Engineering Mathematics
Dr Colin Turner
i
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Queries
Should you have any queries about these notes, you should approach your lecturer or your tutor as soon as possible. Don’t be afraid to ask questions, it is possible you may have found an error, and if you have not, your questions will help your lecturer / tutor understand the problems you are experiencing. As mathematics is cumulative, it will be very hard to continue the module with outstanding problems from the start, a bit of work at this point will make the rest much easier going.
Practice
Mathematics requires practice. No matter how simple a procedure may look when demonstrated in a lecture or a tutorial you can have no idea how well you can perform it until you try. As there is very little opportunity for practice in the university environment it is vitally important that you attempt the questions provided in the tutorial, preferably before attending the relevant tutorial class. Your time with your tutor will be best spent when you arrive at the class with a list of problems you are unable to tackle, the more specific the better. If you find the questions too hard before the tutorial, do not become discouraged, the mere act of thinking about the problem will have a positive affect on your understanding of the problem once explained to you in the tutorial.
Contact Details
My contact details are as follows Name Dr Colin Turner
Room 5F10
Phone 68084 (+44-28-9036-8084 externally) Email [email protected]
Contents
1 Preliminaries 1
1.1 Introduction . . . 1
1.2 Notation . . . 1
1.3 Arithmetic . . . 2
1.3.1 The law of signs . . . 2
1.3.2 Order of precedence . . . 3
1.4 Decimal Places & Significant Figures . . . 3
1.4.1 Decimal Places . . . 3 1.4.2 Significant Figures . . . 4 1.5 Standard Form . . . 5 1.5.1 Standard prefixes . . . 5 2 Number Systems 8 2.1 Natural numbers . . . 8 2.2 Prime numbers . . . 8 2.3 Integers . . . 9 2.4 Real numbers . . . 9 2.5 Rational numbers . . . 9 2.6 Irrational Numbers . . . 10 3 Basic Algebra 11 3.1 Rearranging Equations . . . 11 3.1.1 Example . . . 11 3.1.2 Order of Rearranging . . . 12 3.1.3 Example . . . 13 3.1.4 Example . . . 14 3.1.5 Example . . . 15 3.2 Function Notation . . . 15
CONTENTS iii
3.3 Expansion of Brackets . . . 17
3.3.1 Examples . . . 18
3.3.2 Brackets upon Brackets . . . 18
3.3.3 Examples . . . 19 3.4 Factorization . . . 20 3.4.1 Examples . . . 21 3.5 Laws of Indices . . . 21 3.5.1 Example “proofs” . . . 22 3.5.2 Examples . . . 23 3.6 Laws of Surds . . . 23 3.6.1 Examples . . . 24 3.7 Quadratic Equations . . . 24 3.7.1 Examples . . . 24 3.7.2 Graphical interpretation . . . 25 3.7.3 Factorization . . . 26
3.7.4 Quadratic solution formula . . . 27
3.7.5 The discriminant . . . 28
3.7.6 Examples . . . 28
3.7.7 Special cases . . . 29
3.8 Notation . . . 30
3.8.1 Modulus or absolute value . . . 30
3.8.2 Sigma notation . . . 31
3.8.3 Factorials . . . 32
3.9 Exponential and Logarithmic functions . . . 32
3.9.1 Exponential functions . . . 32
3.9.2 Logarithmic functions . . . 33
3.9.3 Logarithms to solve equations . . . 34
3.9.4 Examples . . . 35 3.9.5 Anti-logging . . . 36 3.9.6 Examples . . . 37 3.10 Binomial Expansion . . . 38 3.10.1 Theory . . . 38 3.10.2 Example . . . 40 3.10.3 Examples . . . 40 3.10.4 High values of n . . . 41 3.11 Arithmetic Progressions . . . 42 3.11.1 Examples . . . 42
