extended real number line - Article Index

Encyclopedia - Infinity: Encyclopedia - Infinity
Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or c...   » Read the article

Encyclopedia - Unit Interval: Encyclopedia - Unit Interval
In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x ...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia - Complete Lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Com...   » Read the article

Encyclopedia - Supremum: Encyclopedia Ii - Supremum - Supremum Of A Set Of Real Numbers
In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real numb...   » Read the article

Encyclopedia - Infimum: Encyclopedia Ii - Infimum - Infima Of Real Numbers
In analysis the infimum or greatest lower bound of a set S of real numbers is denoted by inf(S) and is defined to be the biggest real num...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which sta...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800...   » Read the article

Encyclopedia - Limit Of A Sequence: Encyclopedia Ii - Limit Of A Sequence - Formal Definition
Suppose x1, x2, ... is a sequence of elements in a topological space T. We say that L∈T is a limit of this sequence and write if and...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Construction Of The Lebesgue Integral
The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach the theory of integr...   » Read the article

Encyclopedia - Improper Integral: Encyclopedia Ii - Improper Integral - Infinite Bounds Of Integration
The most basic of improper integrals are integrals such as: As stated above, this need not be defined as an improper integral, since i...   » Read the article

Encyclopedia - Projective Line: Encyclopedia Ii - Projective Line - Homogeneous Coordinates
An arbitrary point in the projective line P1(K) may be given in homogeneous coordinates by a pair [x1:x2] of points in K which are not ...   » Read the article

Encyclopedia - Metric Mathematics: Encyclopedia Ii - Metric Mathematics - Related Concepts And Alternative Axiom Systems
Some authors use the extended real number line and allow the distance function d to attain the value ∞. Such a metric is called an exte...   » Read the article

Encyclopedia - Real Number: Encyclopedia Ii - Real Number - Properties
Real number - Completeness. The main reason for introducing the reals is that the reals contain all limits. More technically, the reals...   » Read the article

Encyclopedia - Kleene Algebra: Encyclopedia Ii - Kleene Algebra - Definition
Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. See [1] for a survey. Here ...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Free Construction And Completion
Complete lattice - Free complete semilattices. As usual, the construction of free objects depends on the chosen class of morphisms. Let...   » Read the article

Encyclopedia - Limit Of A Function: Encyclopedia Ii - Limit Of A Function - Formal Definition
Limit of a function - Functions on metric spaces. Suppose f : (M,dM) -> (N,dN) is a map between two metric spaces, p is a limit...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Mathematical Infinity
Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Use Of Infinity In Common Speech
In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forev...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity In Science Fiction
The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, m...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Formal Definition
A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (infimum, meet) and a least...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Physical Infinity
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i...   » Read the article

Encyclopedia - Real Number: Encyclopedia Ii - Real Number - Definition
Real number - Construction from the rational numbers. The real numbers can be constructed as a completion of the rational numbers. For ...   » Read the article

Encyclopedia - Projective Line: Encyclopedia Ii - Projective Line - Examples
Projective line - Real projective line. The projective line over the real numbers is called the real projective line. It is given by pr...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Use Of Infinity In Common Speech
In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forev...   » Read the article

Encyclopedia - Kleene Algebra: Encyclopedia Ii - Kleene Algebra - Properties
Zero is the smallest element: 0 ≤ a for all a in A. The sum a + b is the least upper bound of a and b: we have a ≤ a + b and b ≤ a ...   » Read the article

Encyclopedia - Limit Of A Function: Encyclopedia Ii - Limit Of A Function - Examples
Limit of a function - Real-valued functions. Limit of a function - Functions on metric spaces. If z is a complex number with |z| <...   » Read the article

Encyclopedia - Limit Of A Function: Encyclopedia Ii - Limit Of A Function - Properties
To say that the limit of a function f at p is L is equivalent to saying for every convergent sequence (xn) in M with limit equal to p, t...   » Read the article

