In abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups, fields, and other algebraic structures. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Roughly speaking, it is the property of having no infinitely large or infinitely small elements (i. e. no nontrivial infinitesimals). Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have This can be made precise in various contexts, for example, for fields with an absolute value, where the ordered field of real numbers is Archimedean, but the field of p-adic numbers with the p-adic absolute value is non-Archimedean. In Mathematics, an absolute value is a function which measures the "size" of elements in a field or Integral domain. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897
An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is called Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is called non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group, and a field with a non-Archimedean absolute value is a non-Archimedean field. In Abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "&le" In Abstract algebra, a branch of Mathematics, an Archimedean group is an Algebraic structure consisting of a set together with a Binary In Mathematics, an Archimedean field is an Ordered field with the Archimedean property, named after the ancient Greek mathematician Archimedes
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Definition
Let x and y be positive elements of a linearly ordered group. In Abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "&le" Then x is infinitesimal with respect to y (or equivalently, y is infinite with respect to x) if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds:
More generally, if K is an algebraic structure with a notion of 'size', for example, a field with an absolute value, a similar definition applies to K. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, If x is infinitesimal with respect to every positive element y, then x is an infinitesimal element. Likewise, if y is infinite with respect to every positive element x, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements.
Remarks
- In a ring, it suffices to consider y = 1 (to define when x is infinitesimal) or x = 1 (to define when y is infinite). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real
- In a field, it is enough to check only one of these conditions (since if x is infinitesimal, then 1/x is infinite, and vice versa). In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division
Archimedean property of the real numbers
In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows. Denote by Z the set consisting of all positive infinitesimals, together with zero. This set is bounded above by 1 (or by any other positive non-infinitesimal, for that matter) and nonempty. Therefore, Z has a least upper bound c. Suppose that the real number c is positive. Is c itself an infinitesimal? If so, then 2c is also an infinitesimal (since n(2c) = (2n)c < 1), but that contradicts the fact that c is an upper bound of Z (since 2c > c when c is positive). Thus c is not infinitesimal, so neither is c/2 (by the same argument as for 2c, done the other way), but that contradicts the fact that among all upper bounds of Z, c is the least (since c/2 < c; but every x > c/2 can't be infinitesimal: nx > nc/2 > 1). Therefore, c is not positive, so c = 0 is the only infinitesimal.
It is interesting to note that the Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context. In Mathematics, constructive analysis is Mathematical analysis done according to the principles of Constructivist mathematics.
Example of a non-Archimedean ordered field
For an example of an ordered field that is not Archimedean, take the field of rational functions with real coefficients. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In (A rational function is any function that can be expressed as one polynomial divided by another polynomial; we will assume in what follows that this has been done in such a way that the leading coefficient of the denominator is positive. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a coefficient is a Constant multiplicative factor of a certain object ) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now f > g if and only if f − g > 0, so we only have to say which rational functions are considered positive. Write the rational function in the form of a polynomial plus a remainder over the denominator, where the degree of the remainder is less than the degree of the denominator (using the Euclidean algorithm for polynomials). In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD Call the function positive if either (1) the leading coefficient of the polynomial part is positive, or (2) the polynomial part is zero and the leading coefficient of the remainder is positive. (One must check that this ordering is well defined and compatible with the addition and multiplication operations. ) By this definition, the rational function 1/x is positive but less than the rational function 1. In fact, if n is any natural number, then n(1/x) = n/x is positive but still less than 1, no matter how big n is. Therefore, 1/x is an infinitesimal in this field.
Origin of the name of the Archimedean property
The concept is named after the ancient Greek geometer and physicist Archimedes of Syracuse. The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in Archimedes stated that for any two line segments, laying the shorter end-to-end only a finite number of times will always suffice to create a segment exceeding the longer of the two in length. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points If we take the shorter line segment to have length x, then any (larger) positive real number y defines a longer line segment, so we recognise Archimedes' claim as the Archimedean property of real numbers. Nonetheless, Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs. Archimedes' use of infinitesimals is the first attested explicit use heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true
Because Archimedes credited it to Eudoxus of Cnidus it is also known as the Eudoxus axiom. Eudoxus of Cnidus ( Greek Εὔδοξος ὁ Κνίδιος (410 or 408 BC &ndash 355 or 347 BC was a Greek Astronomer, Mathematician
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