3.11.2 Sum of an arithmetic progression . . . 42
3.11.3 Example . . . 43
3.11.4 Example . . . 44
CONTENTS iv
3.12.1 Examples . . . 45
3.12.2 Sum of a geometric progression . . . 46
3.12.3 Sum to infinity . . . 46 3.12.4 Example . . . 47 3.12.5 Example . . . 48 4 Trigonometry 49 4.1 Right-angled triangles . . . 49 4.1.1 Labelling . . . 49 4.1.2 Pythagoras’ Theorem . . . 50
4.1.3 Basic trigonometric functions . . . 50
4.1.4 Procedure . . . 51 4.1.5 Example . . . 51 4.2 Notation . . . 52 4.2.1 Example . . . 53 4.3 Table of values . . . 53 4.4 Graphs of Functions . . . 53 4.5 Multiple Solutions . . . 54 4.5.1 CAST diagram . . . 55 4.5.2 Procedure . . . 56 4.5.3 Example . . . 56 4.6 Scalene triangles . . . 58 4.6.1 Labelling . . . 58 4.6.2 Scalene trigonmetry . . . 58 4.6.3 Sine Rule . . . 58 4.6.4 Cosine Rule . . . 59 4.6.5 Example . . . 60 4.7 Radian Measure . . . 61 4.7.1 Conversion . . . 61 4.7.2 Length of Arc . . . 61 4.7.3 Area of Sector . . . 62 4.8 Identities . . . 63 4.8.1 Basic identities . . . 64
4.8.2 Compound angle identities . . . 64
4.8.3 Double angle identities . . . 64
4.9 Trigonmetric equations . . . 65
4.9.1 Example . . . 65
4.9.2 Example . . . 65
CONTENTS v
5 Complex Numbers 68
5.1 Basic Principle . . . 68
5.1.1 Imaginary and Complex Numbers . . . 69
5.2 Examples . . . 69
5.3 Argand Diagram Representation . . . 70
5.4 Algebra of Complex Numbers . . . 70
5.4.1 Addition . . . 71 5.4.2 Subtraction . . . 72 5.4.3 Multiplication . . . 72 5.4.4 Division . . . 74 5.4.5 Examples . . . 74 5.5 Definitions . . . 75 5.5.1 Modulus . . . 75 5.5.2 Conjugate . . . 75 5.5.3 Real part . . . 76 5.5.4 Imaginary part . . . 76 5.6 Representation . . . 76 5.6.1 Cartesian form . . . 76 5.6.2 Polar form . . . 77 5.6.3 Exponential form . . . 78 5.6.4 Examples . . . 79 5.6.5 Examples . . . 80 5.7 De Moivre’s Theorem . . . 80 5.7.1 Examples . . . 82 5.7.2 Roots of Unity . . . 82
5.7.3 Roots of other numbers . . . 83
5.8 Trigonometric functions . . . 84
6 Vectors & Matrices 85 6.1 Vectors . . . 85
6.1.1 Modulus . . . 85
6.1.2 Unit Vector . . . 86
6.1.3 Cartesian unit vectors . . . 86
6.1.4 Examples . . . 87 6.1.5 Signs of vectors . . . 87 6.1.6 Addition . . . 88 6.1.7 Subtraction . . . 88 6.1.8 Zero vector . . . 88 6.1.9 Scalar Product . . . 89 6.1.10 Example . . . 90
CONTENTS vi 6.1.11 Example . . . 91 6.1.12 Vector Product . . . 91 6.1.13 Example . . . 93 6.2 Matrices . . . 93 6.2.1 Square matrices . . . 93
6.2.2 Row and Column vectors . . . 94
6.2.3 Examples . . . 94
6.2.4 Zero and Identity . . . 94
6.3 Matrix Arithmetic . . . 95 6.3.1 Addition . . . 95 6.3.2 Examples . . . 95 6.3.3 Subtraction . . . 95 6.3.4 Examples . . . 95 6.3.5 Multiplication by a scalar . . . 96 6.3.6 Examples . . . 96 6.3.7 Domino Rule . . . 96 6.3.8 Multiplication . . . 97 6.3.9 Examples . . . 97 6.3.10 Exercise . . . 99 6.4 Determinant of a matrix . . . 99 6.4.1 Examples . . . 99
6.4.2 Sign rule for matrices . . . 99
