Sergei Yakovenko's blog: on Math and Teaching

Monday, November 25, 2013

Lecture notes for the first part of the course

Here is the syllabus of the first part of the course, together with the updated lecture notes.

Lecture 1 (2013-Analysis-01)

  • First encounters with infinity
  • Formal language of mathematics: sets, operations, functions, logical constructions, quantifiers…
  • Infinite operations (union, intersection, …)
  • Enumeration. Comparison of sets.
  • Sequences: stabilization, accumulation.

Lecture 2 (2013-Analysis-02)

  • Numbers: how we discover them and how we “build” them.
  • Arithmetic operations. Equations.
  • Adjunction missing solutions of equations that we want to be solvable.
  • Rational numbers: the structure of a field.
  • Why can’t one divide by zero and what is the price to pay if we allow such division
  • Irrational algebraic numbers: how to operate with them

Lecture 3 (2013-Analysis-03)

  • Order on rational numbers.
  • Density.
  • Holes between rational numbers
  • Infinite systems of two-sided inequalities
  • Dedekind cuts and operations on them.
  • Completeness of the new “number system” \mathbb R
  • Uncountability of the set of real numbers \mathbb R.

See you tomorrow to talk about convergence and limits.

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