Kompleksarvudele

Kompleksarvudele is the Estonian term describing the study of complex numbers and their properties. It encompasses both the algebraic manipulation of numbers of the form a + bi, where a and b are real and i^2 = -1, and the geometric interpretation of these numbers on the complex plane. The field integrates arithmetic, geometry, and analysis to describe and solve problems that involve real and imaginary components.

Historically, complex numbers arose during attempts to solve polynomial equations in the early modern period. The geometric interpretation was advanced by Jean-Robert Argand with the Argand plane, and the algebraic foundations were strengthened by Carl Friedrich Gauss, who helped establish the robustness of complex numbers and their roots, culminating in the Fundamental Theorem of Algebra.

Core concepts include the modulus |z| = sqrt(a^2 + b^2) and the argument arg(z) (the angle with the positive real axis), as well as representations such as the polar form z = r(cos theta + i sin theta) and the exponential form z = re^{i theta} using Euler's formula e^{i theta} = cos theta + i sin theta. Operations cover addition, multiplication, division, conjugation, and root extraction, with De Moivre's theorem linking powers and roots to trigonometric forms. The Fundamental Theorem of Algebra states that every non-constant polynomial has a complex root, ensuring a complete field of roots.

Applications span signal processing, quantum mechanics, control theory, and computer graphics. Complex analysis, a central subfield, studies analytic functions on complex domains and yields results such as contour integration and conformal mappings. Complex dynamics, including Mandelbrot and Julia sets, illustrate the rich behavior of iterative complex functions.

See also: complex plane, imaginary unit, Euler's formula, complex analysis.