Polaarimuoto

Polaarimuoto, or polar form, is a representation of a plane point or a complex number by its distance from the origin and the angle it makes with the positive real axis. For a complex number z = x + iy, the modulus r = |z| and the argument θ = Arg(z) determine the polar form z = r(cos θ + i sin θ) = re^{iθ}, with r ≥ 0 and θ ∈ ℝ. Equivalently, in polar coordinates a point has coordinates (r, θ) related to Cartesian coordinates by x = r cos θ and y = r sin θ; here r = sqrt(x^2 + y^2) and θ is commonly taken as the principal value Arg z ∈ (-π, π]. The angle is defined modulo 2π, so θ and θ + 2πk describe the same point or number.

In the complex plane, polar form expresses the same z with a modulus and a direction. Operations follow standard identities: multiplying complex numbers corresponds to multiplying moduli and adding angles; dividing subtracts angles; powering multiplies the angle by the power. Specifically, z1 z2 = (r1 r2) e^{i(θ1+θ2)}, z1 / z2 = (r1/r2) e^{i(θ1-θ2)}, and z^n = r^n e^{inθ}. Roots are given by z^{1/n} = r^{1/n} e^{i(θ+2πk)/n}, for k = 0, 1, ..., n−1.

Polaarimuoto is foundational in complex analysis and appears in applications such as solving polynomial equations, signal processing, phasor representations in electrical engineering, and the geometric interpretation of rotations in the plane. Related concepts include polar coordinates, Euler's formula, and De Moivre's theorem.