Complex Numbers Cheat Sheet - Polar Form, Roots, and Applications

Complex Numbers - Complex numbers extend the real number system by including the imaginary unit (

$i$), where $i^2 = -1$. They are essential in mathematics, engineering, and physics, particularly in solving equations, modeling oscillations, and understanding wave phenomena. This comprehensive cheat sheet covers all the fundamental concepts, operations, and applications of complex numbers.

Basics of Complex Numbers

1. Definition

A complex number is expressed as:

• $z = x + yi$, where:
  • $x$: Real part ($\text{Re}(z) = x$)
  • $y$: Imaginary part ($\text{Im}(z) = y$)
  • $i^2 = -1$

2. Conjugate

The conjugate of $z = x + yi$ is:

• $\overline{z} = x - yi$

3. Modulus

The modulus of $z = x + yi$ is the distance from the origin in the complex plane:

• $|z| = \sqrt{x^2 + y^2}$

4. Argument

The argument ($\arg(z)$) is the angle made by $z$ with the positive real axis:

• $\arg(z) = \tan^{-1}\left(\frac{y}{x}\right)$

Polar and Exponential Forms

1. Polar Form

A complex number can be represented in polar form as:

• $z = r(\cos\theta + i\sin\theta)$, where:
  • $r = |z|$ (modulus)
  • $\theta = \arg(z)$ (argument)

2. Exponential Form (Euler’s Formula)

Using Euler's formula:

• $z = re^{i\theta}$, where $e^{i\theta} = \cos\theta + i\sin\theta$

Operations with Complex Numbers

1. Addition and Subtraction

Add or subtract real and imaginary parts separately:

• $z_1 + z_2 = (x_1 + x_2) + (y_1 + y_2)i$
• $z_1 - z_2 = (x_1 - x_2) + (y_1 - y_2)i$

2. Multiplication

Use distributive property or polar form:

• $z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$

3. Division

In polar form:

• $\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$

4. Powers

Using De Moivre’s theorem:

• $z^n = r^n [\cos(n\theta) + i\sin(n\theta)]$

5. Roots

The $n$-th roots of $z$ are given by:

• $z_k = r^{1/n} [\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n})]$, for $k = 0, 1, ..., n-1$

Argand Diagram

The Argand diagram represents complex numbers graphically:

• The x-axis represents the real part.
• The y-axis represents the imaginary part.
• $z = x + yi$ is a point or vector starting from the origin.

Solving Polynomial Equations

1. Roots of Polynomials

For polynomials with real coefficients:

• Roots occur in conjugate pairs if they are non-real.

2. Multiple Angle Formulas

Complex numbers simplify the manipulation of trigonometric polynomials:

• $z^n + \overline{z}^n = 2\cos(n\theta)$
• $z^n - \overline{z}^n = 2i\sin(n\theta)$

Applications of Complex Numbers

1. Electrical Engineering: Used to analyze AC circuits, impedance, and phase relationships.
2. Fluid Dynamics: Represent potential flow and vortices.
3. Quantum Mechanics: Model wave functions and probability amplitudes.
4. Signal Processing: Fourier transforms and filters.

Summary Table of Key Equations

Concept	Equation
Modulus	(
Argument	$\arg(z) = \tan^{-1}\left(\frac{y}{x}\right)$
Polar Form	$z = r(\cos\theta + i\sin\theta)$
Exponential Form	$z = re^{i\theta}$
Multiplication (Polar)	$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$
Division (Polar)	$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$
De Moivre’s Theorem	$z^n = r^n [\cos(n\theta) + i\sin(n\theta)]$
Roots of $z$	$z_k = r^{1/n} [\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n})]$

FAQs About Complex Numbers

Q1: What is the significance of the imaginary unit ($i$)?
The imaginary unit $i$ ($i^2 = -1$) allows the extension of real numbers to complex numbers, enabling solutions to equations like $x^2 + 1 = 0$.

Q2: How are complex numbers represented graphically?

Complex numbers are represented on the Argand diagram, where the x-axis is the real part, and the y-axis is the imaginary part.

Q3: What is Euler’s formula?

Euler’s formula, $e^{i\theta} = \cos\theta + i\sin\theta$, connects exponential functions with trigonometric functions.

Q4: What is De Moivre’s theorem used for?

De Moivre’s theorem is used to compute powers and roots of complex numbers.

Q5: Why are roots of real polynomials in conjugate pairs?

Polynomials with real coefficients have conjugate roots because the imaginary parts cancel when combined.