This video begins a discussion on the role of irrationality in mathematics, starting with the "square root of 2". The difficulties with this concept go back to the ancient Greeks, as the Pythagoreans realized that the side and diagonal of a square were incommensurable. The Greeks realized that there was no rational number whose square was exactly two, a result which historically appeared in Euclid. In the modern age this idea that there were "irrational numbers" that could be incorporated into the Hindu-Arabic decimal number system was introduced by Stevin in 1585.

There are these days three approaches to "sqrt(2)": an applied one dealing with approximations, an algebraic one involving a finite field extension of the rational numbers, and an analytic one which attempts to apply the square root algorithm to assign to sqrt(2) an infinite decimal. It is this last approach which does not work, leading to serious logical problems with modern analysis.

This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. For those interested in this topic, be sure to watch the further videos in this series which go into considerable detail about the various problems with real numbers as infinite decimals, equivalence classes of Cauchy sequences, and Dedekind cuts. None of them work.

Video Content:
00:00 Introduction
4:06 The Pythagoreans
8:35 There is no rational which squares to 2
11:43 It's wrong to restate that the number square root of 2 is irrational
14:39 An applied approach
17:43 Applied approach is practical and important theoretically
20:53 Three cases arising in geometry
23:37 Algebraic approach
28:18 Analytic approach
35:12 Modern analysis

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Screenshot PDFs for my videos are available at the website [ Ссылка ]. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at [ Ссылка ]

My blog is at [ Ссылка ], where I will discuss lots of foundational issues, along with other things.

Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at [ Ссылка ] Please join us for an exciting new approach to one of mathematics' most important subjects!

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: [ Ссылка ]?
list=PL8403C2F0C89B1333
Year 9 Maths: [ Ссылка ]
Ancient Mathematics: [ Ссылка ]
Wild West Banking: [ Ссылка ]
Sociology and Pure Mathematics: [ Ссылка ]
Old Babylonian Mathematics (with Daniel Mansfield): [ Ссылка ]?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: [ Ссылка ]
Wild Trig: Intro to Rational Trigonometry: [ Ссылка ]
MathFoundations: [ Ссылка ]
Wild Linear Algebra: [ Ссылка ]
Famous Math Problems: [ Ссылка ]
Probability and Statistics: An Introduction: [ Ссылка ]
Boole's Logic and Circuit Analysis: [ Ссылка ]
Universal Hyperbolic Geometry: [ Ссылка ]
Differential Geometry: [ Ссылка ]
Algebraic Topology: [ Ссылка ]
Math Seminars: [ Ссылка ]
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And here are the Wild Egg Maths Playlists:

Triangle Centres: [ Ссылка ]
Six: An elementary course in pure mathematics: [ Ссылка ]
Algebraic Calculus One: [ Ссылка ]
Algebraic Calculus Two: [ Ссылка ]-
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