Number Theory and Geometry through History

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. The book develops an appreciation mathematics by not only looking at the work of individual, including Euclid, Euler, Gauss, etc. but also how mathematics developed from ancient civilizations. Brahmins (Hindu priests) devised our current decimal number system now adopted throughout the world. The concept of limit, which is what calculus is all about, was not alien to ancient civilizations as Archimedes used a method similar to the Riemann sums to compute the surface area and volume of the sphere. No theorem is here cited in a proof that has not been proved earlier in the book. There are some exceptions when it comes to the frontier of current research. Appreciating mathematics requires more than thoughtlessly reciting first the ten by ten, then twenty by twenty multiplication tables. Many find this approach fails to develop an appreciation for the subject. The author was once one of those students. Here he exposes how he found joy in studying mathematics, and how developed a life-long interest in it he hopes to share. The book is suitable for high school teachers, as a textbook for undergraduate students and their instructors. It is a fun book for advanced readership interested in mathematics.

Table of Contents:
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 eπi + 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