# Understanding Mathematics

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## Description

Mathematics is essential to the work of scientists, engineers, and a whole host of other professions, but it can be difficult for non-mathematicians to master. Keith Gregson's near-painless tour of mathematics uses ample examples from the world around us and worked problems to teach the topics that non-mathematicians often struggle with. Gregson explains the fundamentals behind mathematical relationships and working with equations and teaches readers how to craft estimates. Using examples from the environmental and life sciences, individual chapters cover powers and logarithms, calculus, probability and statistics, and matrix algebra. Gregson also explains how to solve iterative problems, laying the groundwork to go from solving simple equations to calculating answers to real-world problems. Featuring end-of-chapter exercises and suggestions for further reading, this succinct account is a great resource for students of biological or environmental sciences as well as professionals seeking to brush up on basic skills.

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## Product details

- Paperback | 176 pages
- 156 x 235 x 12.7mm | 295g
- 20 Dec 2010
- Johns Hopkins University Press
- Baltimore, MD, United States
- English
- Revised
- second edition
- 41 Line drawings, black and white
- 0801897025
- 9780801897023
- 2,923,929

## Table of contents

Preface

1. Fundamentals

1.1. Why Mathematics?

1.2. What's It All About?

1.2.1. x?

1.2.2. Mathematics?

1.2.3. Functions and Equations

1.2.4. Relationships

1.2.5. Why Don't We Speak Mathematics All the Time?

1.2.6. And Why Do I Need to Understand It?

1.3. Working with Equations

1.3.1. Rearranging Equations

1.3.2. Order of Evaluating Algebraic Expressions

1.3.3. Some Useful Algebraic Relationships

1.3.4. A Word about Calculators

14. Exercises

15. Answers

2. Numbers

2.1. Decimal Number Representation

2.1.1. Significant Figures and Decimal Places

2.1.2. Scientific Notation

2.2. Binary and Hexadecimal Numbers

2.2.1. Binary Numbers

2.2.2. Conversion from Decimal to Binary

2.2.3. Hexadecimal Numbers

2.2.4. Conversion from Decimal to Hexadecimal

2.2.5. Binary-Hex Conversion

2.3. Preliminary Calculations: Check the Problem

2.3.1. Dimension Analysis

Application: How Much Rain Flows into the Oceans?

2.3.2. A Rough Calculation on the Back of an Envelope

2.4. Exercises

2.5. Answers

3. Powers and Logarithms

3.1. Powers and Indices

3.1.1. Some General Rules of Powers and Indices

3.1.2. Rules of Powers and Indices: Summary

3.2. Logarithms

3.2.1. What Are Logarithms?

3.2.2. Definition

Common Logarithms

Natural Logarithms

e: An Interesting Number

3.2.3. Mathematical Derivation of the Rules of Logarithms

3.2.4. Calculating Logarithms to a Different Base

3.2.5. Rules of Logarithms: Summary

3.3. Population Dynamics and the Exponential Equation

3.4. Exercises

3.5. Answers

4. Calculations and Applications

4.1. Convert Miles/Hour (mph or miles hour-1) to m s-1

4.2. Body Mass Index (lb/in2)

4.3. The pH of a Solution

4.4. How Many Microbes? The Viable Count Method

4.5. Surface Area of Humans

4.6. Blood Flow in the Arteries

4.7. The Growth of a Bacterial Population

4.8. Light Passing Through a Liquid

4.9. A Water Pollution Incident

4.10. The Best Straight Line

4.10.1. Notation for Sums of Sequences

4.10.2. Fitting the Best Straight Line

4.11. The Michaelis Menton Equation

4.11.1. The Lineweaver-Burke Transformation

4.11.2. The Eadie-Hofsee Transformation

4.11.3. Fitting the Parameters the Modern Way

4.12. Graphs and Functions

4.12.1. Plotting Graphs

4.12.2. Shapes of Some Useful Functions

5. Neat Tricks and Useful Solutions

5.1. The Difference of Two Squares

5.2. Mathematical Induction

5.3. Pythagoras' Theorem

5.4. Pythagoras' Theorem Revisited

5.5. Limits

5.6. Trigonometry: Angles with a Difference

5.6.1. Radians and Degrees

5.6.2. Trigonometric Ratios: Sine, Cosine, Tangent

Application: Radiation on a Surface

Application: What Force on the Biceps?

5.7. Numerical Calculations

5.7.1. Iteration

5.7.2. The Method of Iteration

5.7.3. The Method of Bisection

6. Differential Calculus

6.1. Introduction

6.2. What Is Differentiation?

6.1. Distance and Velocity

6.2. Average Velocity

6.3. Instantaneous Velocity

6.4. The Differential Coefficient of Any Function

6.5. Differentiability

6.6. Evaluation of Some Standard Derivatives

6.7. Derivatives Involving Two Functions

6.8. The Chain Rule

6.9. Optimum Values: Maxima and Minima

Application: How Fast Should a Fish Swim?

