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Topic: Archimedean property


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 PlanetMath: Archimedean property
It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).
This is version 6 of Archimedean property, born on 2002-08-29, modified 2007-01-15.
Archimedean property follows as a consequence, hence is a theorem.
planetmath.org /encyclopedia/ArchimedeanProperty.html   (281 words)

  
 PlanetMath: real number
This definition is well-defined and does not depend on the choice of Cauchy sequences used to represent the equivalence classes.
The real numbers can also be defined as the unique (up to isomorphism) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism.
But the original use of the phrase “complete Archimedean field”; was by David Hilbert, who meant still something else by it.
planetmath.org /encyclopedia/MathbbR.html   (919 words)

  
 NationMaster - Encyclopedia: Archimedean property
In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers.
Roughly speaking, it is the property of having no infinite elements or (non-zero) infinitesimals (this is a precise definition for ordered fields).
The non-existence of nonzero infinitesimal real numbers follows from the least upper bound property of the real numbers, as follows: The set Z of infinitesimals is bounded above (by 1, or by any other positive non-infinitesimal, for that matter) and nonempty (because 0 is infinitesimal); therefore, it has a least upper bound c.
www.nationmaster.com /encyclopedia/Archimedean-property   (1597 words)

  
 PlanetMath: Archimedean semigroup
is a commutative semigroup with the property that for all
This is related to the Archimedean property of positive real numbers
This is version 1 of Archimedean semigroup, born on 2002-11-05.
planetmath.org /encyclopedia/ArchimedeanSemigroup.html   (77 words)

  
 Real closed field
A crucially important property of the real numbers is that it is an archimedean field, meaning it has the archimedean property that for any real number, there is an integer larger than it in absolute value.
A non-archimedean field is, of course, a field that is not archimedean, and there are real closed non-archimedean fields; for example any field of hyperreal numbers is real closed and non-archimedean.
The archimedean property is related to the concept of cofinality.
www.xasa.com /wiki/en/wikipedia/r/re/real_closed_field.html   (913 words)

  
 Quasi-regular polyhedra
The other Archimedean solids have all vertices equivalent but at least two types of faces and edges.
The Archimedean duals have all faces equivalent, but at least two types vertices and edges.
Another fourth property of these quasiregular polyhedra is that in each case they are the intersection of a Platonic solid compounded with its dual.
www.georgehart.com /virtual-polyhedra/quasi-regular-info.html   (613 words)

  
 ECONOPHYSICS TRADING - Science for Forex Management
Property is an asset which forms part of the portfolios of many investors, particularly institutional ones, along with equities and bonds.
A key issue in translating methods of analysis in financial markets to property data is whether they are applicable given the small number of data points available.
The correlations between different types and geographical locations of property tend to have far more true information and be more stable over time than is the case with financial data, despite the large number of observations available with the latter.
econophysics-trading.com /econophy.htm   (3745 words)

  
 Mixing Materials and Mathematics
The model surface with these properties is the helicoid-- the surface swept out by a horizontal line rotating at a constant rate as it moves at constant speed up a vertical axis.
Put a helicoid inside a vertical cylinder filled with fluid and you have an Archimedean screw, a rotation of which translates the fluid up or down.
They meet along the network of lines illustrated in Figure 1b.) It is not minimal (he acknowledges it) and it is not known whether or not there is a minimal surface close to it in the sense that the gyroid is close to the zero set of the equation above.
www.msri.org /about/sgp/david/papers/nature96   (1752 words)

  
 Analysis WebNotes: Chapter 02, Class 06   (Site not responding. Last check: )
Back in the Introduction we tried to prove the Archimedean Property of the real numbers.
Back in the last section we asserted that the rational numbers do not have the least upper bound property, and the reals do have this property.
It is now time to show that the rationals do not have the least upper bound property, and also that there are real numbers which are not rational.
www.math.unl.edu /~webnotes/classes/class06/class06.htm   (816 words)