6.4.3 Order 3 . . . 100 6.4.4 Examples . . . 100 6.4.5 Order 4 . . . 101 6.5 Inverse of a matrix . . . 101 6.5.1 Order 2 . . . 102 6.5.2 Examples . . . 102 6.5.3 Other orders . . . 103 6.5.4 Exercise . . . 103 6.6 Matrix algebra . . . 104 6.6.1 Addition . . . 104 6.6.2 Multiplication . . . 104 6.6.3 Mixed . . . 104 6.7 Solving equations . . . 105 6.7.1 Example . . . 106 6.7.2 Example . . . 107 6.7.3 Row reduction . . . 108 6.8 Row Operations . . . 108 6.8.1 Determinants . . . 109 6.8.2 Example . . . 110
CONTENTS vii
6.9 Solving systems of equations . . . 110
6.9.1 Gaussian Elimination . . . 111
6.9.2 Example . . . 112
6.9.3 Example . . . 114
6.9.4 Example . . . 115
6.9.5 Number of equations vs. unknowns . . . 116
6.10 Inversion by Row operations . . . 117
6.11 Rank . . . 117 6.11.1 Example . . . 117 6.11.2 Systems of equations . . . 119 6.11.3 Example . . . 119 6.11.4 Example . . . 119 6.11.5 Example . . . 120 6.11.6 Summary . . . 121 6.11.7 Exercise . . . 122
6.12 Eigenvalues and Eigenvectors . . . 122
6.12.1 Finding Eigenvalues . . . 122 6.12.2 Example . . . 123 6.12.3 Finding eigenvectors . . . 123 6.12.4 Example . . . 123 6.12.5 Example . . . 124 6.12.6 Other orders . . . 125 6.13 Diagonalisation . . . 125
6.13.1 Powers of diagonal matrices . . . 126
6.13.2 Example . . . 126
6.13.3 Powers of other matrices . . . 127
6.13.4 Example . . . 127
7 Graphs of Functions 128 7.1 Simple graph plotting . . . 128
7.1.1 Example . . . 128 7.1.2 Example . . . 129 7.1.3 Example . . . 130 7.2 Important functions . . . 130 7.2.1 Direct Proportion . . . 130 7.2.2 Inverse Proportion . . . 131
7.2.3 Inverse Square Proportion . . . 132
7.2.4 Exponential Functions . . . 134
7.2.5 Logarithmic Functions . . . 134
CONTENTS viii
7.3.1 Addition or Subtraction . . . 136
7.3.2 Multiplication or Division . . . 136
7.3.3 Adding to or Subtracting from x . . . 136
7.3.4 Multiplying or Dividing x . . . 137
7.4 Even and Odd functions . . . 138
7.4.1 Even functions . . . 138 7.4.2 Odd functions . . . 138 7.4.3 Combinations of functions . . . 139 7.4.4 Examples . . . 140 8 Coordinate geometry 142 8.1 Elementary concepts . . . 142
8.1.1 Distance between two points . . . 143
8.1.2 Example . . . 143
8.1.3 Example . . . 143
8.1.4 Midpoint of two points . . . 144
8.1.5 Example . . . 144
8.1.6 Example . . . 144
8.1.7 Gradient . . . 144
8.1.8 Example . . . 145
8.1.9 Example . . . 145
8.2 Equation of a straight line . . . 145
8.2.1 Meaning of equation of line . . . 146
8.2.2 Finding the equation of a line . . . 146
8.2.3 Example . . . 147
8.2.4 Example . . . 147
9 Differential Calculus 149 9.1 Concept . . . 149
9.2 Notation . . . 149
9.3 Rules & Techniques . . . 150
9.3.1 Power Rule . . . 150
9.3.2 Addition and Subtraction . . . 151
9.3.3 Constants upon functions . . . 151
9.3.4 Chain Rule . . . 151 9.3.5 Product Rule . . . 151 9.3.6 Quotient Rule . . . 152 9.3.7 Trigonometric Rules . . . 152 9.3.8 Exponential Rules . . . 152 9.3.9 Logarithmic Rules . . . 152
CONTENTS ix 9.4 Examples . . . 152 9.4.1 Solutions . . . 153 9.5 Tangents . . . 156 9.5.1 Example . . . 157 9.5.2 Example . . . 157 9.6 Turning Points . . . 158
9.6.1 Types of turning point . . . 158
9.6.2 Finding turning points . . . 159
9.6.3 Classification of turning points . . . 160
9.6.4 Example . . . 161 9.6.5 Example . . . 162 9.6.6 Example . . . 163 9.7 Newton Rhapson . . . 163 9.7.1 Example . . . 164 9.7.2 Example . . . 165 9.8 Partial Differentiation . . . 166 9.8.1 Example . . . 167 9.9 Small Changes . . . 168 9.9.1 Example . . . 168 9.9.2 Example . . . 169 10 Integral Calculus 172 10.1 Concept . . . 172 10.1.1 Constant of Integration . . . 172
