Coursera Flash Sale
40% Off Coursera Plus for 3 Months!
Grab it
Explore fundamental concepts in real number theory and limits through this comprehensive mathematics course that challenges conventional approaches to mathematical foundations. Examine the logical inconsistencies and practical difficulties inherent in traditional treatments of real numbers, starting with an investigation of inconvenient truths about √2 and progressing through measurement theory and interval arithmetic. Master Newton's method for finding zeros and approximating roots, then apply these techniques to solve quadratic and cubic equations and work with algebraic curves. Delve into critical examinations of logical weaknesses in modern pure mathematics and the decline of mathematical rigor, while studying fractions, repeating decimals, and p-adic numbers. Investigate the problems with representing real numbers as infinite decimals and explore the mysteries surrounding π. Analyze the difficulties with limits and Cauchy sequences, then discover the deep structure of rational numbers through the Stern-Brocot tree and its connections to matrices and wedges. Question fundamental assumptions about sequences and infinity, examining what "infinite sequences" actually represent and building understanding of on-sequences and their challenges. Learn about limits through rational polynomial on-sequences and explore extensions of arithmetic to include infinity, culminating in practical applications of extended rational numbers and a thorough examination of what limits truly represent mathematically.
