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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. A small number x is classed as
infinitesimal if the inequality
-
always holds, no matter how large is the 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 in mathematical arguments, although he denied that those
were finished mathematical proofs.
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