Understanding Mathematics

Copyright\n......Page 3
Dedication\n......Page 4
Contents......Page 6
Preface\n......Page 12
1.1 Why Mathematics......Page 14
1.2.2 Mathematics?......Page 15
1.2.3 Functions and Equations......Page 16
1.2.6 ...and why do I need to understand it?......Page 17
1.3.1 Rearranging Equations......Page 18
1.3.2 Order of Evaluating Algebraic Equations......Page 19
1.3.3 Some Useful Algebraic Relationships......Page 20
1.4 Preliminary Calculations - check the problem......Page 21
1.4.2 An example calculation......Page 22
1.4.3 And the back of the envelope......Page 23
2 Numbers\n......Page 26
2.1 Decimal Number Representation......Page 27
2.1.2 Scientific Notation......Page 28
2.2.1 Binary Numbers......Page 29
2.2.3 Hexadecimal Numbers......Page 30
2.2.5 Binary-Hex conversion......Page 31
3.1 Powers and Indices......Page 34
3.1.1 Some general rules of powers and indices......Page 35
3.1.2 Rules of Powers and Indices - Summary......Page 36
3.2 Logarithms......Page 37
3.2.1 What are logarithms?......Page 38
3.2.2 Definition......Page 39
3.2.3 Mathematical Derivation of the Rules of Logarithms\n......Page 40
3.2.4 Calculating logarithms to a different base......Page 42
3.3 Exponential Functions......Page 43
4.2 The pH of a solution......Page 46
4.4 Surface Area of Humans......Page 47
4.6 The Growth of a Bacterial Population......Page 49
4.7 The Beer-Lambert Law......Page 50
4.8 A River Pollution Incident......Page 51
4.9.1 Notation for sums of sequences......Page 52
4.9.2 Fitting the best Straight Line......Page 53
4.10.1 The Lineweaver-Burke transformation......Page 54
4.10.3 Fitting the parameters the Modern Way......Page 55
4.11.1 Plotting Graphs......Page 56
4.11.2 Shapes of some useful functions......Page 58
5.1 The Difference of Two Squares......Page 60
5.2 Mathematical Induction......Page 61
5.3 Pythagoras’ Theorem......Page 63
5.4 Pythagoras’ Theorem revisited......Page 64
5.5 Limits......Page 67
5.6 Trigonometry - angles with a difference......Page 68
5.7 Trigonometric Ratios......Page 69
6.1 Introduction......Page 72
6.1.1 What is differentiation?......Page 73
6.2 Distance and Velocity......Page 74
6.2.2 Instantaneous Velocity......Page 75
6.3 The Differential Coefficient of any function......Page 78
6.3.1 Differentiability......Page 80
6.4.1 The derivative of a sum u(x) + v(x)......Page 81
6.4.2 The derivative of a product u(x)v(x)......Page 82
6.5.1 The derivative of a Constant......Page 83
6.5.3 The derivative of sin x......Page 84
6.5.4 The derivative of a constant times a function of x......Page 85
6.5.5 The derivative of ex......Page 87
6.5.6 The derivatives of ln x and ax......Page 88
6.5.7 The Chain Rule......Page 89
6.6 Optimum values - maxima and minima......Page 90
6.7 Small Errors......Page 94
6.8.3 Maxima and Minima......Page 95
6.9 Applications......Page 96
7.1 Introduction......Page 102
7.2 Integration as the Area under a Curve......Page 103
7.2.1 Area of a Circle 1\n......Page 106
7.2.2 Area of a Circle 2......Page 107
7.3.1 The Chain Rule......Page 108
7.3.2 Integration by Parts......Page 109
7.4.2 Techniques......Page 112
7.5 Applications......Page 113
8.1 Introduction......Page 120
8.2.1 Equality of Matrices......Page 122
8.2.5 Identity Matrix......Page 123
8.2.7 Multiplication of Matrices......Page 124
8.3 Determinants......Page 125
8.3.2 Minors and Cofactors......Page 126
8.4 The Inverse Matrix......Page 127
8.4.1 Solution of Linear Simultaneous Equations......Page 128
8.5.1 Application to Population Dynamics......Page 129
8.5.2 Eigenvalues and eigenvectors......Page 130
9 The End of the Beginning\n......Page 132
9.1 Further Reading......Page 133
Index\n......Page 134