Fundamental Concepts of Mathematics

Fundamental Concepts of Mathematics

2nd Edition - January 1, 1979

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  • Author: R. L. Goodstein
  • eBook ISBN: 9781483154053

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Fundamental Concepts of Mathematics, 2nd Edition provides an account of some basic concepts in modern mathematics. The book is primarily intended for mathematics teachers and lay people who wants to improve their skills in mathematics. Among the concepts and problems presented in the book include the determination of which integral polynomials have integral solutions; sentence logic and informal set theory; and why four colors is enough to color a map. Unlike in the first edition, the second edition provides detailed solutions to exercises contained in the text. Mathematics teachers and people who want to gain a thorough understanding of the fundamental concepts of mathematics will find this book a good reference.

Table of Contents

  • Preface to the Second Edition


    Chapter 1 Numbers for Counting

    Definition of Counting. Addition Positional Notation; Commutative and Associative Properties; Recursive Definition

    Mathematical Induction. Inequality. Subtraction. Multiplication

    Shortcuts in Multiplication. The Distributive Property

    Prime Numbers; the Infinity of Primes

    Division; Quotient and Remainder

    Exponentiation; Representation in a Scale. The Counterfeit Penny Problem. Tetration

    The Arithmetic of Remainders. Rings and Fields. The Fundamental Theorem of Arithmetic

    The equation ax-by-1; the Measuring Problem and the Explorer Problem

    Groups. Isomorphism. Cyclic Groups. Normal Subgroups; the Normalizer, the Center, the Factor Group

    Semi-groups. The Word problem for Semi-groups and for Groups

    Congruences. Format's Theorem. Tests for Divisibility

    Tests for Powers

    Pascal's Triangle; Binomial Coefficients

    Ordinal Numbers; Transfinite Ordinals; Transfinite Induction

    Chapter 2 Numbers for Profit and Loss and Numbers for Sharing

    Positive and Negative Integers. Addition, Subtraction and Multiplication of Integers. The Ring of Integers


    Numbers for Sharing. Addition, Multiplication and Division of Fractions

    Inequalities. Enumeration of Fractions

    Farey Series. Index Laws

    The Field of Rational Numbers. Negative Indices; Fractional Indices. The Square Root of 2. The Extension Field x+y/2

    Polynomials. The Remainder Theorem. Remainder Fields

    Enumeration of Polynomials

    Examples I

    Solutions to Examples I

    Chapter 3 Numbers Unending

    Decimal Fractions; Terminating and Recurring Decimals

    Addition, Subtraction and Multiplication of Decimals

    Irrational Decimals. Positive and Negative Decimals

    Examples II

    Solutions to Examples II

    Convergence; some Important Limits. Generalized Binomial Theorem

    Examples III

    Solutions to Examples III

    Sequence for e. The Exponential Series


    Intervals. Limit Point. Closed Sets and Open Sets. Closure. Interior Points. Denumerable Sets. Finite Sets. Infinite Sets. Sequence. Null Sequence

    Continuity. Functions. Function of a Function. Inverse Functions

    Examples IV

    Solutions to Examples IV

    Integration; Increasing Functions. Integral of a Sum

    Differentiation. Derivative of an Integral; of a Sum, Product, Quotient and Composite Function. The Exponential and Logarithmic Functions. The Logarithmic Series

    The Circular Functions; the Evaluation of π

    Examples V

    Solutions to Examples V

    Pretender Numbers. Dyadic Numbers. Pretender Difference and Convergence; Pretender Limit

    Chapter 4 Sets and Truth Function

    Union and Intersection of Sets. Distributive Law. Complement of a Set. Inclusion; Partial Order. Boolean Arithmetic; Axiomatic Theory

    Sentence Logic. Truth Tables. Representing Functions

    Switching Circuits. Three Pole and Four Pole Switches

    Axiomatic Theory; Consistency, Completeness and Independence of the Axioms. The Deduction Principle. Truth Tables as a Decision Method for Sentence Logic

    Impossibility of a Decision Method for Arithmetic. Incompleteness of Arithmetic

    Hilbert's Tenth Problem

    Examples VI

    Solutions to Examples VI

    Chapter 5 Networks and Maps

    Connectivity. Networks. Königsberg Bridges Problem

    Necessary and Sufficient Conditions for a Traversable Network

    Euler's Formula. Characteristic of a Surface. The Regular Solids

    Map Coloring. Two-color, Three-color, Four- and Five-color Maps

    Maps on Anchor Rings; on Möbius Bands

    Metric Spaces; Neighborhood, Open Set. Limit Point, Closure. Continuous Mappings

    Topological Space; Open Sets, Neighborhoods, Closure

    Continuous Mappings; Necessary and Sufficient Conditions

    Chapter 6 Axiomatic Theory of Sets


    Other Titles in the Series

Product details

  • No. of pages: 334
  • Language: English
  • Copyright: © Pergamon 1979
  • Published: January 1, 1979
  • Imprint: Pergamon
  • eBook ISBN: 9781483154053

About the Author

R. L. Goodstein

About the Editor

I. N. Sneddon

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