Kompleksarvude

Kompleksarvud are numbers of the form a + bi where a and b are real numbers and i satisfies i^2 = -1. They extend the real numbers to a two-dimensional number system, allowing representation as points in the complex plane (also called the Argand plane). The real part is a and the imaginary part is b.

Addition and subtraction are performed componentwise. Multiplication uses i^2 = -1, giving (a+bi)(c+di) = (ac - bd) + (ad + bc)i. Division is achieved by multiplying numerator and denominator by the conjugate c - di, as in (a+bi)/(c+di) = ((a+bi)(c - di))/(c^2 + d^2) when c^2 + d^2 ≠ 0. The conjugate of z = a+bi is z̄ = a - bi, and the modulus |z| = sqrt(a^2 + b^2).

Every nonzero complex number has a polar form r(cos theta + i sin theta) or r e^{i theta}, where r = |z| and theta is the argument arctan2(b, a). Euler's formula links these representations: e^{i theta} = cos theta + i sin theta.

Geometrically, addition corresponds to vector addition in the plane, while multiplication combines rotation and scaling: |z1 z2| = |z1| |z2| and arg(z1 z2) = arg(z1) + arg(z2).

Important theoretical results include the Fundamental Theorem of Algebra: every nonconstant polynomial with complex coefficients has a complex root. For polynomials with real coefficients, nonreal roots occur in conjugate pairs.

Applications span signal processing, electrical engineering, quantum physics, control theory, and applied mathematics. Note: In Estonian, kompleksarvud means complex numbers, and kompleksarvude is the genitive form used in phrases such as “kompleksarvude teooria” (theory of complex numbers).