You are here

Understanding Mathematics

Keith Gregson
Publisher: 
Johns Hopkins University Press
Publication Date: 
2010
Number of Pages: 
153
Format: 
Paperback
Edition: 
2
Price: 
25.00
ISBN: 
9780801897023
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
01/3/2011
]

This is a confused and confusing treatment of lower-division college mathematics. According to the Preface it is aimed at those “who are struggling with mathematics, either as first-year undergraduates taking math as a subsidiary to a science course, or students taking AP courses”. From the body of the book it appears to be a refresher course that is not intended for those who have never seen the material before (it treats two years of college math in 150 pages).

The introductory chapters cover numbers in general, unit conversion, and dimensional analysis. These are not bad, except that the unit conversions are done ad hoc and no method is apparent, and there’s no mention of approximate values or significant digits. For example, on p. 13 we find that the age of the Earth is 4000000000 years, this being claimed to be equivalent to 4 x 109 years. Throughout the book numerical values are rounded off without any discussion.

The two chapters on calculus are the longest and weakest of the book. The explanation of the limit concept (p. 53) doesn’t make any sense, even if you already know what limits are, and it occurs out of context, ten pages before it is used. The weak explanation is not helped by the statement in two places (pp. 56, 73) that the limit of sin θ as θ goes to 0 is θ (not 0). The book makes reference to infinitesimals without explaining them (e.g., p. 98).

The chapters on Matrix Algebra and Statistics are not bad, although very skimpy. The book suggests on p. 122 that the way to solve a system of 100 equations in 100 variables is to find the matrix inverse, which nobody would do in real life. The book states incorrectly on p. 138 that the chi-square test is nonparametric, but its coverage is otherwise reliable.

At times the book seems self-defeating. For example, on p. 25 it states, “To a large extent the overt use of logarithms can now be avoided because of the power of the computer,” and immediately launches into a five-page discussion of the properties of logarithms. The book doesn’t mention that there are many applications where the items of interest grow logarithmically, and in fact it has such an example (the pH of a chemical solution) in the exercises for this section.

Bottom line: An extremely concise, but mathematically shaky, refresher course that might be useful to those who understood the material well at one time. Others should seek lengthier or more specialized texts.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

