Complex Number Studio
Arithmetic, the Argand plane, polar and exponential form, DeMoivre powers, and all n-th roots. Covers §2.11–2.16 of the algebra material.
Complex number
A complex number is an ordered pair of real numbers, z = x + yi, where x = Re z is the real part, y = Im z is the imaginary part, and i² = −1. Every z has a modulus |z| = √(x² + y²) and an argument θ = atan2(y, x), giving the polar form z = r(cos θ + i sin θ) = r eiθ by Euler's formula.
Two-Number Arithmetic
z₁ = x₁ + y₁i and z₂ = x₂ + y₂i
Polar Form · DeMoivre · n-th Roots
Enter z, a power n for zⁿ, and a root order m for the m-th roots of z.
Argand plane: z (steel), zⁿ (red), and the m-th roots of z (gold) equally spaced by 2π/m on a circle of radius |z|1/m.
FIG. T14-1 — ORIGINAL MECHFORGE DIAGRAM
DeMoivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). The m-th roots share modulus r1/m and arguments (θ + 2πk)/m for k = 0, 1, …, m−1 — which is why they always form a regular polygon.