Number Theory and Geometry through History

by J. S. Chahal

Overview

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

Number Theory and Geometry through History develops an appreciation of mathematics by not only looking at the work of individual, including Euclid, Euler, Gauss, and more, 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 here is 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 he developed a lifelong 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 text for advanced readership interested in mathematics.

Product Details

ISBN-13:	9781041011750
Publisher:	CRC Press
Publication date:	05/21/2025
Series:	Textbooks in Mathematics
Pages:	221
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Dr. J. S. Chahal is a professor of mathematics at Brigham Young University. He earned a PhD 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.

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