Number Theory and Geometry through History

Auteur

Dr. Jasbir S. Chahal is a Professor of Mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University. After spending a couple of years at the University of Wisconsin as a postdoc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes in and has published several papers in number theory. For hobbies, he likes to travel and hike. His books, Fundamentals of Linear Algebra, and Algebraic Number Theory, are also published by CRC Press.

Texte du rabat

Developed from a course on the history of mathematics, the book is aimed at school teachers of mathematics who need to learn more about mathematics than its history, and in a way they can communicate to middle and high school students. The author hopes to overcome, through these teachers using this book, math phobia among these students.

Contenu

I Arithmetic

1 What is a Number?

1.1 Various Numerals to Represent

2 Arithmetic in Different Bases

3 Arithmetic in Euclid's Elements

4 Gauss-Advent of Modern Number Theory

4.1 Number Theory of Gauss

4.2 Cryptography

4.3 Complex Numbers

4.4 Application of Number Theory - Construction of Septadecagon

4.5 How Did Gauss Do It?

4.6 Equations over Finite Fields*

4.7 Law of Quadratic Reciprocity*

4.8 Cubic Equations*

4.9 Riemann Hypothesis*

5 Numbers beyond Rationals

5.1 Arithmetic of Rational Numbers

5.2 Real Numbers

II Geometry

6 Basic Geometry

7 Greece: Beginning of Theoretical Mathematics

8 Euclid: The Founder of Pure Mathematics

8.1 Some Comments on Euclid's Proof

9 Famous Problems from Greek Geometry

III Contributions of Some Prominent Mathematicians

10 Fibonacci's Time and Legacy

10.1 Liber Abaci

10.2 Liber Quadratorum

10.3 Equivalent Formulations of the Problems

11 Solution of the Cubic

11.1 Introduction

11.2 History

12 Leibniz, Newton, and Calculus

12.1 Differential Calculus

12.2 Integral Calculus

12.3 Proof of FTC

12.4 Application of FTC

13 Euler and Modern Mathematics

13.1 Algebraic Number Theory

13.2 Analytical Number Theory

13.3 Euler's Discovery of e i + 1 = 0

13.4 Graph Theory and Topology

13.5 Traveling Salesman Problem

13.6 Planar Graphs

13.7 Euler-Poincaré Characteristic

13.8 Euler Characteristic Formula

14 Non-European Roots of Mathematics

15 Mathematics of the 20th Century*

15.1 Hilbert's 23 Problems

1 Riemann Hypothesis

2 Poincaré Conjecture

3 Birch & Swinnerton-Dyer (B&S-D) Conjecture

15.2 Fermat's Last Theorem

15.3 Miscellaneous