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). Structures that lack such elements are called Archimedean; those that possess them are non-Archimedean. In particular, a linearly ordered group that is Archimedean is an Archimedean group, and an ordered field that is Archimedean is an Archimedean field. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... In mathematics, a linearly ordered group is both a group and a linearly ordered set, in which the group operation is in a certain sense compatible with the linear ordering. ... 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. ... An archimedean field is an ordered field with the archimedean property. ...

If x and y are positive numbers (or positive, non-zero elements of any ordered algebraic structure), 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 inequality A negative number is a number that is less than zero, such as −3. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

always holds, no matter how large the number (n) of terms in the sum may be, as long as this is finite. So the algebraic structure is Archimedean if no such x and y exist. (Authorities differ on whether zero is considered to be an infinitesimal; but in any case, zero does not count in the Archimedean property.) In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

Note that there is an absolute definition of infinitesimal and infinite elements in a ring by taking y = 1 (to define when x is infinitesimal) or by taking x = 1 (to define when y is infinite). Thus a ring is Archimedean if it has no infinite or infinitesimal elements. 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). Even without the multiplicative structure of the ring, however, there is a notion of Archimedean and non-Archimedean. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...

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. Suppose that 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 when c is positive). Therefore, c is not positive, so c = 0 is the only infinitesimal. (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, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In mathematics, constructive analysis is mathematical analysis done according to the principles of constructivist mathematics. ...

The concept is named after the ancient Greek geometer and physicist Archimedes of Syracuse. 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. 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. The Ancient Greek world, circa 550 BC Ancient Greece is the period in Greek history which lasted for around one thousand years and ended with the rise of Christianity. ... Archimedes of Syracuse. ... Syracuse (Italian, Siracusa, ancient Syracusa - see also List of traditional Greek place names) is a city on the eastern coast of Sicily and the capital of the province of Syracuse, Italy. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ... Heuristic is the art and science of discovery and invention. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

Example of a nonarchimedean ordered field

For an example of an ordered field that is not Archimedean, take the field of rational functions with real coefficients. (A rational function is any function that can be expressed as one polynomial divided by another polyomial; we will assume in what follows that this has been done in such a way that the leading coefficient of the denominator is positive.) 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). 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. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ...