kompleksiluvuilla

Kompleksiluvuilla is a Finnish phrase that translates to “with complex numbers.” In mathematics, it refers to performing calculations, analysis, or statements within the system of complex numbers. The complex numbers extend the real numbers by including imaginary units and form the set C of all numbers of the form z = a + bi, where a and b are real and i is the imaginary unit with i^2 = -1. Real numbers are a subset of complex numbers with b = 0.

Arithmetic and structure. Complex numbers are closed under addition, subtraction, multiplication, and division by nonzero elements. The complex conjugate of z = a + bi is z̄ = a − bi, and the modulus is |z| = sqrt(a^2 + b^2). Every nonzero complex number z can be written in polar form z = r(cos θ + i sin θ) = r e^{iθ}, where r ≥ 0 is the modulus and θ is the argument. This allows multiplication and division to be handled conveniently by adding or subtracting arguments and multiplying or dividing moduli.

Representations and theory. The Cartesian form (a, b) and the polar form provide complementary viewpoints. Euler’s formula e^{iθ} = cos θ + i sin θ connects complex numbers to trigonometry, and the complex plane provides a geometric interpretation. Complex analysis studies analytic functions on C, with important results such as the Fundamental Theorem of Algebra, which states that every non-constant polynomial with complex coefficients has a complex root.

Usage in Finnish language. The form kompleksiluvuilla is the adessive plural used to denote the domain in which a calculation or discussion occurs, for example “laskutehtävä kompleksiluvuilla.”