Note: R(x) is actually a proper subfield of the hyperreals *R. Real Analysis Study help for Baby Rudin, Part 1.5. The ordered field of real numbers R satisfies the Archimedean property. Given a positive real number x and another real number y, there exists a natural number n so that n*x is greater than y. Principles of Mathematical Analysis (Baby Rudin): [ Ссылка ]

If we only use the ordered field properties of R, it is not sufficient to prove the Archimedean property. We need the Completeness Axiom as well (least upper bound property). There are non-Archimedean ordered fields, and we consider the example of the field of rational functions R(x) with coefficients in R. It is an example of a field of fractions for the integral domain of polynomials with coefficients in R. We can visualize R(x) as corresponding to a subfield of the hyperreal numbers.

#realanalysis #archimedeanproperty #hyperreals

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(0:00) Introductory problem
(0:47) A simpler example visualized on Mathematica
(1:35) This video is about the Archimedean Property
(2:00) General question and possible answer
(3:31) The Archimedean Property of R and proof
(6:10) The Archimedean Property of the rationals Q
(7:29) An important related fact and infinitesimals
(8:13) A non-Archimedean ordered field: R(x) = the field of rational functions with coefficients from R
(12:03) R(x) as a subfield of the hyperreal numbers
(12:47) Visualizing hyperreal numbers in R(x)
(15:57) Relationship to R(x).

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