Complex numbers in polar form offer a powerful way to visualize and manipulate numbers in the complex plane. By representing numbers as distances and angles, we gain insights into their geometric properties and simplify certain calculations.

This representation connects trigonometry with complex algebra, enabling easier multiplication, division, and exponentiation of complex numbers. It also provides a foundation for understanding periodic phenomena and wave-like behavior in various fields of mathematics and science.

Complex Numbers in Polar Form

Complex numbers in the plane

• Complex numbers plotted on a 2D coordinate system called the complex plane
  • Real numbers on the horizontal axis (real axis)
  • Imaginary numbers on the vertical axis (imaginary axis)
• Complex number $z = a + bi$ plotted at the point $(a, b)$
• Absolute value (modulus) of $z = a + bi$ is the distance from the origin to $(a, b)$
  • Formula: $|z| = \sqrt{a^2 + b^2}$ (Pythagorean theorem)
• Vector representation: complex numbers can be viewed as vectors in the complex plane

Rectangular vs polar forms

• Rectangular form: $z = a + bi$
  • $a$ is the real part, $b$ is the imaginary part
• Polar form: $z = r(\cos\theta + i\sin\theta)$ or $z = r \, cis \, \theta$
  • $r$ is the modulus (absolute value)
  • $\theta$ is the argument (also known as the phase angle)
• Converting rectangular to polar:
  • $r = \sqrt{a^2 + b^2}$
  • $\theta = \tan^{-1}(\frac{b}{a})$, adjust quadrant based on signs of $a$ and $b$
• Converting polar to rectangular:
  • $a = r \cos\theta$
  • $b = r \sin\theta$
• Polar coordinates: (r, θ) represent the same point as (a, b) in rectangular coordinates

Multiplication and division in polar form

• Multiplication: $z_1 z_2 = r_1 r_2 \, cis \, (\theta_1 + \theta_2)$
  • Multiply moduli, add arguments
• Division: $\frac{z_1}{z_2} = \frac{r_1}{r_2} \, cis \, (\theta_1 - \theta_2)$
  • Divide moduli, subtract arguments
• Simplifies complex number arithmetic

Powers and roots of complex numbers

• Powers: $z^n = r^n \, cis \, (n\theta)$, $n$ is an integer
• Roots: The $n$th roots of $z = r \, cis \, \theta$:
  • $\sqrt[n]{z} = \sqrt[n]{r} \, cis \, (\frac{\theta + 2k\pi}{n})$, $k = 0, 1, 2, ..., n-1$
  • Generates $n$ equally spaced points on a circle

Applications of De Moivre's Theorem

• De Moivre's Theorem: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$
• Simplifies calculation of powers and roots in polar form
• Useful in solving complex equations and deriving trigonometric identities

Geometry of polar complex numbers

• Modulus: distance from origin
• Argument: angle with positive real axis
• Multiplication: rotation and scaling
• Roots: equally spaced points on a circle
• Unit circle: special case where r = 1, used to visualize trigonometric functions

Equations with polar complex numbers

• Some equations easier to solve using polar form
• Convert to polar, perform operations, convert back to rectangular if needed

Polar form and trigonometry connections

• Polar form closely related to trigonometric functions
• Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$
• Allows derivation of trigonometric identities using complex numbers
  • Example: $\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta$
• Complex exponential: $z = re^{i\theta}$ is an alternative representation of polar form