Inconvenient truths about sqrt(2) | Real numbers and limits Math Foundations 80 | N J Wildberger

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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Elementary Mathematics (K-6) Explained:    / playlist  
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Year 9 Maths:    • Плейлист  
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