  
 NationMaster - Encyclopedia: Extreme value theorem
Then, by the Archimedean property of the real numbers, for every m, there exists an x in [a, b] such that f(x) > m.
In general topology, the extreme value theorem is follows from the general fact that compactness is preserved under continuity, and the fact that a subset of the real line is compact if and only if it is both closed and bounded.
In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them.
www.nationmaster.com /encyclopedia/Extreme-value-theorem   (1109 words)

  
 Index: Platonic and Archimedean Solids (69-79)
These are distinguished by the property that they have equal and regular faces, with the same number of faces meeting at each vertex.
A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, but the numbers of vertices and faces are interchanged).
The seven Archimedean solids not shown here are the truncated dodecahedron, truncated icosahedron, cuboctahedron, rhombicosidodecahedron, truncated icosidodecahedron, snub cube, and snub dodecahedron.
math.arizona.edu /~models/Platonic_and_Archimedean_Solids   (376 words)

  
 Archimedean - Qwika
Archimedean Archimedean means of or pertaining to or named...
Archimedean spiral Three 360° turnings of one arm of an Archimedean spiral An Archimedean spiral (also arithmetic spiral) is a curve...
Archimedean field In mathematics, an Archimedean field is an ordered field with the Archimedean property, named after the ancient Greek mathematician...
www.qwika.com /find/Archimedean   (478 words)

  
 Advanced Calculus
It will take you through the property of real numbers, as a totally ordered complete field and also as the metric completion of the rational numbers.
The Bolzano Weierstrass property and that every Cauchy sequence in the real numbers is convergent are just the manifestation of the complete metric space like the real numbers or R^n.
Though metric property is not explicitly mentioned, the technique from metric space is used as it is most efficient and conceptually clearer.
www.math.nus.edu.sg /~matngtb/Calculus/Calculus2/Calculus2.htm   (1552 words)

  
 Analysis of the Archimedes' Property used for the calculation of the circle perimeter
It is interesting to determined why the result of the classical Pi first determined by Archimedes is untrue and why the modern transcendental Pi that was based on the Archimedes' method is also untrue due to the same error analysis.
At first it is presented the spatial property of the space according to the first postulate of the Dakhiometry on the matter.
The quantum consciousness is the property of the Space to be a whole of the absolute determined locations.
www.dakhi.com /somen6.php   (677 words)

  
 ARCHIMEDEAN PROPERTY : Encyclopedia Entry
Specifically, it is the property of having no (non-zero) infinitesimals.
If x is a positive number (or positive element of any ordered algebraic structure), then x is infinitesimal if there exists a positive element y such that, for every natural number n, the multiple nx is less than y.
(The Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.)
www.bibleocean.com /OmniDefinition/Archimedean_property   (597 words)

  
 Platonic and Archimedean Polyhedra
The purple Archimedean solids have the interesting property of having right-handed and left-handed forms.
Several Archimedean solids can be broken down into parts that can be rotated against each other to produce new polyhedra with less symmetry.
All of these rotations will also produce some vertices with different arrangements of the constituent polygons except one, the "pseudo-rhombicuboctohedron," derived from the rhombicubotohedron, where the arrangement of all the vertices is retained (but there are differing arrangements of the polygons around each square).
www.friesian.com /polyhedr.htm   (633 words)

  
 Science Fair Projects - Archimedean property
In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals.
Structures that lack infinitesimals are called Archimedean; those that possess infinitesimals are non-Archimedean.
The non-existence of nonzero infinitesimal real numbers follows from the least upper bound property of the real numbers, as follows.
www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Archimedean_property   (406 words)