6.10. Small Errors

6.11. Summary Notes on Differentiation

6.11.1. Standard Derivatives

6.11.2. Rules for Differentiation

6.11.3. Maxima and Minima

6.12. Applications

Equation for Radioactive Decay

Half-Life

Fitting the Best Line: The Method of Least Squares

Cylinder of Minimum Surface Area

6.13. Exercises

6.14. Answers

7. Integral Calculus

7.1. Introduction

7.2. Integration as the Area under a Curve

7.2.1. Area of a Circle 1

7.2.2. Area of a Circle 2

7.3. Techniques of Integration

7.3.1. The Chain Rule

7.3.2. Integration by Parts

7.4. Summary Notes on Integration

7.4.1. Standard Integrals

7.4.2. Techniques

7.5. Applications

Mean Value

Surfaces and Volumes of Revolution

Equations of Motion

Pollution of a Lake

7.6. Exercises

7.7. Answers

8. Matrix Algebra

8.1. Introduction

8.2. What Is a Matrix?

8.3. Developing the Algebra

8.3.1. Equality of Matrices

8.3.2. Addition of Matrices

8.3.3. Subtraction of Matrices

8.3.4. Zero or Null Matrix

8.3.5. Transpose Matrix

8.3.6. Identity Matrix

8.3.7. Multiplication by a Scalar

8.3.8. Matrix Multiplication

8.4. Applications

Population Dynamics

Using Matrix Multiplication to Rotate Coordinates

Finding Pathways

8.5. Determinants

8.5.1. The Determinant of a 3 3 Matrix

8.5.2. Minors and Cofactors

8.5.3. Area of a Triangle

8.5.4. Some Properties of Determinants

8.6. The Inverse Matrix

8.6.1. Solution of (Lots of) Simultaneous Equations

8.6.2. Eigenvalues and Eigenvectors

9. Statistics

9.1. Introduction

9.2. The Statistical Method

9.3. Basic Statistics

9.3.1. Mean

9.3.2. Variance

9.3.3. Standard Deviation

9.3.4. Standard Error (of ...)

9.4. The Normal Frequency Distribution

9.5. The t-Test: Are Two Means Different?

9.6. How to Perform a t-Test

9.7. Is the Data from a Normal Distribution?

9.8. The 2 Test for Frequencies

9.8.1. Degrees of Freedom

9.8.2. Contingency Tables: Are Hair and Eye Color Related?

9.9. The Mann-Whitney Test: Are Two Samples Different?

9.10. One-Tailed Tests

10. The End of the Beginning

Further Reading

Index

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1. Fundamentals

1.1. Why Mathematics?

1.2. What's It All About?

1.2.1. x?

1.2.2. Mathematics?

1.2.3. Functions and Equations

1.2.4. Relationships

1.2.5. Why Don't We Speak Mathematics All the Time?

1.2.6. And Why Do I Need to Understand It?

1.3. Working with Equations

1.3.1. Rearranging Equations

1.3.2. Order of Evaluating Algebraic Expressions

1.3.3. Some Useful Algebraic Relationships

1.3.4. A Word about Calculators

14. Exercises

15. Answers

2. Numbers

2.1. Decimal Number Representation

2.1.1. Significant Figures and Decimal Places

2.1.2. Scientific Notation

2.2. Binary and Hexadecimal Numbers

2.2.1. Binary Numbers

2.2.2. Conversion from Decimal to Binary

2.2.3. Hexadecimal Numbers

2.2.4. Conversion from Decimal to Hexadecimal

2.2.5. Binary-Hex Conversion

2.3. Preliminary Calculations: Check the Problem

2.3.1. Dimension Analysis

Application: How Much Rain Flows into the Oceans?

