kompleksarvudes

Kompleksarvud are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit with i^2 = -1. The real part is Re(z) = a and the imaginary part is Im(z) = b. Complex numbers can be written in Cartesian form a + bi or in polar form r(cos θ + i sin θ), with r ≥ 0 and θ as the argument of z. In exponential form, z = r e^{iθ}, using Euler’s formula e^{iθ} = cos θ + i sin θ.

Geometrically, complex numbers correspond to points in the complex plane (also called the Argand plane). The modulus is |z| = sqrt(a^2 + b^2), and the argument θ describes the direction of the vector from the origin to (a, b). The conjugate of z = a + bi is z* = a − bi, and z z* = |z|^2.

Arithmetic with complex numbers follows familiar rules: addition and subtraction are performed componentwise, (a + bi) + (c + di) = (a + c) + (b + d)i. Multiplication gives (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division can be done by multiplying numerator and denominator by the conjugate of the denominator: (a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c^2 + d^2).

Complex numbers form a field, meaning they are closed under addition, subtraction, multiplication, and division (except by zero). They underpin results such as De Moivre’s theorem and Euler’s formula, and they are central to the fundamental theorem of algebra: every non-constant polynomial with complex coefficients has a root in the complex numbers.

Historically, complex numbers emerged in solving polynomial equations, with geometric interpretation developed by Argand and later formalized by Gauss and others. Today they are widely used in mathematics, physics, engineering, and signal processing.