Constructions of natural numbers, integers and rationals

Real numbers - motivation and construction

Modern algebra - abstractions and generalizations (groups, rings, fields)

Examples of use of algebra: an outline of an argument that the solution of the old problems of antiquity is impossible with just the compass and straighedge (squaring the circle, doubling the cube, trisecting an angle)

Mathematics as the tool for analysing the natural phenomena (rate of change, optimal solutions, etc.)

Developement of mathematical analysis - limits, continuity, derivation and integration