The book is based on lecture notes of a course 'from elementary number theory to an introduction to matrix theory' given at the Technion to gifted high school students. It is problem based, and covers topics in undergraduate mathematics that can be introduced in high school through solving challenging problems. These topics include Number theory, Set Theory, Group Theory, Matrix Theory, and applications to cryptography and search engines.
Author(s): Abraham Berman
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 162
City: Singapore
Contents
Introduction
Introduction to the Course
1 Algebraic Structures
1.1 Groups, Fields and Rings
1.2 Polynomials
1.3 Hints
1.4 Solutions
1.5 Notes
1.5.1 Prestigious prizes in mathematics
2 The Natural Numbers
2.1 What is Induction?
2.2 Mathematical Induction
2.3 Easy to State Open Problems
2.4 Tiling and Geometry Problems
2.5 Hints
2.6 Solutions
2.7 Notes
2.7.1 Fermat’s Last Theorem
2.7.2 The Catalan Conjecture
2.7.3 Euler
2.7.4 Euclid
2.7.5 Gauss
2.7.6 Newton and Leibniz
2.7.7 Collatz and Tau
2.7.8 Sylvester–Gallai theorem
3 The Integers
3.1 The Greatest Common Divisor
3.2 Congruence
3.3 Hints
3.4 Solutions
4 The Real Numbers
4.1 Sequences and Rational Numbers
4.2 Irrational Numbers
4.3 Hints
4.4 Solutions
4.5 Notes
5 Introduction to Set Theory
5.1 Countable Sets
5.2 Uncountable Sets
5.3 Hints
5.4 Solutions
5.5 Notes
5.5.1 Cantor, Fraenkel, Russel and Zermelo
5.5.2 Hilbert’s 23 Problems
5.5.3 Gödel and Cohen
5.5.4 Bernstein and Schröder
6 The Pigeonhole Principle and the Base 2 Number System
6.1 The Pigeonhole Principle
6.2 The Base 2 Number System
6.3 Hints
6.4 Solutions
6.5 Notes
6.5.1 Dirichlet
6.5.2 Ask Marilyn
6.5.3 The Erdos–Szekeres Theorem
7 Introduction to Group Theory
7.1 Subgroups
7.2 Lagrange’s, Euler’s and Fermat’s Theorems
7.3 The RSA Public Key Cybersystem
7.4 Permutations
7.5 Hints
7.6 Solutions
8 Introduction to Matrix Theory
8.1 Matrices
8.2 Graphs and Matrices
8.3 Hints
8.4 Solutions
9 Fibonacci Numbers, Determinants and Eigenvalues
9.1 The Fibonacci Sequence
9.2 Determinants
9.3 Eigenvalues and Eigenvectors
9.4 The Zeckendorf Representation of the Natural Numbers
9.5 Hints
9.6 Solutions
9.7 Notes
9.7.1 Fibonacci
9.7.2 The golden ratio
9.7.3 Cayley and Hamilton
9.7.4 The Friendship Theorem
10 The Mathematics Behind Google’s Page Rank and a Game of Numbers
10.1 Page Rank
10.2 Back to the Numbers on the Pentagon Problem
10.3 Notes
10.3.1 Perron and Frobenius
10.3.2 Brin and Page
10.3.3 Alon, Peres, Mozes and Eriksson
Bibliography
Index
