kompleksset

Kompleksset, often denoted by ℂ, is the set of all complex numbers, where a complex number is expressed in the form a + bi. Here a and b are real numbers and i represents the imaginary unit, defined by the property i² = −1. The complex numbers extend the real number system to provide solutions to equations that are unsolvable over the reals, such as x² + 1 = 0.

The structure of ℂ is that of a two‑dimensional vector space over the real numbers, with the real axis representing the real parts and the imaginary axis representing the imaginary parts. Addition and multiplication are defined componentwise, following the distributive, associative, and commutative laws. Every non‑zero complex number possesses a multiplicative inverse, making ℂ a field. In polar form, a complex number r e^{iθ} highlights its magnitude r ≥ 0 and argument θ, leading to Euler’s formula e^{iθ} = cosθ + i sinθ.

Topologically, ℂ is homeomorphic to the Euclidean plane ℝ², allowing the application of many real‑analysis concepts. Algebraically, complex numbers are algebraically closed: every non‑constant polynomial with complex coefficients has a zero in ℂ. They play foundational roles in complex analysis, Fourier analysis, control theory, and many areas of physics and engineering.