Reaalijoukon

Reaalijoukko is a Finnish mathematical term referring to the set of real numbers. In most contexts it denotes the collection of all numbers that can be represented on the real number line, including both rational and irrational numbers. The standard notation for this set is R, sometimes written as the real line when described geometrically.

The real numbers form a complete ordered field. This means they support the usual arithmetic operations (addition, subtraction, multiplication, division by nonzero) and are equipped with a total order that is compatible with these operations. A key feature is the completeness axiom: every nonempty subset that has an upper bound has a least upper bound (supremum) within the real numbers. This property underpins limits, continuity, and many results in analysis.

Within the reaalijoukko, the integers and rationals are dense, and the set also includes irrational numbers such as pi and sqrt(2). Real numbers can be represented by decimal expansions; rationals have terminating or repeating decimals, while irrationals have non-terminating, non-repeating expansions. The real numbers can be visualized as the real number line, where every point corresponds to a real value and distance is measured by the absolute difference d(x,y)=|x−y|.

Real numbers are foundational in calculus, real analysis, and many areas of applied mathematics. They provide the setting for limits, sequences, continuity, integration, and differentiation. The term reaali- in Finnish reflects the traditional distinction between real numbers and other number systems, such as complex numbers.

See also: Real numbers, Real analysis, Number line, Decimal expansion, Continuity.