The Handbook of Essential Mathematics

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Section I: Formulas with
Select Processes
Index to Processes Page 06
1. Algebra 13
1.1. What is a Variable? 13
1.2. Field Axioms 14
1.3. Divisibility Tests 15
1.4. Subtraction, Division, Signed Numbers 16
1.5. Rules for Fractions 18
1.6. Partial Fractions 19
1.7. Rules for Exponents 20
1.8. Rules for Radicals 21
1.9. Factor Formulas 22
1.10. Laws of Equality 24
1.11. Laws of Inequality 26
1.12. Order of Operations 27
1.13. Three Meanings of ‘Equals’ 27
1.14. The Seven Parentheses Rules 28
1.15. Rules for Logarithms 30
1.16. Complex Numbers 31
1.17. What is a Function? 32
1.18. Function Algebra 33
1.19. Quadratic Equations & Functions 34
1.20. Cardano’s Cubic Solution 36
1.21. Theory of Polynomial Equations 37
1.22. Determinants and Cramer’s Rule 39
1.23. Binomial Theorem 40
1.24. Arithmetic Series 41
1.25. Geometric Series 41
1.26. Boolean Algebra 42
1.27. Variation Formulas 43

2. Classical and Analytic Geometry 44
2.1. The Parallel Postulates 44
2.2. Angles and Lines 44
2.3. Triangles 45
2.4. Congruent Triangles 46
2.5. Similar Triangles 47
2.6. Planar Figures 47
2.7. Solid Figures 49
2.8. Pythagorean Theorem 50
2.9. Heron’s Formula 52
2.10. Golden Ratio 53
2.11. Distance and Line Formulas 54
2.12. Formulas for Conic Sections 55
2.13. Conic Sections
3. Trigonometry 57
3.1. Basic Definitions: Functions & Inverses 57
3.2. Fundamental Definition-Based Identities 58
3.3. Pythagorean Identities 58
3.4. Negative Angle Identities 58
3.5. Sum and Difference Identities 58
3.6. Double Angle Identities 60
3.7. Half Angle Identities 60
3.8. General Triangle Formulas 60
3.9. Arc and Sector Formulas 62
3.10. Degree/Radian Relationship 62
3.11. Addition of Sine and Cosine 63
3.12. Polar Form of Complex Numbers 63
3.13. Rectangular to Polar Coordinates 64
3.14. Trigonometric Values from Right Triangles 64
4. Elementary Vector Algebra 65
4.1. Basic Definitions and Properties 65
4.2. Dot Products 65
4.3. Cross Products 66
4.4. Line and Plane Equations 67
4.5. Miscellaneous Vector Equations 67

5. Elementary Calculus 68
5.1. What is a Limit? 68
5.2. What is a Differential? 69
5.3. Basic Differentiation Rules 70
5.4. Transcendental Differentiation 70
5.5. Basic Antidifferentiation Rules 71
5.6. Transcendental Antidifferentiation 72
5.7. Lines and Approximation 73
5.8. Interpretation of Definite Integral 73
5.9. Fundamental Theorem of Calculus 75
5.10. Geometric Integral Formulas 75
5.11. Select Elementary Differential Equations 76
5.12. Laplace Transform: General Properties 77
5.13. Laplace Transform: Specific Transforms 78
6. Money and Finance 80
6.1. What is Interest? 80
6.2. Simple Interest 81
6.3. Compound and Continuous Interest 81
6.4. Effective Interest Rates 82
6.5. Present-to-Future Value Formulas 82
6.6. Present Value of a “Future Deposit Stream” 82
6.7. Present Value of a “ “ with Initial Lump Sum 83
6.8. Present Value of a Continuous “ “ 83
6.9. Types or Retirement Savings Accounts 84
6.10. Loan Amortization 85
6.11. Annuity Formulas 86
6.12. Markup and Markdown 86
6.13. Calculus of Finance 86
7. Probability and Statistics 87
7.1. Probability Formulas 87
7.2. Basic Concepts of Statistics 88
7.3. Measures of Central Tendency 89
7.4. Measures of Dispersion 90
7.5. Sampling Distribution of the Mean 91
7.6. Sampling Distribution of the Proportion 92

