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Real Numbers. The Archimedes approach reinforced with Mathematica
Author

Evgenii M. Vorob'ev
Education level

College
Description

The Archimedes method for calculating π is well known. It is suprising that Archimedes obtained the approximate value of π but he had no clear idea of the real numbers. In his time, real numbers were no more than irrational numbers, i.e. numbers not represented by fractions p/q. Presumably, Archimedes considered {an} and {bn} as infinite sequences of approximate values of π.

From our point of view, the Archimedean pair ({an}, {bn}) is, in fact, the number {bn} itself.

This goal of this tutorial is to extend this treatment to all real numbers. We would like to consider real numbers as appropriately defined Archimedean pairs of rational sequences.
Subjects

*Mathematics
*Mathematics > Geometry
*Mathematics > Number Theory
Keywords

Archimedean pairs, rational sequences, Pythagorean theorem, natural logarithms, Cantor's axiom
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Real Numbers.nb (140.7 KB) - Mathematica Notebook