# 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.