Sergei Yakovenko's blog: on Math and Teaching

Tuesday, November 10, 2009

Lecture 2 (Thu, Nov 12, 13:30 – 16:30)

Existence of limits and completeness of the real numbers system

  • Monotonicity and its implications.
  • Nested intervals and their common point
  • Boundedness as another property stable by finite alterations
  • Converging subsequense of  a bounded sequence
  • But why we are so sure that there are no gaps on the real line? And what is a real line?

Construction of the number system: from natural numbers toward scary numbers

  • Completion by algebraic operations: from \mathbb N to \mathbb Q via \mathbb Z. Everything you need to solve linear equations
  • Problems  with quadratic equations: irrationalities and negative discriminants. An idea of algebraic number.
  • Problems with transition to limit: the ubiquitous \pi and much, much more
  • Infinite decimal fractions: completion by “adding limits of monotone sequences”.
  • Operations with real numbers: ordered field. Completeness “axiom”.
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