Basic Mathematics (Book)
The textbook Basic Mathematics by Serge Lang provides an overview of mathematical topics usually encountered through the end of high school/secondary school, specifically arithmetic, algebra, trigonometry, logic, and geometry. It serves as a solid review no matter how far along one may be in their studies, be it just beginning or returning to strengthen one's foundations.
| Basic Mathematics | |
| |
| Information | |
|---|---|
| Author | Serge Lang |
| Language | English |
| Publisher | Springer |
| Publication Date | 1 July 1988 |
| Pages | 496 |
| ISBN-10 | 0387967877 |
| ISBN-13 | 978-0387967875 |
Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.
Table of ContentsEdit
| Chapter/Section # | Title | Page # |
|---|---|---|
| PART I: ALGEBRA | ||
| Chapter 1: Numbers | ||
| 1 | The integers | 5 |
| 2 | Rules for addition | 8 |
| 3 | Rules for multiplication | 14 |
| 4 | Even and odd integers; divisibility | 22 |
| 5 | Rational numbers | 26 |
| 6 | Multiplicative inverses | 42 |
| Chapter 2: Linear Equations | ||
| 1 | Equations in two unknowns | 53 |
| 2 | Equations in three unknowns | 57 |
| Chapter 3: Real Numbers | ||
| 1 | Addition and multiplication | 61 |
| 2 | Real numbers: positivity | 64 |
| 3 | Powers and roots | 70 |
| 4 | Inequalities | 75 |
| Chapter 4: Quadratic Equations | ||
| Interlude: On Logic and Mathematical Expressions | ||
| 1 | On reading books | 93 |
| 2 | Logic | 94 |
| 3 | Sets and elements | 99 |
| 4 | Notation | 100 |
| PART II: INTUITIVE GEOMETRY | ||
| Chapter 5: Distance and Angles | ||
| 1 | Distance | 107 |
| 2 | Angles | 110 |
| 3 | The Pythagoras theorem | 120 |
| Chapter 6: Isometries | ||
| 1 | Some standard mappings of the plane | 133 |
| 2 | Isometries | 143 |
| 3 | Composition of isometries | 150 |
| 4 | Inverse of isometries | 155 |
| 5 | Characterization of isometries | 163 |
| 6 | Congruences | 166 |
| Chapter 7: Area and Applications | ||
| 1 | Area of a disc of radius r | 173 |
| 2 | Circumference of a circle of radius r | 180 |
| PART III: COORDINATE GEOMETRY | ||
| Chapter 8: Coordinates and Geometry | ||
| 1 | Coordinate systems | 191 |
| 2 | Distance between points | 197 |
| 3 | Equation of a circle | 203 |
| 4 | Rational points on a circle | 206 |
| Chapter 9: Operations on Points | ||
| 1 | Dilations and reflections | 213 |
| 2 | Addition, subtraction, and the parallelogram law | 218 |
| Chapter 10: Segments, Rays, and Lines | ||
| 1 | Segments | 229 |
| 2 | Rays | 231 |
| 3 | Lines | 236 |
| 4 | Ordinary equation for a line | 246 |
| Chapter 11: Trigonometry | ||
| 1 | Radian measure | 249 |
| 2 | Sine and cosine | 252 |
| 3 | The graphs | 264 |
| 4 | The tangent | 266 |
| 5 | Addition formulas | 272 |
| 6 | Rotations | 277 |
| Chapter 12: Some Analytic Geometry | ||
| 1 | The straight line again | 281 |
| 2 | The parabola | 291 |
| 3 | The ellipse | 297 |
| 4 | The hyperbola | 300 |
| 5 | Rotation of hyperbolas | 305 |
| PART IV: MISCELLANEOUS | ||
| Chapter 13: Functions | ||
| 1 | Definition of a function | 313 |
| 2 | Polynomial functions | 318 |
| 3 | Graphs of functions | 330 |
| 4 | Exponential function | 333 |
| 5 | Logarithms | 338 |
| Chapter 14: Mappings | ||
| 1 | Definition | 345 |
| 2 | Formalism of mappings | 351 |
| 3 | Permutations | 359 |
| Chapter 15: Complex Numbers | ||
| 1 | The complex plane | 375 |
| 2 | Polar form | 380 |
| Chapter 16: Induction and Summations | ||
| 1 | Induction | 383 |
| 2 | Summations | 388 |
| 3 | Geometric series | 396 |
| Chapter 17: Determinants | ||
| 1 | Matrices | 401 |
| 2 | Determinants of order 2 | 406 |
| 3 | Properties of 2 x 2 determinants | 409 |
| 4 | Determinants of order 3 | 414 |
| 5 | Properties of 3 x 3 determinants | 418 |
| 6 | Cramer's Rule | 424 |
| Index | 429 | |
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