Complex numbers expand our mathematical toolkit beyond real numbers. They introduce a new dimension, combining real and imaginary parts on a 2D plane. This allows us to solve equations that were previously impossible and model more complex phenomena.

In the complex plane, we can visualize, add, multiply, and transform these numbers geometrically. This visual approach helps us understand their properties and operations, making abstract concepts more concrete and intuitive.

Complex Numbers and the Complex Plane

Complex numbers on the plane

• 2D coordinate system real numbers on horizontal axis imaginary numbers on vertical axis
• Horizontal axis represents real part of complex number
• Vertical axis represents imaginary part of complex number
• Complex number $z = a + bi$ plotted as point $(a, b)$
  • $a$ real part determines horizontal position
  • $b$ imaginary part determines vertical position
• This representation is also known as an Argand diagram

Absolute value of complex numbers

• Absolute value (modulus) of complex number $z = a + bi$ denoted as $|z|$ calculated using formula $|z| = \sqrt{a^2 + b^2}$
• Represents distance between complex number and origin on complex plane
• Measure of magnitude of complex number regardless of direction
• Always non-negative real number

Polar form of complex numbers

• Polar form of complex number $z$ is $z = r(\cos\theta + i\sin\theta)$ or $z = re^{i\theta}$
  • $r$ magnitude (absolute value) of complex number
  • $\theta$ angle (argument) in radians measured counterclockwise from positive real axis
• Magnitude $r$ calculated using formula $r = \sqrt{a^2 + b^2}$
• Angle $\theta$ calculated using formula $\theta = \tan^{-1}(\frac{b}{a})$ with adjustments based on quadrant of complex number
• The angle $\theta$ between -π and π is called the principal argument

Rectangular vs polar forms

• Rectangular form $(a + bi)$ to polar form $(re^{i\theta})$:
  1. Calculate magnitude $r = \sqrt{a^2 + b^2}$
  2. Calculate angle $\theta = \tan^{-1}(\frac{b}{a})$ adjusting for quadrant
• Polar form $(re^{i\theta})$ to rectangular form $(a + bi)$:
  1. Calculate real part $a = r\cos\theta$
  2. Calculate imaginary part $b = r\sin\theta$

Multiplication in polar form

• Multiplication in polar form $(r_1e^{i\theta_1})(r_2e^{i\theta_2}) = r_1r_2e^{i(\theta_1 + \theta_2)}$
  • Multiply magnitudes add angles
• Division in polar form $\frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$
  • Divide magnitudes subtract angles

Powers and roots in polar form

• Raise complex number $z = re^{i\theta}$ to power $n$: $z^n = r^ne^{in\theta}$
  • Raise magnitude to power $n$ multiply angle by $n$
• Find $n$-th roots of complex number $z = re^{i\theta}$: $\sqrt[n]{z} = \sqrt[n]{r}e^{i(\frac{\theta + 2k\pi}{n})}$ where $k = 0, 1, ..., n-1$
  • Take $n$-th root of magnitude divide angle (plus multiples of $2\pi$) by $n$

Applications of De Moivre's Theorem

• De Moivre's Theorem for complex number $z = r(\cos\theta + i\sin\theta)$ and integer $n$: $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$
• Useful for:
  • Raising complex numbers to powers
  • Finding roots of complex numbers
  • Solving equations involving complex numbers

Visualization on complex plane

• Addition of complex numbers:
  • Represent addition as vector sum resultant vector obtained by placing tail of one vector at head of other
• Multiplication by complex number:
  • Multiplication by $e^{i\theta}$ rotates complex number by angle $\theta$ counterclockwise around origin
  • Multiplication by real number scales magnitude of complex number without changing angle
• Powers and roots of complex numbers:
  • Raising complex number to power corresponds to rotating and scaling number on complex plane
  • Taking $n$-th root of complex number corresponds to dividing angle by $n$ taking $n$-th root of magnitude resulting in $n$ evenly spaced points on circle centered at origin

Complex Exponential and Unit Circle

• Euler's formula relates complex exponentials to trigonometric functions: $e^{i\theta} = \cos\theta + i\sin\theta$
• The unit circle in the complex plane is described by complex numbers of the form $e^{i\theta}$
• Complex exponentials are used to represent rotations and periodic phenomena
• The complex exponential form $re^{i\theta}$ combines magnitude and angle in a compact notation