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Contents

1. INTRODUCTORY TOPICS ............................................ 1
1-1. The Real-Number System ................................ 1
1-2. Fundamental Assumptions ................................ 1
1-3. Operations With Zero ................................... 5
1-4. Reciprocals ............................................. 5
1-5. The Real-Number Scale .................................. 6
1-6. Rules of Signs ........................................... 7
1-7. Fundamental Operations on Fractions ..................... 9
1-8. Order Relations for Real Numbers ........................ 12
1-9. Absolute Value .......................................... 14
1-10. Inequalities Involving Absolute Values ..................... 14
1-11. Positive Integral Exponents .............................. 16
1-12. Algebraic Expressions .................................... 17
1-13. Equations and Identities .................................. 18
1-14. Symbols of Grouping ..................................... 18
1-15. Order of Fundamental Operations ......................... 20
1-16. Addition and Subtraction of Algebraic Expressions ...... 21
1-17. Multiplication of Algebraic Expressions .................... 22
1-18. Special Products ......................................... 22
1-19. Division of Algebraic Expressions ......................... 23
1-20. Factoring .............................................. 25
1-21. Important Type Forms for Factoring ...................... 27
1-22. Greatest Common Divisor . ............................... 30
1-23. Least Common Multiple .................................. 32
1-24. Reduction of Fractions ................................... 33
1-25. Signs Associated With Fractions .......................... 34
1-26. Addition and Subtraction of Fractions ..................... 36
1-27. Multiplication and Division of Fractions .................. 39
1-28. Complex Fractions ....................................... 40
1-29. Linear Equations ........................................ 42
1-30. Linear Equations in One Unknown ......................... 43


2. THE FUNCTION CONCEPT ......................................... 49
2-1. Rectangular Coordinate Systems in a Plane ................. 49
2-2. Distance Between Two Points ............................ 50
2-3. Functions ............................................... 52
2-4. Functional Notation ...................................... 55
2-5. Some Special Functions .................................. 57
2-6. Variation ............................................... 57
2-7. Classification of Functions ................................ 61


3. TRIGONOMETRIC FUNCTIONS .................................. 63
3-1. The Point Function P(t) ................................. 63
3-2. Definitions of the Trigonometric Functions ................. 64
3-3. Identities ............................................... 68
3-4. Tables of Trigonometric Functions ......................... 71
3-5. Positive and Negative Angles and Standard Position ......... 75
3-6. Measurement of Angles ................................... 76
3-7. The Relation Between Radians and Degrees ................. 77
3-8. Arc Length and Area of a Sector .......................... 78
3-9. Trigonometric Functions of Angles ........................ 81
3-10. Tables of Natural Trigonometric Functions of Angles ........ 82


4. TIIE LAWS OF EXPONENTS ........................................ 86
4-1. Positive Integral Exponents ............................... 86
4-1. Meaning of a^0 ........................................... 88
4-1. Negative Exponents ...................................... 90
4-1. Scientific Notation ....................................... 92
4-1. Rational Exponents ...................................... 92
4-1. The Factorial Symbol .................................... 97
4-1. The Binomial Theorem ................................... 97
4-1. General Term in the Binomial Expansion .................. 99


5. LOGARITHMS .................................................... 101
5-1. Definition of a Logarithm ................................. 101
5-2. Laws of Logarithms ..................................... 102
5-3. Systems of Logarithms ................................... 105
5-4. Common Logarithms ..................................... 105
5-5. Rules for Characteristic and Mantissa ...................... 106
5-6. How to Write Logarithms ................................ 108
5-7. How to Use a Table of Mantissas ......................... 108
5-8. Logarithmic Computation ................................. 110
5-9. Change of Base .......................................... 113


6. RIGHT TRIANGLES AND VECTORS ................................... 115
6-1. Rounding Off Numbers ................................... 115
6-2. Trigonometric Functions of Acute Angles .................. 116
6-3. Procedures for Solving Right Triangles .................... 117
6-4. Angles of Elevation and Depression ....................... 120
6-5. Bearing in Navigation and Surveying ...................... 121
6-6. Projections .............................................. 122
6-7. Scalar and Vector Quantities .............................. 125
6-8. Logarithms of Trigonometric Functions .................... 131
6-9. Logarithmic Solution of Right Triangles .................... 133


7. TRIGONOMETRIC FUNCTIONS OF SUMS AND DIFFERENCES .............. 135
7-1. Derivation of the Addition Formulas ...................... 135
7-2. The Double-Angle Formulas .............................. 139
7-3. The Half-Angle Formulas ................................ 140
7-4. Products of Two Functions Expressed as Sums, and Sums Expressed as Products ......... 143


8. GRAPHS OF TRIGONOMETRIC FUNCTIONS; INVERSE FUNCTIONS AND THEIR GRAPHS ......... 146
8-1. Variation of the Trigonometric Functions .................. 146
8-2. The Graph of the Sine Function ........................... 117
8-3. The Graphs of the Cosine and Tangent Functions ............ 148
8-4. Periodicity, Amplitude, and Phase ......................... 149
8-5. Inverse Functions ...................................... 155
8-6. Inverses of the Trigonometric Functions ................... 156


9. LINEAR EQUATIONS AND GRAPHS .................................. 163
9-1. Solution of Simultaneous Equations ................... 163
9-2. Algebraic Solution of Linear Equations in Two Unknowns... 166
9-3. Linear Equations in Three Unknown ...................... 167
9-4. Graphs of Linear Functions ............................. 169
9-5. Intercepts .............................................. 170
9-6. Graphical Solution of Linear Equations in Two Unknowns ... 171


