Real Number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; <), up to an isomorphism,[a] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.

The set of all real numbers is uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers {c} and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted {{0}}, 'aleph-naught'). The statement that there is no subset of the reals with cardinality strictly greater than {0} and strictly smaller than {c} is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC), the standard foundation of modern mathematics, in the sense that some models of ZFC satisfy CH, while others violate it.

https://en.wikipedia.org/wiki/Real_number

Real Line
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In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.

Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, {\displaystyle \mathbb {R} } \mathbb {R} , the letter “R” in blackboard bold). However, it is sometimes denoted R1 in order to emphasize its role as the first Euclidean space.

This article focuses on the aspects of R as a geometric space in topology, geometry, and real analysis. The real numbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For more information on R in all of its guises, see real number.

https://en.wikipedia.org/wiki/Real_line

Compactification
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Compactification (mathematics) From Wikipedia, the free encyclopedia Jump to navigationJump to search In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.[1] A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".

Contents 1 An example 2 Definition 2.1 Alexandroff one-point compactification 2.2 Stone–Cech compactification 3 Projective space 4 Compactification and discrete subgroups of Lie groups 5 Other compactification theories 6 References An example Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by 8. The resulting compactification can be thought of as a circle (which is compact as a closed and bounded subset of the Euclidean plane). Every sequence that ran off to infinity in the real line will then converge to 8 in this compactification.

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (-p,p) on the x-axis; then bend the ends of this interval upwards (in positive y-direction) and move them towards each other, until you get a circle with one point (the topmost one) missing. This point is our new point 8 "at infinity"; adding it in completes the compact circle.

A bit more formally: we represent a point on the unit circle by its angle, in radians, going from -p to p for simplicity. Identify each such point ? on the circle with the corresponding point on the real line tan(?/2). This function is undefined at the point p, since tan(p/2) is undefined; we will identify this point with our point 8.

Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism between the real line and the unit circle without 8. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. It is also possible to compactify the real line by adding two points, +8 and -8; this results in the extended real line.

https://en.wikipedia.org/wiki/Compactification_(mathematics)

Point at infinity
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In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring.

In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).

In the case of a hyperbolic space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric.

https://en.wikipedia.org/wiki/Point_at_infinity