Archimedean Property of the Real Numbers R, a Non-Archimedean Ordered Field, and Hyperreal Numbers

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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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  1. Bill Kinney
  2. archimedean property
  3. archimedean property proof
  4. archimedean property of r
  5. archimedean property of real numbers
  6. real analysis
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Комментарии
Автор

This lecture totally blows my mind, Dr Kinney. But it also explains the Archimedean Property and hyperreal numbers better than anything I have seen or read anywhere else. A true tour de force, sir! 👍

punditgi
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It is really a great video. Learned a lot of things

educationdunia
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It seems like 𝛆 = 1/𝛚 and 𝛚 = 1/𝛆, so a hyperreal number could be represented as a polynomial ax^n + bx^(n-1) + ... + cx^2 + dx^1 + ex^0 + fx^-1 + gx^-2 + ... + hx^-(m-1) + ix^-m, where x^0 is just 1, x^1 is 𝛚, and x^-1 is 𝛆.

angeldude
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Thankyou sooo much for sharing this content... If you are sharing much about non archimedean analysis, it will be very useful... Feeling very difficult to understand the basic concepts of Non archimedean... so will you please, ??

saranyan