Lukujoukot

Lukujoukot, or number sets, are foundational concepts in mathematics used to group numbers by shared properties. They help organize arithmetic and analysis by providing standard domains for operations and relations. Common lukujoukot include natural numbers, integers, rational numbers, real numbers and complex numbers.

The natural numbers, denoted N, are typically the nonnegative integers or, in some conventions, the positive integers. Integers, Z, extend N by including negative numbers and zero. Rational numbers, Q, consist of fractions formed by quotients of integers with nonzero denominators. Real numbers, R, include all rational numbers along with irrational numbers, which cannot be expressed as simple fractions. Complex numbers, C, comprise all numbers of the form a + bi, where a and b are real numbers and i^2 = -1.

These sets are commonly related by inclusions, often written as N ⊆ Z ⊆ Q ⊆ R ⊆ C, though conventions may vary, particularly regarding whether 0 is included in N. Real numbers are sometimes described as a complete ordered field, while complex numbers form an algebraically closed field. Each set has characteristic properties: N is closed under addition and multiplication; Z is a ring; Q is a field; R is complete and ordered; C allows complex arithmetic.

Beyond these basics, other number systems such as irrational, algebraic, transcendental, or p-adic numbers extend or refine the concept of Lukujoukot.