  
 Molnar's ERA - Reading Assignments (ch. 1)
Notes as you read the section: S is used throughout as the set of all numbers n for which a particular statement about n is true.
The Well-Ordering Property, however, is applied to the set of positive integers which are not in S.
The Archimedean Property of the Real Numbers says that "If a and b are positive real numbers, then there exists a positive integer n such that na>b".
www.stolaf.edu /people/molnar/theory/244/reading1.html   (782 words)

  
 dating Archimedean_field - dating-report.com   (Site not responding. Last check: )
In mathematics, an Archimedean field is an ordered field with the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse.
Then, an Archimedean field F is one such that for any x in F there exists n in the natural numbers N for which x
Archimedean fields are important in the axiomatic construction of real numbers.
www.dating-report.com /Archimedean_field   (307 words)

  
 Judicial Tyranny at yazadjal.com
If they are seeing property development today, the resultant gains should not be denied to these victims of ‘rural development’.
Similarly, a healthy mix of commercial and residential properties in a locality is a natural outcome of liberty, and is in the interest of the residents.
It hardly matters whether they are urban villager or builder, the fact is that that they have violated the law of land and should not be spared at all.
www.yazadjal.com /2006/03/14/judicial-tyranny   (5086 words)

  
 Four Dimensional Figures Page
Below are tabulated the six Archimedean prisms and antiprisms that occur as cells of the nonprismatic convex Archimeadean polychora.
(This property is called transitivity: the vertices of the polytope are transitive under its symmetry group.) It should be obvious that all the edges of any uniform polytope in n dimensions have the same length, and that all the faces (elements of dimension 2) are always regular polygons.
The existence of exactly those 18 convex uniform polyhedra is a fundamental property of three-dimensional Euclidean space, and the existence of the 64 corresponding polychora is likewise a fundamental property of four-dimensional Euclidean space.
members.aol.com /Polycell/uniform.html   (4265 words)

  
 Archimedean group
In abstract algebra, a branch of mathematics, an Archimedean group is an algebraic structure consisting of a set together with a binary operation and binary relation satisfying certain axioms detailed below.
We can also say that an Archimedean group is a linearly ordered group for which the Archimedean property holds.
For example, the set R of real numbers together with the operation of addition and usual ordering relation (≤) is an Archimedean group.
www.kiwipedia.com /archimedean-group.html   (86 words)

  
 The Myth of the Right to Life
But it is dubious - to say the least - that it has a right to go on using the mother's body, or resources, or to burden her in any way in order to sustain its own life.
The flip-side of this perception is that life is communal property.
Given such lack of equivocation, the amount of dilemmas and controversies surrounding the right to life is, therefore, surprising.
samvak.tripod.com /life.html   (2841 words)

  
 Read This: Real Analysis - A Historical Approach
Yes, of course, I knew that Newton and Leibniz were the "parents" of calculus, that Archimedes must have had something to do with the Archimedean Property, but I never took the time to find out what each of these people actually did.
I wondered how they actually reasoned, what their mathematical statements sounded like, how rigorous their arguments were, given their knowledge at the time.
And so the first five chapter of the book came as a wonderful surprise to me. Not only did I find out that Archimedes did discover his namesake property, but also what he was doing when he stumbled onto it, and how he felt the need for "epsilon-proofs" centuries before they were developed.
www.maa.org /reviews/stahl.html   (566 words)

  
 Math Forum Discussions
Certain properties of the real line play a key role.
is essentially the same as the Archimedean property.
is with these properties-- not just that they're there, but how they
www.mathforum.org /kb/message.jspa?messageID=297813&tstart=0   (400 words)

  
 math lessons - Archimedean property
In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals.
Structures that lack infinitesimals are called Archimedean; those that possess infinitesimals are non-Archimedean.
The non-existence of nonzero infinitesimal real numbers follows from the least upper bound property of the real numbers, as follows.
www.mathdaily.com /lessons/Archimedean_property   (266 words)

  
 The Dispatch - Serving the Lexington, NC - News   (Site not responding. Last check: )
An algebraic structure is Archimedean if it has no infinite elements and it has no infinitesimal elements.
Even without the multiplicative structure of the ring, however, the more formal defintion of Archimedean and non-Archimedean holds.
Origin of the name of the Archimedean property
www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Archimedean_principle   (714 words)

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