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.