10. DETERMINANTS ................................................. 173
10-1. Determinants of the Second Order ......................... 173
10-2. Determinants of the Third Order .......................... 175
10-3. Properties of Determinants .............................. 177
10-4. Solution of Three Simultaneous Linear Equations in Three Unknowns ........... 181
10-5. Systems of Three Linear Equations in Three Unknowns When D = 0 .............. 183
10-6. Homogeneous Equations .................................. 184
10-7. Sum and Product of Determinants ......................... 185


11. COMPLEX NUMBERS ............................................. 189
11-1. The Complex Number System ............................ 189
11-2. The Standard Notation for Complex Numbers ............... 191
11-3. Operations on Complex Numbers in Standard Form ........ 192
11-4. Graphical Representation ................................. 195
11-5. Trigonometric Representation ............................. 196
11-6. Multiplication and Division in Trigonometric Form .......... 198
11-7. De Moivre's Theorem ..................................... 199
11-8. Roots of Complex Numbers ............................... 200


12. EQUATIONS IN QUADRATIC FORM .................................. 204
12-1. Quadratic Equations in One Unknown ...................... 204
12-2. Solution of Quadratic Equations by Factoring ............... 204
12-3. Completing the Square ................................... 206
12-4. Solution of Quadratic Equations by the Quadratic Formula ..... 209
12-5. Equations Involving Radicals ............................. 212
12-6. Equations in Quadratic Form ........................... 211
12-7. The Discriminant ....................................... 215
12-8. Sum and Product of the Roots ............................ 217
12-9. Graphs of Quadratic Functions ............................ 218
12-10. Quadratic Equations in Two Unknowns .................... 221
12-11. Graphical Solutions of Systems of Equations Involving Quadratics ........ 224
12-12. Algebraic Solutions of Systems Involving Quadratics ........ 227
12-13. Exponential and Logarithmic Equations .................... 231
12-14. Graphs of Logarithmic and Exponential Functions .......... 233


13. THEORY OF EQUATIONS ........................................... 235
13-1. Introductory Remarks .................................... 235
13-2. Synthetic Division ....................................... 235
13-3. The Remainder Theorem .................................. 240
13-4. The Fundamental Theorem of Algebra ..................... 241
13-5. Pairs of Complex Roots of an Equation ..................... 243
13-6. The Graph of a Polynomial for Large Values of x ........... 244
13-7. Roots Between a and b If f(a) and f(b) Have Opposite Signs ..... 245
13-8. Rational Roots ........................................... 245


14. INEQUALITIES ................................................... 248
14-1. Introduction ............................................. 248
14-2. Properties of Inequalities ................................. 248
14-3. Solution of Conditional Inequalities ........................ 249
14-4. Absolute Inequalities ..................................... 254


15. PROGRESSIONS ................................................. 256
15-1. Sequences and Series ..................................... 256
15-2. Arithmetic Progressions .................................. 260
15-3. The General Term of an Arithmetic Progression ............ 260
15-4. Sum of the First n Terms of an Arithmetic Progression ...... 261
15-5. Arithmetic Means ........................................ 262
15-6. Harmonic Progressions ................................... 264
15-7. Geometric Progressions ................................... 265
15-8. The General Term of a Geometric Progression ............... 265
15-9. Sum of the First n Terms of a Geometric Progression ........ 266
15-10. Geometric Means ........................................ 267
15-11. Infinite Geometric Progression ............................ 268
15-12. Repeating Decimals ...................................... 269
15-13. The Binomial Series ...................................... 271


16. MATHEMATICAL INDUCTION ....................................... 273
16-1. Method of Mathematical Induction ......................... 273
16-2. Proof of the Binomial Theorem for Positive Integral Exponents ... 275


17. PERMUTATIONS, COMBINATIONS, AND PROBABILITY ................... 278
17-1. Fundamental Principle ................................... 278
17-2. Permutations ............................................ 279
17-3. Permutations of n Things Not All Different ................. 280
17-4. Combinations ............................................ 281
17-5. Binomial Coefficieuts ..................................... 282
17-6. Mathematical Probability ................................. 283
17-7. Most Probable Number and Mathematical Expectation ....... 284
17-8. Statistical, or Empirical, Probability ....................... 284
17-9. Mutually Exclusive Events ................................ 285
17-10. Dependent and Independent Events ........................ 286
17-11. Repeated Trials .......................................... 287


18. SOLUTION OF THE GENERAL TRIANGLE .............................. 289
18-1. Classes of Problems ...................................... 289
18-2. The Law of Sines ...................................... 289
18-3. Solution of Case I by the Law of Sines: Given One Side and Two Angles ............... 290
18-4. Solution of Case II by the Law of Sines: Given Two Sides and the Angle Opposite One of Them ......... 291
18-5. The Law of Cosines ..................................... 296
18-6. Solution of Case llI and Case IV by the Law of Cosines ....... 297
18-7. The Law of Tangents .................................... 298
18-8. The Half-Angle Formulas ................................ 300
18-9. Area of a Triangle ....................................... 302


APPENDIX
A. Tables ..................................................... 807
B. Answers to Odd-Numbered Problems ......................... 337


INDEX ............................................................. 353