduces to  $z^{\frac{1}{3}}={\left|z\right|}^{\frac{1}{3}}e^{\frac{j\theta}{3}+2\pi}={\left|z\right|}^{\frac{1}{3}}e^{\frac{j\theta}{3}}$.
which is the same as Root 1. So there's no need to continue adding 2$\pi$ past the Root 3. Any other roots found by adding 2$\pi$ past this point are redundant.
A2: To answer the general case of Q2, we substitute $n$ for 3 to find that the $n$ roots of $z^{\frac{1}{n}}$ are:
 $z^{\frac{1}{n}}={\left|z\right|}^{\frac{1}{n}}e^{\frac{j(\theta+2k\pi)}{n}}$ where $k$ is an integer in the range 0, 1, ... , $n-1$

Complex Multiplication/Division

Given:
$z_1=a_1+jb_1=\left|z_1\right|e^j\theta_1$
$z_2=a_2+jb_2=\left|z_2\right|e^j\theta_2$

Q1: What is $z_1z_2$ in Cartesian form?
A1: Using the FOIL method and collecting the real and imaginary parts:
$z_1z_2=(a_1a_2-b_1b_2)+j(a_1b_2+a_2b_1)$

Q2: What is $z_1z_2$ in polar form?
A2: $z_1z_2=\left|z_1\right|\left|z_2\right|e^{j(\theta_1+\theta_2)}$

Q3: What is $\frac{z_1}{z_2}$ in Cartesian form?
A3: Multiplying both the numerator and denominator by $z_2^*$:
$\frac{z_1}{z_2}=\frac{(a_1a_2+b_1b_2)+j(a_2b_1-a_1b_2)}{a_2^2+b2_2}$

Q4: What is $\frac{z_1}{z_2}$ in polar form?
A4: $\frac{z_1}{z_2}=\frac{\left|z_1\right|}{\left|z_2\right|}e^{j(\theta_1-\theta_2)}$

Complex Addition/Subtraction

Given:
$z_1=a_1+jb_1$
$z_2=a_2+jb_2$

Q1: How do I add complex numbers in Cartesian form?
A1: Complex numbers are added component-wise:
$z_1+z_2=(a_1+a_2)+j(b_1+b_2)$

Q2: How do I subtract complex numbers in Cartesian form?
A2: Complex numbers are subtracted component-wise:
$z_1-z_2=(a_1-a_2)+j(b_1-b_2)$

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. $b=\left|z\right|\sin{\theta}$