Innovative Textbooks

Publishers of the Modules in Mathematics

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There are currently 13 Modules in Mathematics, all authored by Steven Roman. The titles are listed below, followed by a brief description of each, with tables of contents. Click on a title to see the description, or simply use the navigation keys to browse the entire list.

Table of Contents

The Mathematics of Social Science

Third Edition, 93 pages, ISBN 1-878015-20-6

An elementary discussion of how mathematics may be used in the social sciences. Requires only basic arithmetic skills. Chapter 1 contains a discussion of how to form a group ranking of products, based on a set of individual rankings. The chapter concludes with a brief discussion of the famous Arrow Impossibility Theorem. Chapter 2 is devoted to measuring an individual's power, or influence, in a group setting. For instance, how much more power does a permanent member of the U.N. security council have than a temporary member? Chapter 3 contains a discussion of various methods for apportioning seats in the U.S. House of Representatives—a very important contemporary political problem. The final section gives a fascinating historical perspective on this 200 year old problem.

Chapter 1 Group Preferences
1.1 Picking a Winner
1.2 Picking a Group Ranking
Chapter 2 Power
2.1 The Banzhaf Power Index
2.2 Computing Banzhaf Indices
Chapter 3 Fair Apportionment
2.1 Quota Methods of Apportionment
2.2 Divisor Methods of Apportionment
2.3 The Apportionment Paradoxes
2.4 A Brief History of Apportionment

Graph Theory

Third Edition, 95 pages, ISBN 1-878015-24-9

Requiring no formal mathematical background, this modules provides a wide-ranging introduction to graph theory and its applications to various areas of applied science.

Chapter 1 Traversing Graphs
1.1 Eulerian Graphs
1.2 Hamiltonian Graphs
Chapter 2 Trees
2.1 Binary Search Trees
2.2 The Minimum Connector Problem
Chapter 3 Additional Applications
3.1 Project Management
3.2 Orienting the Edges of a Graph
3.3 The Traveling Salesman Problem

Codes and Coding

Third Edition, 68 pages, ISBN 1-878015-21-4

A survey of the contemporary topic of codes and coding. Prerequisites are minimal, since all of the necessary mathematics is developed in the module. Chapter 1 contains a discussion of the ubiquitous check digit codes that are used for error detection, and can be found in a wide variety of common circumstances, such as Universal Product Codes (Bar Codes), credit card numbers, bank check numbers, driver's licence numbers and ISBN's. We compare various commonly used methods and show which ones work best for detecting errors. Chapter 2 contains a discussion of the most famous code of all — the Hamming code for error correction. The final chapter discusses the Huffman coding scheme, which is used to encode data for space saving purposes (rather than for error detection or correction).

Chapter 1 Coding for Error Detection
1.1 Check Digit Methods
1.2 Comparing Check Digit Methods
Chapter 2 Coding for Error Correction
2.1 Hamming Codes
2.2 Another Approach to Hamming Codes
2.3 Why Hamming's Method Works
Chapter 3 Coding for Efficiency
3.1 Variable Length Codes
3.2 Huffman Encoding


Third Edition, 54 pages, ISBN 1-878015-19-2

This module requires no special mathematical background. Its aim is to acquaint the student with the basics of symbolic logic, such as how to correctly use DeMorgan's Laws, what the difference is between a conditional statement and its converse and how to recognize when an argument is logically valid. The module concludes with a brief discussion of how logic can be used to design circuits.

Chapter 1 Elementary Logic
1.1 Symbolic Form
1.2 Truth Tables
1.3 Logical Equivalence
Chapter 2 Applications of Logic
2.1 Valid Arguments
2.2 Logic Circuits

Counting and Probability

Third Edition, 102 pages, ISBN 1-878015-18-4

An introduction to counting techniques and elementary probability that requires only high school algebra.

Chapter 1 Elementary Counting
1.1 The Fundamental Counting Principle
1.2 Permutations
1.3 Combinations
Chapter 2 Elementary Probability
2.1 An Introduction to Probability
2.2 Independent Events; Binomial Probability
2.3 Conditional Probability; Expected Value
Chapter 3 Applications of Probability
3.1 Games of Chance
3.2 Single Gene Inheritance

Probability and Statistics

Second Edition, 90 pages, ISBN 1-878015-17-6

This module contains less probability than Counting and Probability, second edition or alternate second edition. However, it does include a chapter on rudimentary statistics.

