Inconvenient Truths About the Square Root of 2 - Real Numbers and Limits

Explore the historical and mathematical complexities surrounding the concept of the square root of 2 in this 42-minute lecture from the MathFoundations series. Delve into the ancient Greek discovery of incommensurability, examine three modern approaches to understanding √2, and critically analyze the logical challenges posed by irrational numbers in contemporary mathematics. Learn about the Pythagoreans' realization, Euclid's contribution, and Stevin's introduction of irrational numbers into the decimal system. Investigate applied, algebraic, and analytic methods for dealing with √2, and understand why the analytic approach leads to significant problems in modern analysis. Gain insights into the foundational issues in mathematics and prepare for further exploration of the logical difficulties associated with real numbers, Cauchy sequences, and Dedekind cuts.

Introduction
The Pythagoreans
There is no rational which squares to 2
It's wrong to restate that the number square root of 2 is irrational
An applied approach
Applied approach is practical and important theoretically
Three cases arising in geometry
Algebraic approach
Analytic approach
Modern analysis