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Understanding Mathematics: From Counting to Calculus
Table of Contents
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Chapter
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1
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13
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2
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14
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3
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15
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4
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16
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Measurement of Geometric Figures
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5
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17
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6
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18
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7
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19
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8
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20
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Finding Geometry Using Algebra
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9
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21
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10
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Solving One Algebraic Equation
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22
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11
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23
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12
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24
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Section Title
1-1 Why Anyone Needs This Math Book
1-2 What This Book Provides
1-3 The Layout of This Text
1-4 Who Would Benefit from This Book
1-5 Why Buy One Book that Contains 12 Years of Mathematics?
1-6 This Math is Really Used in Practice
1-7 What the Reader Should Already Know
Chapter 2 - Numbers
Section Title
2-1 The Definition of a Number
2-2 Numbers are Universal
2-3 Counting
2-4 Naming Digits
2-5 Decimals
2-6 Common Symbols
2-7 Summary of Numbers
Chapter 3 - Addition and Subtraction
Section Title
3-1 Explaining Addition
3-2 Addition of Single Digit Integers
3-3 Commutative Law of Addition
3-4 Understanding Variables
3-5 Associative Law of Addition
3-6 Explaining Subtraction
3-7 Subtraction of Single Digit Integers
3-8 Absolute Values
3-9 Addition and Subtraction are Complementary
3-10 Addition of Multi-Digit Integers
3-11 Separating Multi-Digit Numbers for Addition
3-12 Adding Decimals
3-13 Subtraction of Multi-Digit Integers
3-14 Subtraction Resulting in Negative Answers
3-15 Subtracting Decimals
3-16 Summary of Addition and Subtraction
Chapter 4 - Multiplication and Division
Section Title
4-1 Explaining Multiplication
4-2 Operators for Multiplication
4-3 Multiplication of Single Digit Integers
4-4 Commutative Law of Multiplication
4-5 Associative Law of Multiplication
4-6 Distributive Law of Multiplication
4-7 Variations of the Distributive Law of Multiplication
4-8 Multiplication of Multi-Digit Integers
4-9 Separating Multi-Digit Numbers for Multiplication
4-10 Explaining Division
4-11 Division of Single Digit Integers
4-12 Division of Multi-Digit Numbers
4-13 Separating Multi-Digit Numbers for Division
4-14 Division with Remainders
4-15 Multiplying Decimals
4-16 Dividing Decimals
4-17 Rounding and Truncation
4-18 A Pause for Congrats!
Section Title
5-1 Defining Fractions
5-2 Reading Fractions
5-3 Types of Fractions
5-4 Converting Fractions to Decimals
5-5 Converting Decimals to Fractions
5-6 Multiplying Fractions
5-7 Making Different but Equal Fractions
5-8 Simplifying Fractions with Factorization and Cancellation
5-9 Prime Numbers
5-10 Prime Factorization
5-11 Reducing Fractions to Lowest Terms
5-12 Lowest Common Denominators (LCD)
5-13 Adding and Subtracting Fractions
5-14 Conversion of Mixed Numbers
5-15 Dividing Fractions
5-16 Summary of Fractions