Chapter 1 Elementary Counting
1.1 The Fundamental Counting Principle
1.2 Permutations
1.3 Combinations
Chapter 2 Elementary Probability
2.1 An Introduction to Probability
2.2 Expected Value
2.3 Binomial Probabilities
2.4 An Introduction to Genetics
Chapter 3 Elementary Statistics
3.1 Charting Data
3.2 Measures of Central Tendency
3.3 Measures of Dispersion

The Mathematics of Finance

Second Edition, 66 pages, ISBN 1-878015-23-0

Prerequisites are intermediate algebra. The goals of this module are to introduce the basic terminology related to interest, loans, leases and bonds; to show how various quantities, such as the monthly payments on a loan, can be computed using mathematical formulas; and to show how business calculators are designed to make these computations easier. Examples are done using a scientific calculator, the TI BA II Plus and the HP 10B. This module would make a nice supplement to a course in business calculus.

A Note on Calculators
Chapter 1 A Review of Exponents and Logarithms
1.1 Exponents and Logarithms
Chapter 2 Money, Money, Money
2.1 The Compounding of Interest
2.2 The Time Value of Money; Loans and Leases
2.3 A Closer Look at Annuities; Amortization Schedules
2.4 Bonds
Appendix: More on Exponents


Third Edition, 42 pages, ISBN 1-878015-16-8

An introduction to the fascinating field of secret messages, requiring no formal mathematical prerequisites for the first chapter, and an acquaintance with exponents for the second chapter. Chapter 1 describes some traditional, pre World War 2 methods for encoding messages. In Chapter 2, the author discusses one of the most used contemporary methods for encoding — the RSA method. At present, this method is believed to be secure, but may prove otherwise if efficient methods for factoring large numbers are ever discovered!

Chapter 1 Traditional Cryptology
1.1 Substitution Ciphers
1.2 The Vigenere and Playfair Ciphers
Chapter 2 Modern Cryptology
2.1 Public Key Cryptology -The RSA System

Applications of Exponents and Logarithms

Fourth Edition, 49 pages, ISBN 1-878015-22-2

This module shows how exponents and logarithms play a role in the processes of growth and decay. Prerequisites are intermediate algebra. After a short review of logarithms, there follows a discussion of compound interest. The next section is devoted to the time value of money, including how to compute the payments on an auto loan or home mortgage. Then comes a discussion of famous logarithmic scales, such as the Richter scale. The final section concerns the exponential growth of organisms and the decay of radioactive substances.

Chapter 1 A Review of Exponents and Logarithms
Chapter 2 Applications of Exponents and Logarithms
2.1 The Compounding of Interest
2.2 The Time Value of Money
2.3 The Logarithmic Scale
2.4 Exponential Growth and Decay

Modern Geometry

Second Edition, 51 pages, ISBN 1-878015-25-7

An introduction to some of the most interesting topics in modern geometry, including tilings of the plane and fractals.

Chapter 1 Polygons
1.1 Polygons and Symmetry
1.2 Tiling the Plane with Polygons
1.3 Polyhedra
Chapter 2 Fractals
2.1 Fractals

Topics in Mathematics

Second Edition, 52 pages, ISBN 1-878015-26-5

Each chapter of this module is independent of the others, and contains a different topic in mathematics. The only prerequisite is intermediate algebra. This module would make a nice supplement to a precalculus course.

Chapter 1 Mathematical Induction
1.1 Mathematical Induction
Chapter 2 Sequences and Series
2.1 Infinite Sequences
2.2 The Partial Sums of a Sequence
2.3 Infinite Series
Chapter 3 Finite and Infinite Sets
3.1 One-to-One Correspondences
3.2 Infinite Sets

Systems of Linear Equations

84 pages, ISBN 1-878015-11-7

An introduction to matrices and systems of linear equations, with an emphasis on applications. Prerequisites are high school algebra only.

Chapter 1 Systems of Linear Equations
1.1 Triangular Systems of Equations
1.2 Gaussian Elimination
1.3 Matrices and Gaussian Elimination
1.4 Applications Involving Systems of Linear Equations
Chapter 2 Matrices
2.1 The Algebra of Matrices
2.2 Solving Systems of Equations Using Matrix Algebra
2.3 The Adjacency Matrix of a Graph
Chapter 3 Topics
3.1 Linear Programming
3.2 Determinants and Cramer's Rule

Basic Algebra

126 pages, ISBN 1-878015-10-9

The first two chapters of a standard college algebra book, this module is designed for self-study and as a supplement to a calculus course for those students who need a little review or reference in algebra.

Chapter 1 Elementary Algebra
1.1 The Real Number System
1.2 Exponents and Radicals
1.3 The Complex Number System (Optional)
1.4 Complex Division (Optional)
1.5 Algebraic Expressions, Polynomials and Factoring
1.6 Rational and Other Expressions
1.7 Direct and Inverse Variation
Chapter 2 Equations and Inequalities
2.1 Solving Equations
2.2 Applications Involving Linear Equations
2.3 Quadratic Equations: Real Solutions
2.4 Quadratic Equations: Complex Solutions (Optional)
2.5 Applications Involving Quadratic Equations
2.6 Miscellaneous Equations
2.7 Linear and Absolute Value Inequalities
2.8 Quadratic and Other Inequalities
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