Encyclopedia - Kleene Algebra: Encyclopedia Ii - Kleene Algebra - Examples
Let Σ be a finite set (an "alphabet") and let A be the set of all regular expressions over Σ. We consider two such regular expressions ...   » Read the article

Encyclopedia - Improper Integral: Encyclopedia Ii - Improper Integral - Vertical Asymptotes At Bounds Of Integration
Consider This integral involves a function with a vertical asymptote at x = 0. One can evaluate this integral by evaluating from b to ...   » Read the article

Encyclopedia - Limit Of A Sequence: Encyclopedia Ii - Limit Of A Sequence - Properties
Consider the following function: f(x)=xn if n-1<x≤n. Then the limit of the sequence of xn is just the limit of f(x) at infinity. A f...   » Read the article

Encyclopedia - Limit Of A Sequence: Encyclopedia Ii - Limit Of A Sequence - History
The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involved limiting processes. Leucippus, Democritus, Antiphon,...   » Read the article

Encyclopedia - Projective Line: Encyclopedia Ii - Projective Line - Symmetry Group
Quite generally, the group of Möbius transformations with coefficients in K acts on the projective line P1(K). This group action is tran...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Alternative Formulations
If f is non-negative, then ∫f dμ is precisely the area under the curve as measured by the product measure μ × λ where λ is the Leb...   » Read the article

Encyclopedia - Metric Mathematics: Encyclopedia Ii - Metric Mathematics - Relation Of Norms And Metrics
Given a normed vector space (X,||.||) we can define a metric on X by d(x,y):=||x-y||. The metric d is called induced by ||.||. Converse...   » Read the article

Encyclopedia - Infimum: Encyclopedia Ii - Infimum - Infima In Partially Ordered Sets
The definition of infima easily generalizes to subsets of arbitrary partially ordered sets and as such plays a vital role in order theory...   » Read the article

Encyclopedia - Supremum: Encyclopedia Ii - Supremum - Suprema Within Partially Ordered Sets
Least upper bounds are important concepts in order theory, where they are also called joins (especially in lattice theory). As in the spe...   » Read the article

Encyclopedia - Metric Mathematics: Encyclopedia Ii - Metric Mathematics - Equivalence Of Metrics
For a given set X two metrics d1 and d2 are called topological equivalent (uniformly equivalent) if the identity mapping id: (X,d1) → ...   » Read the article

Encyclopedia - Metric Mathematics: Encyclopedia Ii - Metric Mathematics - Notes
These conditions express intuitive notions about the concept of distance. For example, that the distance between distinct points is posit...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Morphisms Of Complete Lattices
The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are charact...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Representation
There are various other mathematical concepts that can be used to represent complete lattices. One means of doing so is the Dedekind-MacN...   » Read the article

Encyclopedia - Metric Mathematics: Encyclopedia Ii - Metric Mathematics - Definition
A metric on a set X is a function (called the distance function or simply distance) d : X × X → R (where R is the set of real num...   » Read the article

Encyclopedia - Supremum: Encyclopedia Ii - Supremum - Comparison With Other Order Theoretical Notions
Supremum - Greatest elements. The difference between the supremum of a set and the greatest element of a set may not be immediately obv...   » Read the article

Encyclopedia - Supremum: Encyclopedia Ii - Supremum - Least-upper-bound Property
The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. ...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Mathematical Infinity
Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Use Of Infinity In Common Speech
In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forev...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Physical Infinity
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Proof Techniques
To illustrate some of the proof techniques used in Lebesgue integration theory, we sketch a proof of the above mentioned Lebesgue monoton...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Basic Theorems Of The Lebesgue Integral
The Lebesgue integral does not distinguish between functions which only differ on a set of μ-measure zero. To make this precise, functio...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Introduction
The integral of a function f between limits a and b can be interpreted as the area under the graph of f. This is easy to understand for f...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Limitations Of The Riemann Integral
Here we discuss the limitations of the Riemann integral and the greater scope offered by the Lebesgue integral. We presume a working unde...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Infinity In Science Fiction
The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, m...   » Read the article