Understanding Pure Mathematics
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English
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Entries: 143
A.J Sadler
D.W.S Thorning
core math
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Understanding Pure Mathematics
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143
0. Introductory Work
Theory
Page 5
0.1 The Number Line
Theory
Page 5
0.2 Inequalities
Theory
Page 6
0.3 Basic Algebraic Manipulation
Theory
Page 9
0.4 Surds
Theory
Page 12
0.5 Quadratics
Theory
Page 14
0.6 Trigonometric Ratios of Acute and Obtuse Angles
Theory
Page 17
0.7 Solution of Triangles
Theory
Page 20
0.8 Three-Dimensional Trigonometry
Theory
Page 26
1. Functions
Theory
Page 33
1.1 Basic Concepts
Theory
Page 33
1.2 Composite Functions
Theory
Page 38
1.3 The Inverse of a Function
Theory
Page 38
1.4 Ordered Pairs
Theory
Page 42
1.5 The Graph of a Function
Theory
Page 43
1.6 Some Further Considerations
Theory
Page 47
2. Vectors
Theory
Page 51
2.1 Basic Concepts
Theory
Page 51
2.2 Position Vectors and Unit Vectors
Theory
Page 56
2.3 The Scalar Product
Theory
Page 62
2.4 Geometrical Proofs by Vector Methods
Theory
Page 67
2.5 The Vector Equation of a Straight Line
Theory
Page 69
3. Coordinate Geometry I
Theory
Page 75
3.1 Loci
Theory
Page 75
3.2 Distance Between Two Points
Theory
Page 76
3.3 Mid-Point of a Line Joining Two Points
Theory
Page 78
3.4 Gradient of a Line Joining Two Points
Theory
Page 81
3.5 Parallel and Perpendicular Lines
Theory
Page 83
3.6 The Equation $y=mx+c$
Theory
Page 85
3.7 Finding the Cartesian Equation of a Straight Line
Theory
Page 91
3.8 Relationships Shown as Linear Graphs
Theory
Page 93
3.9 Intersection of Lines
Theory
Page 98
4. Trigonometry I
Theory
Page 103
4.1 Trigonometric Ratios of Any Angle
Theory
Page 103
4.2 Trigonometric Graphs
Theory
Page 105
4.3 Graph Sketching
Theory
Page 110
4.4 Solving Trigonometric Equations
Theory
Page 114
4.5 Pythagorean Relationships
Theory
Page 118
4.6 Secant, Cosecant and Cotangent
Theory
Page 119
4.7 Compound Angles
Theory
Page 122
4.8 Circular Measure
Theory
Page 128
5. Algebra I
Theory
Page 135
5.1 Indices
Theory
Page 135
5.2 Variation
Theory
Page 137
5.3 Elementary Theory of Logarithms
Theory
Page 140
5.4 Quadratics
Theory
Page 145
5.5 Polynomials in General
Theory
Page 156
6. Matrices
Theory
Page 163
6.1 Basic Concepts
Theory
Page 163
6.2 Transformations
Theory
Page 167
6.3 Further Considerations
Theory
Page 174
6.4 General Reflections and Rotations
Theory
Page 182
7. Permutations and Combinations
Theory
Page 192
7.1 Successive Operations
Theory
Page 192
7.2 Permutations
Theory
Page 193
7.3 Combinations
Theory
Page 202
7.4 Selections of Any Size from a Group
Theory
Page 207
7.5 Division into Groups
Theory
Page 208
8. Series and the Binomial Theorem
Theory
Page 212
8.1 Arithmetic Progressions
Theory
Page 212
8.2 Geometric Progressions
Theory
Page 216
8.3 Summation of Other Series and Proof by Induction
Theory
Page 221
8.4 The Binomial Theorem for a Positive Integral Index
Theory
Page 226
8.5 The Binomial Theorem for Any Rational Index
Theory
Page 232
9. Probability
Theory
Page 239
9.1 Simple Probability
Theory
Page 239
9.2 Sum and Product Laws
Theory
Page 242
9.3 Further Conditional Probability
Theory
Page 250
9.4 Probability Involving Permutations and Combinations
Theory
Page 251