2.3.2. A Rough Calculation on the Back of an Envelope

2.4. Exercises

2.5. Answers

3. Powers and Logarithms

3.1. Powers and Indices

3.1.1. Some General Rules of Powers and Indices

3.1.2. Rules of Powers and Indices: Summary

3.2. Logarithms

3.2.1. What Are Logarithms?

3.2.2. Definition

Common Logarithms

Natural Logarithms

e: An Interesting Number

3.2.3. Mathematical Derivation of the Rules of Logarithms

3.2.4. Calculating Logarithms to a Different Base

3.2.5. Rules of Logarithms: Summary

3.3. Population Dynamics and the Exponential Equation

3.4. Exercises

3.5. Answers

4. Calculations and Applications

4.1. Convert Miles/Hour (mph or miles hour-1) to m s-1

4.2. Body Mass Index (lb/in2)

4.3. The pH of a Solution

4.4. How Many Microbes? The Viable Count Method

4.5. Surface Area of Humans

4.6. Blood Flow in the Arteries

4.7. The Growth of a Bacterial Population

4.8. Light Passing Through a Liquid

4.9. A Water Pollution Incident

4.10. The Best Straight Line

4.10.1. Notation for Sums of Sequences

4.10.2. Fitting the Best Straight Line

4.11. The Michaelis Menton Equation

4.11.1. The Lineweaver-Burke Transformation

4.11.2. The Eadie-Hofsee Transformation

4.11.3. Fitting the Parameters the Modern Way

4.12. Graphs and Functions

4.12.1. Plotting Graphs

4.12.2. Shapes of Some Useful Functions

5. Neat Tricks and Useful Solutions

5.1. The Difference of Two Squares

5.2. Mathematical Induction

5.3. Pythagoras' Theorem

5.4. Pythagoras' Theorem Revisited

5.5. Limits

5.6. Trigonometry: Angles with a Difference

5.6.1. Radians and Degrees

5.6.2. Trigonometric Ratios: Sine, Cosine, Tangent

Application: Radiation on a Surface

Application: What Force on the Biceps?

5.7. Numerical Calculations

5.7.1. Iteration

5.7.2. The Method of Iteration

5.7.3. The Method of Bisection

6. Differential Calculus

6.1. Introduction

6.2. What Is Differentiation?

6.1. Distance and Velocity

6.2. Average Velocity

6.3. Instantaneous Velocity

6.4. The Differential Coefficient of Any Function

6.5. Differentiability

6.6. Evaluation of Some Standard Derivatives

6.7. Derivatives Involving Two Functions

6.8. The Chain Rule

6.9. Optimum Values: Maxima and Minima

Application: How Fast Should a Fish Swim?

6.10. Small Errors

6.11. Summary Notes on Differentiation

6.11.1. Standard Derivatives

6.11.2. Rules for Differentiation

6.11.3. Maxima and Minima

6.12. Applications

Equation for Radioactive Decay

Half-Life

Fitting the Best Line: The Method of Least Squares

Cylinder of Minimum Surface Area

6.13. Exercises

6.14. Answers

7. Integral Calculus

7.1. Introduction

7.2. Integration as the Area under a Curve

7.2.1. Area of a Circle 1

7.2.2. Area of a Circle 2

7.3. Techniques of Integration

7.3.1. The Chain Rule

7.3.2. Integration by Parts

7.4. Summary Notes on Integration

7.4.1. Standard Integrals

7.4.2. Techniques

7.5. Applications

Mean Value

Surfaces and Volumes of Revolution

Equations of Motion

Pollution of a Lake

7.6. Exercises

7.7. Answers

8. Matrix Algebra

8.1. Introduction

8.2. What Is a Matrix?

8.3. Developing the Algebra

8.3.1. Equality of Matrices

8.3.2. Addition of Matrices

8.3.3. Subtraction of Matrices

8.3.4. Zero or Null Matrix

8.3.5. Transpose Matrix

8.3.6. Identity Matrix

8.3.7. Multiplication by a Scalar

8.3.8. Matrix Multiplication

8.4. Applications

Population Dynamics

Using Matrix Multiplication to Rotate Coordinates

Finding Pathways

8.5. Determinants

8.5.1. The Determinant of a 3 3 Matrix

8.5.2. Minors and Cofactors

8.5.3. Area of a Triangle

8.5.4. Some Properties of Determinants

8.6. The Inverse Matrix

8.6.1. Solution of (Lots of) Simultaneous Equations

8.6.2. Eigenvalues and Eigenvectors

9. Statistics

9.1. Introduction

9.2. The Statistical Method

9.3. Basic Statistics

9.3.1. Mean

9.3.2. Variance

9.3.3. Standard Deviation

9.3.4. Standard Error (of ...)

9.4. The Normal Frequency Distribution

9.5. The t-Test: Are Two Means Different?

9.6. How to Perform a t-Test

9.7. Is the Data from a Normal Distribution?

9.8. The 2 Test for Frequencies

9.8.1. Degrees of Freedom

9.8.2. Contingency Tables: Are Hair and Eye Color Related?

9.9. The Mann-Whitney Test: Are Two Samples Different?

9.10. One-Tailed Tests

10. The End of the Beginning

Further Reading

Index

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## Review quote

Recommended. Choice 2011

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## About Keith Gregson

Keith Gregson is a faculty member with the School of Biosciences at Nottingham University.

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