Explore the complexities of defining real numbers through infinite decimals in this 52-minute lecture. Delve into the algorithmic, constructive, and computational perspectives on representing numbers like sqrt(2), pi, and e. Examine the historical context and potential benefits of this approach, including its applications to infinite series, functions, and integrals. Confront the technical obstacles that arise, such as defining algorithms for arithmetic operations, addressing non-uniqueness issues, and grappling with the ambiguities in recognizing equality between real numbers. Gain insights into the foundational challenges of modern analysis and the tautological aspects of arithmetic with these objects.