Faster access than browser!
47 relations: Abstract algebra, Algebraic structure, Ancient Greece, Archimedean group, Archimedes, Coefficient, Cofinal (mathematics), Construction of the real numbers, Constructive analysis, David Hilbert, Dense set, Euclid's Elements, Eudoxus of Cnidus, Field (mathematics), Group (mathematics), Heuristic, Hilbert's axioms, Hyperreal number, Infimum and supremum, Infinitesimal, Least-upper-bound property, Linearly ordered group, Local field, Magnitude (mathematics), Mathematical analysis, Mathematical proof, Monoid, Natural number, Neal Koblitz, Non-standard analysis, On the Sphere and Cylinder, Order type, Ordered field, Ostrowski's theorem, Otto Stolz, P-adic number, Polynomial, Proof by contradiction, Rational function, Real number, Ring (mathematics), Syracuse, Sicily, The Method of Mechanical Theorems, Triangle inequality, Ultrametric space, Upper and lower bounds, Valuation ring.
Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
New!!: Archimedean property and Abstract algebra · See more »
Algebraic structure
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.
New!!: Archimedean property and Algebraic structure · See more »
Ancient Greece
Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 13th–9th centuries BC to the end of antiquity (AD 600).
New!!: Archimedean property and Ancient Greece · See more »
Archimedean group
In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other.
New!!: Archimedean property and Archimedean group · See more »
Archimedes
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
New!!: Archimedean property and Archimedes · See more »
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
New!!: Archimedean property and Coefficient · See more »
Cofinal (mathematics)
In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition: This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”.
New!!: Archimedean property and Cofinal (mathematics) · See more »
Construction of the real numbers
In mathematics, there are several ways of defining the real number system as an ordered field.
New!!: Archimedean property and Construction of the real numbers · See more »
Constructive analysis
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.
New!!: Archimedean property and Constructive analysis · See more »
David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
New!!: Archimedean property and David Hilbert · See more »
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
New!!: Archimedean property and Dense set · See more »
Euclid's Elements
The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.
New!!: Archimedean property and Euclid's Elements · See more »
Eudoxus of Cnidus
Eudoxus of Cnidus (Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato.
New!!: Archimedean property and Eudoxus of Cnidus · See more »
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
New!!: Archimedean property and Field (mathematics) · See more »
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
New!!: Archimedean property and Group (mathematics) · See more »
Heuristic
A heuristic technique (εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.
New!!: Archimedean property and Heuristic · See more »
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry.
New!!: Archimedean property and Hilbert's axioms · See more »
Hyperreal number
The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.
New!!: Archimedean property and Hyperreal number · See more »
Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
New!!: Archimedean property and Infimum and supremum · See more »
Infinitesimal
In mathematics, infinitesimals are things so small that there is no way to measure them.
New!!: Archimedean property and Infinitesimal · See more »
Least-upper-bound property
In mathematics, the least-upper-bound property (sometimes the completeness or supremum property) is a fundamental property of the real numbers and certain other ordered sets.
New!!: Archimedean property and Least-upper-bound property · See more »
Linearly ordered group
In abstract algebra a linearly ordered or totally ordered group is a group G equipped with a total order "≤", that is translation-invariant.
New!!: Archimedean property and Linearly ordered group · See more »
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.
New!!: Archimedean property and Local field · See more »
Magnitude (mathematics)
In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind.
New!!: Archimedean property and Magnitude (mathematics) · See more »
Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
New!!: Archimedean property and Mathematical analysis · See more »
Mathematical proof
In mathematics, a proof is an inferential argument for a mathematical statement.
New!!: Archimedean property and Mathematical proof · See more »
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
New!!: Archimedean property and Monoid · See more »
Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
New!!: Archimedean property and Natural number · See more »
Neal Koblitz
Neal I. Koblitz (born December 24, 1948) is a Professor of Mathematics at the University of Washington.
New!!: Archimedean property and Neal Koblitz · See more »
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.
New!!: Archimedean property and Non-standard analysis · See more »
On the Sphere and Cylinder
On the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BC.
New!!: Archimedean property and On the Sphere and Cylinder · See more »
Order type
In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: X → Y such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in the correct order).
New!!: Archimedean property and Order type · See more »
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.
New!!: Archimedean property and Ordered field · See more »
Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
New!!: Archimedean property and Ostrowski's theorem · See more »
Otto Stolz
Otto Stolz (3 July 1842 – 23 November 1905) was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals.
New!!: Archimedean property and Otto Stolz · See more »
P-adic number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.
New!!: Archimedean property and P-adic number · See more »
Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
New!!: Archimedean property and Polynomial · See more »
Proof by contradiction
In logic, proof by contradiction is a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition.
New!!: Archimedean property and Proof by contradiction · See more »
Rational function
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
New!!: Archimedean property and Rational function · See more »
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
New!!: Archimedean property and Real number · See more »
Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
New!!: Archimedean property and Ring (mathematics) · See more »
Syracuse, Sicily
Syracuse (Siracusa,; Sarausa/Seragusa; Syrācūsae; Συράκουσαι, Syrakousai; Medieval Συρακοῦσαι) is a historic city on the island of Sicily, the capital of the Italian province of Syracuse.
New!!: Archimedean property and Syracuse, Sicily · See more »
The Method of Mechanical Theorems
The Method of Mechanical Theorems (Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes.
New!!: Archimedean property and The Method of Mechanical Theorems · See more »
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
New!!: Archimedean property and Triangle inequality · See more »
Ultrametric space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\.
New!!: Archimedean property and Ultrametric space · See more »
Upper and lower bounds
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.
New!!: Archimedean property and Upper and lower bounds · See more »
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.
New!!: Archimedean property and Valuation ring · See more »
Redirects here:
Archimedean axiom, Archimedean field, Archimedean ordered field, Archimedean ring, Archimedean semi-group, Archimedean semigroup, Archimedes axiom, Archimedes property, Archimedes' theorem, Archimedian Principle, Axiom of Archimedes, Axiom of archimedes, Axiom of continuity, Continuity axiom, Eudoxus axiom, Eudoxus' axiom, Non-Archimedean field, Non-archimedean field, Nonarchimedean field, Nonarchimedean ordered field, Theorem of Eudoxus.
References
[1] https://en.wikipedia.org/wiki/Archimedean_property