10.2 Rules & Techniques . . . 173
10.2.1 Power Rule . . . 173
10.2.2 Addition & Subtraction . . . 173
10.2.3 Multiplication by a constant . . . 174
10.2.4 Substitution . . . 174
10.2.5 Limited Chain Rule . . . 174
10.2.6 Logarithm rule . . . 175 10.2.7 Partial Fractions . . . 176 10.2.8 Integration by Parts . . . 177 10.2.9 Other rules . . . 178 10.3 Examples . . . 179 10.3.1 Examples . . . 179 10.3.2 Examples . . . 180 10.3.3 Example . . . 182 10.3.4 Example . . . 183 10.3.5 Examples . . . 184
CONTENTS x 10.4 Definite Integration . . . 185 10.4.1 Notation . . . 186 10.4.2 Concept . . . 186 10.4.3 Areas . . . 186 10.4.4 Example . . . 186 10.4.5 Example . . . 186 10.4.6 Volumes of Revolution . . . 187 10.4.7 Example . . . 187 10.4.8 Mean Values . . . 188 10.4.9 Example . . . 188 10.4.10 Example . . . 188 10.4.11 RMS Values . . . 189 10.4.12 Example . . . 189 10.4.13 Example . . . 190 10.5 Numerical Integration . . . 191 10.5.1 Simpson’s rule . . . 192 10.5.2 Example . . . 192 11 Power Series 194 11.1 Definition . . . 194 11.1.1 Convergence . . . 194 11.2 Maclaurin’s Expansion . . . 195
11.2.1 Odd and Even . . . 195
11.2.2 Example . . . 196 11.2.3 Exercise . . . 196 11.2.4 Example . . . 196 11.2.5 Exercise . . . 197 11.2.6 Example . . . 197 11.3 Taylor’s Expansion . . . 197 11.3.1 Example . . . 198
11.3.2 Identification of Turning Points . . . 198
12 Differential Equations 200 12.1 Concept . . . 200 12.2 Exact D.E.s . . . 200 12.2.1 Example . . . 201 12.2.2 Example . . . 201 12.2.3 Example . . . 202
12.3 Variables separable D.E.s . . . 203
CONTENTS xi
12.3.2 Example . . . 204
12.4 First order linear D.E.s . . . 206
12.4.1 Example . . . 207
12.4.2 Example . . . 208
12.5 Second order D.E.s . . . 209
12.5.1 Homogenous D.E. with constant coefficients . . . 209
12.5.2 Example . . . 211
12.5.3 Example . . . 211
12.5.4 Example . . . 212
12.5.5 Example . . . 212
12.5.6 Example . . . 213
13 Differentiation in several variables 214 13.1 Partial Differentiation . . . 214 13.1.1 Procedure . . . 214 13.1.2 Examples . . . 215 13.1.3 Notation . . . 216 13.1.4 Higher Derivatives . . . 216 13.1.5 Example . . . 217 13.2 Taylor’s Theorem . . . 218 13.3 Stationary Points . . . 218 13.3.1 Types of points . . . 218 13.3.2 Finding points . . . 219 13.3.3 Classifying points . . . 219 13.3.4 Summary . . . 220 13.3.5 Example . . . 221 13.4 Implicit functions . . . 222 13.5 Lagrange Multipliers . . . 223 13.5.1 Example . . . 224 13.6 Jacobians . . . 225 13.6.1 Differential . . . 226 13.7 Parametric functions . . . 226 13.7.1 Example . . . 227 13.8 Chain Rule . . . 227
14 Integration in several variables 229 14.1 Double integrals . . . 229
14.1.1 Example . . . 229
14.2 Change of order . . . 230
CONTENTS xii 14.3.1 Example . . . 232 14.3.2 Example . . . 232 14.4 Triple integrals . . . 233 14.4.1 Example . . . 233 14.5 Change of variable . . . 234 14.5.1 Polar coordinates . . . 235 14.5.2 Example . . . 235
14.5.3 Cylindrical Polar Coordinates . . . 236
14.5.4 Spherical Polar Coordinates . . . 236
