Explore the intricacies of fractions and repeating decimals in this comprehensive 49-minute video lecture from the Math Foundations series. Delve into the fundamental challenges surrounding real numbers, distinguishing between computable and uncomputable irrational numbers. Learn algorithms for converting between fractions and repeating decimals, and discover the nuanced operations of addition and multiplication for repeating decimals. Examine the logical flaws in current real number theory and consider a shift towards rational number-based mathematics. Topics covered include rational numbers in the continuum, approaches to real numbers, decimal arithmetic, and the relationship between fractions and repeating decimals. Gain valuable insights into the future of mathematics and challenge traditional perspectives on number systems.