Polar Coordinates and Complex Numbers

Polar Coordinates

A Point can be located by (r, θ) where r is the distance from the origin and θ is the Angle from the positive x-axis.

Converting Between Systems

• Polar to Rectangular: x = r cos θ, y = r sin θ
• Rectangular to Polar: r = √(x² + y²), θ = tan⁻¹(y/x)

Complex Numbers and the Complex Plane

A complex number z = a + bi can be plotted on the complex Plane (a on the real axis, b on the imaginary axis).

• Modulus: |z| = √(a² + b²) (distance from origin)
• Argument: θ = tan⁻¹(b/a) (Angle from positive real axis)
• Polar form: z = r(cos θ + i sin θ)

Connection to Geometry

Complex numbers provide an algebraic framework for transformations (rotation = multiplication by e^(iθ), dilation = multiplication by a scalar).