15 Fourier Series 237 15.1 Periodic functions . . . 237 15.1.1 Example . . . 237 15.1.2 Example . . . 238 15.2 Sets of functions . . . 239 15.2.1 Orthogonal functions . . . 239 15.2.2 Orthonormal functions . . . 239 15.2.3 Norm of a function . . . 239 15.3 Fourier concepts . . . 239 15.3.1 Fourier coefficents . . . 239 15.3.2 Fourier series . . . 240 15.3.3 Convergence . . . 240 15.4 Important functions . . . 240 15.4.1 Trigonometric system . . . 241 15.4.2 Exponential system . . . 242 15.5 Trigonometric expansions . . . 242 15.5.1 Even functions . . . 242 15.5.2 Odd functions . . . 243 15.5.3 Other Ranges . . . 243 15.6 Harmonics . . . 243
15.6.1 Odd and Even Harmonics . . . 244
15.6.2 Trigonometric system . . . 244 15.6.3 Exponential system . . . 245 15.6.4 Percentage harmonic . . . 245 15.7 Examples . . . 245 15.7.1 Example . . . 245 15.8 Exponential Series . . . 246
CONTENTS xiii 16 Laplace transforms 248 16.1 Definition . . . 248 16.1.1 Example . . . 248 16.1.2 Example . . . 249 16.1.3 Example . . . 249 16.1.4 Inverse Transform . . . 250 16.1.5 Elementary properties . . . 250 16.1.6 Example . . . 250 16.2 Important Transforms . . . 251
16.2.1 First shifting property . . . 251
16.2.2 Further Laplace transforms . . . 253
16.3 Transforming derivatives . . . 254 16.3.1 First derivative . . . 254 16.3.2 Second derivative . . . 254 16.3.3 Higher derivatives . . . 254 16.4 Transforming integrals . . . 254 16.5 Differential Equations . . . 255 16.5.1 Example . . . 255 16.5.2 Example . . . 256 16.5.3 Example . . . 258 16.5.4 Example . . . 260 16.5.5 Exercise . . . 261 16.5.6 Example . . . 261 16.5.7 Example . . . 262 16.6 Other theorems . . . 263 16.6.1 Change of Scale . . . 263
16.6.2 Derivative of the transform . . . 263
16.6.3 Convolution Theorem . . . 264
16.6.4 Example . . . 264
16.6.5 Example . . . 264
16.6.6 Example . . . 265
16.7 Heaviside unit step function . . . 265
16.7.1 Laplace transform of u(t − c) . . . 268
16.7.2 Example . . . 268
16.7.3 Example . . . 270
16.7.4 Delayed functions . . . 270
16.7.5 Example . . . 271
16.8 The Dirac Delta . . . 272
16.8.1 Delayed impulse . . . 273
16.8.2 Example . . . 274
CONTENTS xiv
16.9.1 Impulse Response . . . 276
16.9.2 Initial value theorem . . . 277
16.9.3 Final value theorem . . . 277
17 Z-transform 279 17.1 Concept . . . 279
17.2 Important Z-transforms . . . 281
17.2.1 Unit step function . . . 281
17.2.2 Linear function . . . 282
17.2.3 Exponential function . . . 283
17.2.4 Elementary properties . . . 283
17.2.5 Real translation theorem . . . 283
18 Statistics 285 18.1 Sigma Notation . . . 285
18.1.1 Example . . . 285
18.2 Populations and Samples . . . 287
18.2.1 Sampling . . . 287
18.3 Parameters and Statistics . . . 288
18.4 Frequency . . . 288 18.5 Measures of Location . . . 288 18.5.1 Arithmetic Mean . . . 289 18.5.2 Mode . . . 289 18.5.3 Median . . . 289 18.5.4 Example . . . 289 18.5.5 Example . . . 290 18.6 Measures of Dispersion . . . 290 18.6.1 Range . . . 291 18.6.2 Standard deviation . . . 291 18.6.3 Inter-quartile range . . . 292 18.7 Frequency Distributions . . . 292 18.7.1 Class intervals . . . 292 18.8 Cumulative frequency . . . 293
18.8.1 Calculating the median . . . 293
18.8.2 Calculating quartiles . . . 294
18.8.3 Calculating other ranges . . . 294
18.9 Skew . . . 294
18.10Correlation . . . 294
18.10.1 Linear regression . . . 295
