A real number is a number belonging to the system denoted by (\mathbb{R}), which contains all rational numbers and irrational numbers. Geometrically, real numbers correspond to points on a number line. Abstractly, they form a complete ordered field: addition, subtraction, multiplication, and division by nonzero elements obey the familiar arithmetic laws, while completeness ensures that the system has no gaps in a precise mathematical sense. These properties distinguish the real numbers from the rational numbers and underpin mathematical analysis. (math.ucdavis.edu)
Membership and representation
The real numbers contain the natural numbers, integers, and rational numbers as successively larger subsystems: [ \mathbb{N}\subsetneq\mathbb{Z}\subsetneq\mathbb{Q}\subsetneq\mathbb{R}. ] A rational number has the form (p/q), where (p) and (q) are integers and (q\ne0). An irrational number cannot be expressed in this way; (\sqrt{2}) is a standard example. Every real number belongs to exactly one of these two classes. (math.ucdavis.edu)
Every real number admits a decimal representation. Rational numbers have terminating or eventually repeating decimal expansions, whereas irrational numbers have expansions that neither terminate nor eventually repeat. Decimal representations are not always unique: for example, [ 0.999\ldots=1.000\ldots. ] This is an exact equality, not an approximation. A terminating expansion, continued with zeros, can also be expressed using a corresponding infinite tail of nines. Apart from this ambiguity, decimal expansions uniquely identify real numbers. (math.northwestern.edu)
Algebra and order
Real-number arithmetic satisfies associativity and commutativity for addition and multiplication, together with distributivity. The additive identity is zero, and the multiplicative identity is one. Every real number has an additive inverse, and every nonzero real number has a multiplicative inverse. Division by zero is undefined within the field. (math.ucdavis.edu)
The order is total: for any (a,b\in\mathbb{R}), exactly one of (a<b), (a=b), or (a>b) holds. It is compatible with arithmetic: adding the same number preserves an inequality, and multiplying by a positive number preserves its direction. Multiplication by a negative number reverses the direction. Consequently, every real square is nonnegative, so the equation (x^2=-1) has no real solution. The larger system of complex numbers supplies such solutions but cannot be ordered in a way compatible with its field operations. (math.ucdavis.edu)
Completeness
The defining completeness property is the least-upper-bound property. Every nonempty set of real numbers that is bounded above has a least upper bound, or supremum, in (\mathbb{R}). This bound need not belong to the set itself: the open interval ((0,1)), for example, has supremum (1) but no greatest element. (math.ucdavis.edu)
The rational numbers lack this property. The set of positive rational numbers whose squares are less than (2) is bounded above, but its least upper bound would be (\sqrt{2}), which is not rational. Thus density alone does not eliminate gaps: between any two distinct real numbers lies a rational number, yet the rationals are not complete. (math.ucdavis.edu)
Completeness implies the Archimedean property: for every real (x), some positive integer exceeds (x). There are therefore no infinitely large real numbers, nor positive real infinitesimals smaller than every (1/n). The symbols (+\infty) and (-\infty), when introduced in extended number systems, are not ordinary real numbers. (people.math.wisc.edu)
Construction and characterization
Two standard constructions build the real numbers from the rationals. A Dedekind cut is a nonempty proper subset of (\mathbb{Q}) that is downward closed and has no greatest element. It encodes a boundary, including boundaries that no rational number occupies. The cut corresponding to a real (r) contains precisely the rational numbers less than (r). (arxiv.org)
The other construction uses Cauchy sequences of rational numbers: sequences whose terms become arbitrarily close to one another sufficiently far along the sequence. Two such sequences represent the same real number when their difference tends to zero. Arithmetic is defined through operations on representative sequences. (arxiv.org)
In classical mathematics, these constructions produce equivalent systems. The real field is unique up to an order-preserving field isomorphism: any two complete ordered fields have the same mathematical structure, despite possible differences in how their elements are constructed. (people.math.wisc.edu)
Size and the continuum
In set theory, (\mathbb{R}) is uncountable, unlike (\mathbb{N}), (\mathbb{Z}), and (\mathbb{Q}). Cantor’s diagonal argument shows that no sequence can list every real number even within ((0,1)). Given a proposed list of decimal expansions, one constructs a decimal differing from the (n)th entry at its (n)th digit, choosing digits to avoid the ambiguity of trailing nines. The resulting number is absent from the list. (stanford.edu)
This distinction demonstrates that having a rational number between every pair of real numbers does not make the two sets equally large. Countability concerns whether all elements can be enumerated; density concerns their placement in the ordered line. (stanford.edu)
Limits and numerical computation
Every Cauchy sequence of real numbers converges to a real limit. This supports the treatment of convergence in calculus and the study of continuous functions. It does not mean that every sequence converges: an unbounded sequence or an oscillating sequence may have no real limit. (math.ucdavis.edu)
Computer representations must be distinguished from the abstract system. Fixed-precision floating-point arithmetic represents only a finite selection of values. Many real numbers, including the rational number (1/10), cannot be represented exactly in standard binary floating-point formats. Stored approximations and rounded operations can therefore behave differently from exact real arithmetic, even when displayed decimal results appear simple. (docs.python.org)