  1. Fundamentals
    1. 1.1 Why Mathematics?
    2. 1.2 What’s It All About?
      1. 1.2.1 x?
      2. 1.2.2 Mathematics?
      3. 1.2.3 Functions and Equations
      4. 1.2.4 Relationships
      5. 1.2.5 Why Don’t We Speak Mathematics All the Time?
      6. 1.2.6 And Why Do I Need to Understand It?
    3. 1.3 Working with Equations
      1. 1.3.1 Rearranging Equations
      2. 1.3.2 Order of Evaluating Algebraic Expressions
      3. 1.3.3 Some Useful Algebraic Relationships
      4. 1.3.4 A Word about Calculators
    4. 1.4 Exercises
    5. 1.5 Answers
  2. Numbers
    1. 2.1 Decimal Number Representation
      1. 2.1.1 Significant Figures and Decimal Places
      2. 2.1.2 Scientific Notation
    2. 2.2 Binary and Hexadecimal Numbers
      1. 2.2.1 Binary Numbers
      2. 2.2.2 Conversion from Decimal to Binary
      3. 2.2.3 Hexadecimal Numbers
      4. 2.2.4 Conversion from Decimal to Hexadecimal
      5. 2.2.5 Binary-Hex Conversion
    3. 2.3 Preliminary Calculations: Check the Problem
      1. 2.3.1 Dimension Analysis
        • Application: How Much Rain Flows into the Oceans?
      2. 2.3.2 A Rough Calculation on the Back of an Envelope
    4. 2.4 Exercises
    5. 2.5 Answers
  3. Powers and Logarithms
    1. 3.1 Powers and Indices
      1. 3.1.1 Some General Rules of Powers and Indices
      2. 3.1.2 Rules of Powers and Indices: Summary
    2. 3.2 Logarithms
      1. 3.2.1 What Are Logarithms?
      2. 3.2.2 Definition
        • Common Logarithms
        • Natural Logarithms
        • e: An Interesting Number
      3. 3.2.3 Mathematical Derivation of the Rules of Logarithms
      4. 3.2.4 Calculating Logarithms to a Different Base
      5. 3.2.5 Rules of Logarithms: Summary
    3. 3.3 Population Dynamics and the Exponential Equation
    4. 3.4 Exercises
    5. 3.5 Answers
  4. Calculations and Applications
    1. 4.1 Convert Miles/Hour (mph or miles hour-1) to m s-1
    2. 4.2 Body Mass Index (lb/in2)
    3. 4.3 The pH of a Solution
    4. 4.4 How Many Microbes? The Viable Count Method
    5. 4.5 Surface Area of Humans
    6. 4.6 Blood Flow in the Arteries
    7. 4.7 The Growth of a Bacterial Population
    8. 4.8 Light Passing Through a Liquid
    9. 4.9 A Water Pollution Incident
    10. 4.10 The Best Straight Line
      1. 4.10.1 Notation for Sums of Sequences
      2. 4.10.2 Fitting the Best Straight Line
    11. 4.11 The Michaelis Menton Equation
      1. 4.11.1 The Lineweaver–Burke Transformation
      2. 4.11.2 The Eadie–Hofsee Transformation
      3. 4.11.3 Fitting the Parameters the Modern Way
    12. 4.12 Graphs and Functions
      1. 4.12.1 Plotting Graphs
      2. 4.12.2 Shapes of Some Useful Functions
  5. Neat Tricks and Useful Solutions
    1. 5.1 The Difference of Two Squares
    2. 5.2 Mathematical Induction
    3. 5.3 Pythagoras’ Theorem
    4. 5.4 Pythagoras’ Theorem Revisited
    5. 5.5 Limits
    6. 5.6 Trigonometry: Angles with a Difference
      1. 5.6.1 Radians and Degrees
      2. 5.6.2 Trigonometric Ratios: sine, cosine, tangent
        • Application: Radiation on a Surface
        • Application: What Force on the Biceps?
    7. 5.7 Numerical Calculations
      1. 5.7.1 Iteration
      2. 5.7.2 The Method of Iteration
      3. 5.7.3 The Method of Bisection
  6. Differential Calculus
    1. 6.1 Introduction
    2. 6.2 What Is Differentiation?
    3. 6.3 Distance and Velocity
      1. 6.3.1 Average Velocity
      2. 6.3.2 Instantaneous Velocity
    4. 6.4 The Differential Coefficient of Any Function
    5. 6.5 Differentiability
    6. 6.6 Evaluation of Some Standard Derivatives
    7. 6.7 Derivatives Involving Two Functions
    8. 6.8 The Chain Rule
    9. 6.9 Optimum Values: Maxima and Minima
      • Application: How Fast Should a Fish Swim?
    10. 6.10 Small Errors
    11. 6.11 Summary Notes on Differentiation
      1. 6.11.1 Standard Derivatives
      2. 6.11.2 Rules for Differentiation
      3. 6.11.3 Maxima and Minima
    12. 6.12 Applications
      • Equation for Radioactive Decay
      • Half-Life
      • Fitting the Best Line: The Method of Least Squares
      • Cylinder of Minimum Surface Area
    13. 6.13 Exercises
    14. 6.14 Answers
  7. Integral Calculus
    1. 7.1 Introduction
    2. 7.2 Integration as the Area under a Curve
      1. 7.2.1 Area of a Circle 1
      2. 7.2.2 Area of a Circle 2
    3. 7.3 Techniques of Integration
      1. 7.3.1 The Chain Rule
      2. 7.3.2 Integration by Parts
    4. 7.4 Summary Notes on Integration
      1. 7.4.1 Standard Integrals
      2. 7.4.2 Techniques
    5. 7.5 Applications
      • Mean Value
      • Surfaces and Volumes of Revolution
      • Equations of Motion
      • Pollution of a Lake
    6. 7.6 Exercises
    7. 7.7 Answers
  8. Matrix Algebra
    1. 8.1 Introduction
    2. 8.2 What Is a Matrix?
    3. 8.3 Developing the Algebra
      1. 8.3.1 Equality of Matrices
      2. 8.3.2 Addition of Matrices
      3. 8.3.3 Subtraction of Matrices
      4. 8.3.4 Zero or Null Matrix
      5. 8.3.5 Transpose Matrix
      6. 8.3.6 Identity Matrix
      7. 8.3.7 Multiplication by a Scalar
      8. 8.3.8 Matrix Multiplication
    4. 8.4 Applications
      • Population Dynamics
      • Using Matrix Multiplication to Rotate Coordinates
      • Finding Pathways
    5. 8.5 Determinants
      1. 8.5.1 The Determinant of a 3 x 3 Matrix
      2. 8.5.2 Minors and Cofactors
      3. 8.5.3 Area of a Triangle
      4. 8.5.4 Some Properties of Determinants
    6. 8.6 The Inverse Matrix
      1. 8.6.1 Solution of (Lots of) Simultaneous Equations
      2. 8.6.2 Eigenvalues and Eigenvectors
  9. Statistics
    1. 9.1 Introduction
    2. 9.2 The Statistical Method
    3. 9.3 Basic Statistics
      1. 9.3.1 Mean
      2. 9.3.2 Variance
      3. 9.3.3 Standard Deviation
      4. 9.3.4 Standard Error (of ...)
    4. 9.4 The Normal Frequency Distribution
    5. 9.5 The t-Test: Are Two Means Different?
    6. 9.6 How to Perform a t-Test
    7. 9.7 Is the Data from a Normal Distribution?
    8. 9.8 The χ2 Test for Frequencies
      1. 9.8.1 Degrees of Freedom
      2. 9.8.2 Contingency Tables: Are Hair and Eye Color Related?
    9. 9.9 The Mann–Whitney Test: Are Two Samples Different?
    10. 9.10 One-Tailed Tests
  10. The End of the Beginning
  11. Further Reading
  12. Index

Dummy View - NOT TO BE DELETED