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. For example, see the Archimedean group. A small number x is classed as infinitesimal if the inequality always holds, no matter ho ...

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Archimedes (Greek: Αρχιμηδης ) (287 BC–212 BC) was an ancient mathematician, physicist, engineer, astronomer and philosopher born in the Greek seaport colony of Syracuse. He is considered by some math historians to be one of history's greatest mathematicians, along with possibly Newton, Gauss and Euler. Archimedes - Discoveries and inventions. Archimedes became a popular figure as a result of his involvement in the defense of Syracuse against the Roman siege in the First and Second Punic Wars. He ...

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• Archimedes - Discoveries and inventions
• Archimedes - Writings by Archimedes
• Archimedes - Quotes about Archimedes
• Archimedes - Named after Archimedes
• Archimedes - Notes

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Archimedes became a popular figure as a result of his involvement in the defense of Syracuse against the fluff-cow siege in the First and Second Puniiic Wars. He is reputed to have held the Romans at bay with war machines of his own design; to have been able to move a full-size ship complete with crew and cargo by pulling a single rope[1]; to have discovered the principles of density and buoyancy, also known as Archimedes' principle, while taking a bath (thereupon taking to the streets naked calling "WAA WAA" - "I have found it!"); and to have i ...

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Archimedes, Archimedes - Discoveries and inventions, Archimedes - Writings by Archimedes, Archimedes - Quotes about Archimedes, Archimedes - Named after Archimedes, Archimedes - Notes

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In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase. Algebraic structure - In the ...

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• Algebraic structure - In the sense of universal algebra
• Algebraic structure - Allowing axioms other than identities
• Algebraic structure - Allowing additional structure
• Algebraic structure - Categories

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In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. A number x is an infinitesimal iff for every integer n, |nx| is less than 1, no matter how large n is. In that case, 1/x is larger than any positive real number. Nonzero infinitesimals, obviously, are not real numbers, so "operations" on them are not familiar. Infinitesimal - History of the infinitesimal. The first mathematician ...

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• Infinitesimal - History of the infinitesimal
• Infinitesimal - Modern uses of infinitesimals

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In mathematics, an Archimedean field is an ordered field with the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse. In an ordered field F we can define the absolute value of an element x in F in the usual way by setting |x| = x for nonnegative x and |x| = −x for negative x. Then, an Archimedean field F is one such that for any x in F there exists n in the natu ...

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The proof consists of two parts — first, the proof of the existence of q and r, and secondly, the proof of the uniqueness of q and r. Division algorithm - Existence. Consider the set We claim that S contains at least one nonnegative integer. There are two cases to consider. If d < 0, then −d > 0, and by the Archimedean property, there is a nonnegative integer n such that (−d)n ≥ −< ...

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Division algorithm, Division algorithm - Statement of theorem, Division algorithm - Examples, Division algorithm - Proof, Division algorithm - Existence, Division algorithm - Uniqueness, Division algorithm - Generalisations

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In the subsequent, we use the notation na (where n is in the set N of natural numbers) for the sum of a with itself n times. An Archimedean group (G, +, ≤) is a linearly ordered group subject to the following condition: for any a and b in G which are greater than 0, the inequality na ≤ b fo ...

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Archimedean group, Archimedean group - Definition, Archimedean group - Examples of Archimedean groups, Archimedean group - Examples of non-Archimedean groups, Archimedean group - Theorems

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Proofs at a more advanced level draw on the axiomatic foundations of mathematics. They use careful and sound definitions of integers, fractions, real numbers, infinity, limits, and equality. The validity of manipulations at the elementary level is a logical consequence of these foundations. One requirement is to characterize numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractiona ...

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Proof that 0.999... equals 1, Proof that 0.999... equals 1 - Elementary proofs, Proof that 0.999... equals 1 - Fraction proof, Proof that 0.999... equals 1 - Algebra proof, Proof that 0.999... equals 1 - Advanced proofs, Proof that 0.999... equals 1 - Order proof, Proof that 0.999... equals 1 - Limit proof, Proof that 0.999... equals 1 - Generalizations, Proof that 0.999... equals 1 - Definitions and justifications

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List of order topics - Well-orders. Well-founded relation Ordinal number Well partial order List of order topics - Completeness properties. Semilattice Lattice (Directed) complete partial order, (d)cpo Bounded complete Complete lattice Knaster-Tarski theorem Infinite divisibility List of order topics - Orders with further algebraic operations. < ...

