Archimedean property

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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.

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

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.)

It is sometimes seen in the form:

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.

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; but every x > c/2 can't be infinitesmal: nx > nc/2 > 1). 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.)

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.

[edit] 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. (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.) 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.