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Fundamental Concepts of Mathematics: Volume 2

by Steven K. Atiyah

702 pages; quality trade paperback (softcover); illustrated; two volumes; catalogue #05-0767; ISBN 1-4120-5866-x; US$59.95, C$75.00, EUR48.75, £33.78

The title of the book reflects very well its content. This two-volume set is extremely useful to all those who are interested in learning mathematics well.

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About the Book      About the Author      Excerpts      Catalogue Information

About the Book

This book covers quite a number of basic concepts of mathematics. It is well-structured, easy to read, and has numerous examples and adequate exercises

It consists of two parts. Part One consists of an introduction discussing the nature of mathematics - induction vs. deduction; the validity of an argument; the criteria for a proper definition; and a mathematical model testing completeness, consistency, and independence of a a set of axioms.

The next three chapters deal primarily with arithmetic- the laws of exponents; common fractions, decimal fractions, and percentages; rules of divisibility 2 through 12 and beyond; finding the GCD and LCM; using the Euclidean algorithm to find the GCD of two large numbers; finding the number of positive divisors of a given number; determining whether a given number is prime or composite, or whether it is rational or irrational; and many other basic rules and concepts of arithmetic.

The theory of sets is covered in chapter 4. Also covered here is the principle of duality, the binomial expansion theorem, and a proof showing that a repeating decimal is a rational number.

Chapter 5 discusses and proves the properties of the real numbers.

In Chapter 6, numbers are expressed in expanded form, and the processes of addition, subtraction, multiplication, and division are demonstrated using these expanded forms to justify the process of place-value carry-overs.

Mensuration, dimensional analysis, and measurement are discussed in chapter 7. The British and metric systems are introduced, and a bridge between them is established. The meaning of accurate, and precise measurements is given; and the greatest possible error and relative error of measurements are explained.

The second part of this book consists of detailed discussion of logic. Among the topics are quantifiers; application of logic to electric circuitry; and methods of proof. Following the chapter on logic is a chapter on relations, functions, and graphs - types of functions; algebra of functions; a practical method of graphing inverse relations; and simply, partially, and totally ordered sets are only a few examples of what was discussed. In chapter 10, patterns of sequences and other number bases are discussed - arithmetic, geometric, and other patterns of sequences are investigated.

Basic number theory proofs are discussed in chapter 11. All the rules of divisibility and what stated without proof in Part One are proven here.

Application problems in probability and consumer math are given in the last chapter. These include simple and compound interest laws; percent markups and markdowns; annuity; combination and permutation, and permutation of things where some are identical; and multistage experiments with tree diagrams.

It is to be noted here that what has been described in Part One and Part Two is but a small fraction of what the manuscript actually contains.

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About the Author

Education: The author has a B.S. in Chemical engineering with emphasis on petroleum refining; an M.S. and beyond in theoretical and applied math; and a ph.D. in Mathematics Education.

Experience: Two years of teaching Math in a H.S. setting and 42 years of teaching graduate and under-graduate math courses in college and universities.

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Excerpts

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Catalogue Information

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