Università Roma Tre

NUMBERS
Counting at the beginning: first ideas about numbers. How to write numbers in history: Summers, Babylonians, Egyptians, Greeks and Romans the positional notation, the axiom of induction and Peano axioms. Cardinality, Cantor diagonal argument. Addition and multiplication. The ordering of natural numbers. Divisibility. The integers. The Euclidean division theorem, basis representation theorem for numbers. The GCD and Euclid algorithm. Prime numbers, their infinity, Eratosthenes, unique factorization theorem, Congruences and modular arithmetics. Rational numbers. The representation of rationals. Operations on rationals. Order and density of rationals .Cardinality of rationals. Decimal representation of rationals, base 10 and base 3 numbers representation. Irrationality of square root of 2, and of a prime. Continuity and the set of real numbers.

EUCLIDEAN GEOMETRY
The origin, Euclid and the axioms. Hilbert's observations. The first book of Euclid: Triangles and Pitaghoras Theorem. The geometric algebra and the second book of Elements. The sixth book of Elements and Thales. Polygons. Convex subsets and polygons, regular polygons. Equivalence in geometry: isoperimetry and equi-extension problems. Solid shapes in space. Analytic geometry: the cartesian plane, the equation of a straight line, the distance of two points and the equation of a circumference.

Giorgio Israel, Ana Millán Gasca Pensare in matematica, 2012, ed Zanichelli.
Ana Millán Gasca Numeri e Forme , 2016, ed Zanichelli.

Altro: Simonetta Di Sieno - Sandro Levi, Aritmetica di base, Mcgraw-Hill ed.,

The student is free of using these or any other book or web source she/he prefers, paying attention to their reliability.