Chapter 6 - Percentages
Section Title
6-1 Defining Percentages
6-2 Operations using Percentages
6-3 Finding the Percentage of a Number
6-4 Percentages Shortcuts
6-5 Mean, Median, and Mode
6-6 Summary of Percentages
Section Title
7-1 Defining Negative Numbers
7-2 Negative Numbers are Necessary
7-3 Addition and Subtraction with Negative Numbers
7-4 Multiplying and Dividing Negative Numbers
7-5 Performing Operations with More than 3 Numbers
7-6 Summary of Negative Number Operations
Section Title
8-1 Defining Exponents
8-2 Multiplying with Positive Exponents
8-3 Using Negative Exponents
8-4 More Rules for Using Exponents
8-5 Summary of Exponential Rules
8-6 Fractional Exponents
8-7 Scientific and Exponential Notation
8-8 Summary of Exponents
Section Title
9-1 Review of the Past and Goals for the Future
9-2 Definitions for Algebra
9-3 Operations with Variables
9-4 Order of Operations
9-5 Summary
Chapter 10 - Solving One Algebraic Equation
Section Title
10-1 Introduction
10-2 Algebraic Definitions
10-3 Two Basic Methods for Solving An Algebraic Equation
10-4 Isolating the Variable
10-5 Factorization of Quadratic Equations by Inspection
10-6 Factorization Using the Quadratic Formula
10-7 Factorization of Higher Order Polynomials
10-8 Reducing an Algebraic Expression to Lowest Terms
10-9 Reducing Algebraic Equations Containing Absolute Values
10-10 Summary
Chapter 11 - Solving a System of Equations
Section Title
11-1 Introduction
11-2 Substitution of Equations
11-3 Elimination of Variables
11-4 Using Matrices
11-5 Solving Systems with Matrix Reduction
11-6 Solving Systems with Cramer’s Rule
11-7 Summary
Section Title
12-1 Introduction
12-2 Steps to Solving Any Word Problem
12-3 Word Problem Examples
12-4 Summary
Section Title
13-1 Introduction
13-2 Explaining Functions
13-3 Dependent and Independent Variables
13-4 Examples Using Functions
13-5 Inverse Functions
13-6 Summary
Section Title
14-1 Introduction
14-2 Plotting Points on the Cartesian Coordinate System
14-3 Plotting Linear Equations
14-4 Plotting Systems of Linear Equations
14-5 Plotting Quadratic Equations
14-6 Plotting Polynomials of Higher Degrees
14-7 Plotting Polynomials with Multiple Variables
14-8 Summary of Graphing
Section Title
15-1 Introduction
15-2 Lines and Angles
15-3 Triangles
15-4 Congruent Triangles
15-5 Polygons
15-6 Circles
15-7 Summary
Chapter 16 - Measurement of Geometric Figures
Section Title
16-1 Introduction
16-2 Lengths, Perimeters and Areas of Polygons
16-3 Circle Measurements
16-4 Volumes and Surface Areas
16-5 Summary
Section Title
17-1 Introduction
17-2 Trigonometric Functions
17-3 Commonly Used Triangles
17-4 Oblique Triangle Relationships
17-5 Trigonometry and Graphing
17-6 Plotting Sine and Cosine Values
17-7 Using Radians
17-8 Summary
Section Title
18-1 Introduction
18-2 Understanding Logarithms
18-3 Calculating Logarithms
18-4 Uses of Logarithms
18-5 Finding the Natural Logarithm
18-6 Summary
Section Title
19-1 Introduction
19-2 Defining Imaginary Numbers
19-3 Adding and Subtracting Imaginary Numbers
19-4 Complex = Real + Imaginary
19-5 Graphing Complex Numbers
19-6 Summary
Chapter 20 - Finding Geometry Using Algebra
Section Title
20-1 Introduction
20-2 Finding Geometry with Complex Exponents
20-3 Using Complex Exponents
20-4 Summary
Chapter 21 - Introducing Calculus
Section Title
21-1 What More is There to Learn?
21-2 An Essential Introduction to Calculus
21-3 Summary of the Introduction
Section Title
22-1 Introduction
22-2 Development of the Derivative Using Limits
22-3 Usefulness of the Derivative
22-4 Finding the Derivative of a Function
22-5 Derivatives of Special Functions
22-6 The Chain Rule
22-7 L’Hopital’s Rule
22-8 Maxima and Minima
22-9 Newton’s Method
22-10 Summary of Derivatives
Section Title
23-1 Introduction
23-2 Development of the Integral Using Limits
23-3 Finding the Integral from the Derivative
23-4 Solving Integrals
23-5 Summary of Integrals
Chapter 24 - A Final Word on Mathematics
Section Title
24-1 How Computers and Calculators Understand Math
24-2 Continued Learning
24-3 Now What?
Appendix A: Trigonometric Identities
Appendix B: Table of Trigonometric Functions
Index