Syllabus
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
Annotation
This course provides an introduction to the concepts of modern mathematics which are essential for modern reasoning and exact science. We will discuss the development of algebra and illustrate it with the outline of impossibility arguments for the old Greek problems of the "squaring of the circle" (whether by the compass and straightedge one can draw a ractangle with the same area as the given circle) and "angle trisection" (whether every angle can be trisected using the compass and straightedge).
We will also discuss the introduction of infinitesimal methods: the concepts of limit, continuity, derivation and integration which are essential to analyzing natural phenomena and everyday processes (rate of change, optimal solutions to given problems, etc.).