Section II: Tables
1. Numerical 94
1.1. Factors of Integers 1 through 192 94
1.2. Prime Numbers less than 1000 96
1.3. Roman Numeral and Arabic Equivalents 96
1.4. Nine Elementary Memory Numbers 97
1.5. American Names for Large Numbers 97
1.6. Selected Magic Squares 97
1.7. Thirteen-by-Thirteen Multiplication Table 101
1.8. The Random Digits of PI 102
1.9. Standard Normal Distribution 103
1.10. Two-Sided Student’s t Distribution 104
1.11. Date and Day of Year 105
2. Physical Sciences 106
2.1. Conversion Factors in Allied Health 106
2.2. Medical Abbreviations in Allied Health 107
2.3. Wind Chill Table 108
2.4. Heat Index Table 108
2.5. Temperature Conversion Formulas 109
2.6. Unit Conversion Table 109
2.7. Properties of Earth and Moon 112
2.8. Metric System 113
2.9. British System 114
Section III: Applications in
Personal Finance
1. The Algebra of Interest 118
1.1. What is Interest? 118
1.2. Simple Interest 120
1.3. Compound Interest 122
1.4. Continuous Interest 124
1.5. Effective Interest Rate 129

2. The Algebra of the Nest Egg 135
2.1. Present and Future Value 135
2.2. Growth of an Initial Lump Sum Deposit 138
2.3. Growth of a Deposit Stream 142
2.4. The Two Growth Mechanisms in Concert 147
2.5. Summary 151
3. The Algebra of Consumer Debt 153
3.1. Loan Amortization 153
3.2. Your Home Mortgage 162
3.3. Car Loans and Leases 173
3.4. The Annuity as a Mortgage in Reverse 183
4. The Calculus of Finance 185
4.1. Jacob Bernoulli’s Differential Equation 185
4.2. Differentials and Interest Rate 187
4.3. Bernoulli and Money 188
4.4. Applications 191
Appendices
A. Greek Alphabet 200
B. Mathematical Symbols 201
C. My Most Used Formulas 204

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关键词：mathematics Mathematic Essential Thematic handbook handbook The Essential mathematics

Introduction
Formulas! They seem to be the bane of every beginning
mathematics student who has yet to realize that formulas are
about structure and relationship—and not about memorization.
Granted, formulas have to be memorized; for, it is partly through
memorization that we eventually become ‘unconsciously
competent’: a true master of our skill, practicing it in an almost
effortless, automatic sense. In mathematics, being ‘unconsciously
competent’ means we have mastered the underlying algebraic
language to the same degree that we have mastered our native
tongue. Knowing formulas and understanding the reasoning
behind them propels one towards the road to mathematical
fluency, so essential in our modern high-tech society.
The Handbook of Essential Mathematics contains three
major sections. Section I, “Formulas”, contains most of the
mathematical formulas that a person would expect to encounter
through the second year of college regardless of major. In
addition, there are formulas rarely seen in such compilations,
included as a mathematical treat for the inquisitive. Section I also
includes select mathematical processes, such as the process for
solving a linear equation in one unknown, with a supporting
examples. Section II, “Tables”, includes both ‘pure math’ tables
and physical-science tables, useful in a variety of disciples ranging
from physics to nursing. As in Section I, some tables are included
just to nurture curiosity in a spirit of fun. In Sections I and II, each
formula and table is enumerated for easy referral. Section III,
“Applications in Personal Finance”, is a small textbook within a
book where the language of algebra is applied to that everyday
financial world affecting all of us throughout our lives from birth to
death. Note: The idea of combining mathematics formulas with
financial applications is not original in that my father had a similar
type book as a Purdue engineering student in the early 1930s.
I would like to take this opportunity to thank Mr. Al
Giambrone—Chairman of the Department of Mathematics, Sinclair
Community College, Dayton, Ohio—for providing requiredmemorization
formula lists for 22 Sinclair mathematics courses
from which the formula compilation was partially built.

John C. Sparks
March 2006