10. Calculus I: Differentiation
Theory
Page 259
10.1 The Gradient of a Curve
Theory
Page 259
10.2 Differentiation of $x^n$
Theory
Page 262
10.3 Maximum, Minimum and Points of Inflexion
Theory
Page 266
10.4 Small Changes
Theory
Page 273
10.5 More About Points of Inflexion
Theory
Page 276
11. Sketching Functions I
Theory
Page 279
11.1 General Methods
Theory
Page 279
11.2 Further Considerations
Theory
Page 286
11.3 Simple Transformations
Theory
Page 288
11.4 $\frac{1}{f(x)}$
Theory
Page 292
12. Calculus II: Integration
Theory
Page 297
12.1 The Reverse of Differentiation
Theory
Page 297
12.2 The Area Under a Curve
Theory
Page 302
12.3 Volume of Revolution
Theory
Page 309
12.4 Two Applications
Theory
Page 312
12.5 Mean Value of a Function
Theory
Page 316
13. Calculus III: Further Techniques
Theory
Page 321
13.1 Function of a Function
Theory
Page 321
13.2 Integration of $\frac{f'(x)}{(f(x))^n}$
Theory
Page 324
13.3 Related Rates of Change
Theory
Page 326
13.4 Parameters
Theory
Page 329
13.5 Product Rule
Theory
Page 332
13.6 Quotient Rule
Theory
Page 333
13.7 Implicit Functions
Theory
Page 335
14. Sketching Functions II
Theory
Page 339
14.1 Rational Functions
Theory
Page 339
14.2 Inequalities
Theory
Page 348
15. Trigonometry II
Theory
Page 355
15.1 Sum and Product Formulae
Theory
Page 355
15.2 $a\cos x\pm b\sin x$
Theory
Page 357
15.3 Inverse Trigonometric Functions
Theory
Page 360
15.4 General Solutions
Theory
Page 365
15.5 Small Angles
Theory
Page 369
15.6 Differentiation of Trigonometric Functions
Theory
Page 371
15.7 Integration of Trigonometric Functions
Theory
Page 375
15.8 Differentiation of Inverse Trigonometric Functions
Theory
Page 378
16. Coordinate Geometry II
Theory
Page 384
16.1 Angle Between Two Straight Lines
Theory
Page 384
16.2 Distance of a Point from a Line
Theory
Page 386
16.3 The Circle
Theory
Page 389
16.4 The Parabola
Theory
Page 396
16.5 The Ellipse and the Hyperbola
Theory
Page 400
16.6 Polar Coordinates
Theory
Page 405
17. Three-Dimensional Work: Vectors and Matrices
Theory
Page 413
17.1 Three-Dimensional Work
Theory
Page 413
17.2 Differentiation and Integration of Vectors
Theory
Page 417
17.3 Equations of a Straight Line in 3-D
Theory
Page 419
17.4 Equations of a Plane
Theory
Page 427
17.5 $3\times3$ Matrices
Theory
Page 437
17.6 Sets of Linear Equations
Theory
Page 443
18. Algebra II
Theory
Page 455
18.1 Partial Fractions
Theory
Page 455
18.2 Partial Fractions and Series
Theory
Page 459
18.3 Complex Numbers
Theory
Page 464
19. Exponential and Logarithmic Functions
Theory
Page 481
19.1 Differentiating the Exponential Function
Theory
Page 481
19.2 The Logarithmic Function
Theory
Page 485
19.3 The Exponential Series
Theory
Page 493
19.4 The Logarithmic Series
Theory
Page 494
20. Calculus IV: Further Integration
Theory
Page 500
20.1 Change of Variable
Theory
Page 500
20.2 Integration by Parts
Theory
Page 506
20.3 General Methods
Theory
Page 510
20.4 Differential Equations
Theory
Page 513
21. Numerical Methods
Theory
Page 528
21.1 The Trapezium Rule
Theory
Page 528
21.2 Simpson's Rule
Theory
Page 529
21.3 Taylor's Theorem
Theory
Page 531
21.4 Maclaurin's Theorem
Theory
Page 532
21.5 Solution of Equations
Theory
Page 533
21.6 $e^z$, $\cos z$ and $\sin z$ for Complex $z$
Theory
Page 539
22. Answers
Theory
Page 544
23. Index
Theory
Page 604
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