Archimedean property

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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

<math>\left|x\right|+\cdots+\left|x\right|<1<math>

always holds, no matter how large is the finite number n of terms in this sum.

The non-existence of nonzero infinitesimal real numbers follows from the least upper bound property of the real numbers, as follows. If nonzero infinitesimals exist, then the set of all of them has a least upper bound c. Either c is infinitesimal or it is not. If c is infinitesimal, then so is 2c, but that contradicts the fact that c is an upper bound of the set of all infinitesimals (unless c is 0, so that 2c is no bigger than c). If c is not infinitesimal, then neither is c/2, but that contradicts the fact that among all upper bounds, c is the least (unless c is 0, so that c/2 is no smaller than c).

Archimedes of Syracuse 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. Nonetheless, Archimedes used infinitesimals brilliantly in mathematical arguments, although he denied that those were finished mathematical proofs.ru:Аксиома Архимеда

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