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List of order topics, List of order topics - Basic concepts, List of order topics - Distinguished elements of partial orders, List of order topics - Subsets of partial orders, List of order topics - Special types of partial orders, List of order topics - Well-orders, List of order topics - Completeness properties, List of order topics - Orders with further algebraic operations, List of order topics - Orders in abstract algebra, List of order topics - Functions between partial orders, List of order topics - Completions and free constructions, List of order topics - Domain theory, List of order topics - Orders in mathematical logic, List of order topics - Orders in topology

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In universal algebra, one studies algebraic structures consisting of a set and a collection of operations defined on the set which are required to satisfy certain identities. Simple structures Set: a set is a degenerate algebraic structure, one that has zero operations defined on it Pointed set: a set S with a distinguished element s of S Unary system: a set S with a unary operation, i.e. a function S → S Pointed unary system: a unary system with a distinguish ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

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The first mathematician to make use of infinitesimals was Archimedes, although he did not believe in their existence. See how Archimedes used infinitesimals. The Archimedean property is the property of an ordered algebraic structure of having no nonzero infinitesimals. When Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go: To find the derivative f'(x) of the function f(x) = x², let dx be an infinitesimal. The ...

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Infinitesimal, Infinitesimal - History of the infinitesimal, Infinitesimal - Modern uses of infinitesimals

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The first mathematician to make use of infinitesimals was Archimedes, although he did not believe in their existence. See how Archimedes used infinitesimals. The Archimedean property is the property of an ordered algebraic structure of having no nonzero infinitesimals. When Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go: To find the derivative f'(x) of the function f(x) = x², let dx be an infinitesimal. The ...

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Infinitesimal, Infinitesimal - History of the infinitesimal, Infinitesimal - Modern uses of infinitesimals

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Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation of the derivative of f(x) = x² can be justified with a minor modification: we have to talk about the standard part of the difference quotient, and the standard part of x + dx is x. Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical ...

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Infinitesimal, Infinitesimal - History of the infinitesimal, Infinitesimal - Modern uses of infinitesimals

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Specifically, the division algorithm states that given two integers a and d, with d ≠ 0 There exists unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d. The integer q is called the quotient r is called the remainder d is called the divisor ...

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Division algorithm, Division algorithm - Statement of theorem, Division algorithm - Examples, Division algorithm - Proof, Division algorithm - Existence, Division algorithm - Uniqueness, Division algorithm - Generalisations

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An ordered group (G, +, ≤) defined as follows is not Archimedean: G = R × R. Let a = (u, v) and b = (x, y) then a + b = (u + x, v + y) a ≤ b iff v < y or (v = y and u ≤ x). Proof: Consider the elements (1, 0) and (0, 1). For ...

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Archimedean group, Archimedean group - Definition, Archimedean group - Examples of Archimedean groups, Archimedean group - Examples of non-Archimedean groups, Archimedean group - Theorems

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Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation of the derivative of f(x) = x² can be justified with a minor modification: we have to talk about the standard part of the difference quotient, and the standard part of x + dx is x. Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical ...

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Infinitesimal, Infinitesimal - History of the infinitesimal, Infinitesimal - Modern uses of infinitesimals

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One broadening of the concept of algebraic structure is to study sets with operations that must satisfy axioms other than identities. Integral domain: a ring with 0 ≠ 1 that has no zero divisors other than 0 Division ring: an integral domain with an inverse operation (the inverse operation is not defined on the whole set) Field: a commutative division ring Although these structures undoubtedly have an algebraic flavor, they suffer from defects not found in universal algebra. For example, there doe ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

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Every algebraic structure has its own notion of homomorphism, a function that is compatible with the given operation(s). In this way, every algebraic structure defines a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This category, being a concrete category, may be regarded as a category of sets with extra structure in the category-theoretic sense. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a ca ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

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Elementary proofs assume that manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition. Proof that 0.999... equals 1 - Fraction proof. The standard method used to convert the fraction 1⁄3 to decimal form is long division, and the well-known result is 0.3333…, with the digit 3 repeating. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…; but 3 × 1⁄3 equals 1, so it must be the case that 0.9999… = 1. < ...

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Proof that 0.999... equals 1, Proof that 0.999... equals 1 - Elementary proofs, Proof that 0.999... equals 1 - Fraction proof, Proof that 0.999... equals 1 - Algebra proof, Proof that 0.999... equals 1 - Advanced proofs, Proof that 0.999... equals 1 - Order proof, Proof that 0.999... equals 1 - Limit proof, Proof that 0.999... equals 1 - Generalizations, Proof that 0.999... equals 1 - Definitions and justifications

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