CONTENTS xv 19 Probability 297 19.1 Events . . . 297 19.1.1 Probability of an Event . . . 297 19.1.2 Exhaustive lists . . . 297 19.2 Multiple Events . . . 298 19.2.1 Notation . . . 298
19.2.2 Relations between events . . . 298
19.3 Probability Laws . . . 299
19.3.1 A or B (mutually exclusive events) . . . 299
19.3.2 not A . . . 299
19.3.3 1 event of N . . . 300
19.3.4 n events of N . . . 300
19.3.5 Examples . . . 300
19.3.6 A and B (independent events) . . . 300
19.3.7 Example . . . 301
19.3.8 A or B or C or ... . . 301
19.3.9 A and B and C and ... . . 301
19.3.10 Example . . . 302 19.3.11 A or B revisited . . . 302 19.3.12 Example . . . 302 19.3.13 A and B revisited . . . 303 19.3.14 Conditional probability . . . 303 19.3.15 Example . . . 304 19.3.16 Bayes Theorem . . . 304
19.4 Discrete Random Variables . . . 304
19.4.1 Notation . . . 304
19.4.2 Expected Value . . . 305
19.4.3 Variance . . . 306
19.4.4 Example . . . 307
19.5 Continuous Random Variables . . . 308
19.5.1 Definition . . . 309
19.5.2 Probability Density Function . . . 309
20 The Normal Distribution 310 20.1 Definition . . . 310
20.2 Standard normal distribution . . . 311
20.2.1 Transforming variables . . . 311
20.2.2 Calculation of areas . . . 312
20.2.3 Example . . . 312
CONTENTS xvi
20.2.5 Sampling distribution . . . 313
20.3 The central limit theorem . . . 314
20.4 Finding the Population mean . . . 315
20.5 Hypothesis Testing . . . 315
20.5.1 Two tailed tests . . . 316
20.6 Difference of two normal distributions . . . 317
A Statistical Tables 318
List of Tables
1.1 Basic notation . . . 2
1.2 The law of signs . . . 2
1.3 Order of precedence . . . 3
1.4 SI prefixes for large numbers . . . 6
1.5 SI prefixes for small numbers . . . 7
3.1 The laws of indices . . . 21
3.2 The laws of surds . . . 23
3.3 Examples of quadratic equations . . . 25
3.4 Laws of Logarithms . . . 34
3.5 Pascal’s Triangle . . . 39
4.1 Table of trigonometric values . . . 54
4.2 Conversion between degrees and radians . . . 62
5.1 Powers of j . . . 73
6.1 Matrix algebra - Addition . . . 104
6.2 Matrix algebra - Multiplication . . . 105
6.3 Matrix algebra - Mixed operations . . . 105
7.1 Adding and Subtracting even and odd functions . . . 139
7.2 Multiplying even and odd functions . . . 140
9.1 Second derivative test . . . 160
9.2 First derivative turning point classification . . . 161
15.1 Symmetry in Fourier Series . . . 244
16.1 Common Laplace transforms . . . 252
16.2 Further Laplace transforms . . . 253
LIST OF TABLES xviii
18.1 An example of class intervals . . . 293
19.1 Probabilities for total of two rolled dice . . . 305
19.2 Calculating E(X) and var (X) for two rolled dice. . . 308
A.1 Table of Φ(x) (Normal Distribution) . . . 319
A.2 Table of χ2 distribution (Part I) . . . 320
List of Figures
2.1 The real number line . . . 9
3.1 The quadratic equation . . . 25
4.1 Labelling right-angled triangles . . . 50
4.2 Generating the trigonometric graphs . . . 55
4.3 The graphs of sin θ and cos θ . . . 56
4.4 The “CAST” diagram . . . 57
4.5 Labelling a scalene triangle . . . 59
4.6 Length of Arc, Area of Sector . . . 63
5.1 The Argand diagram . . . 71
5.2 Polar representation of a complex number . . . 77
6.1 Vector Addition . . . 88 6.2 Vector Subtraction . . . 89 7.1 The graph of x2+ 2x − 3 . . . 129 7.2 The graph of 2x + 3 . . . 130 7.3 The graph of 1x . . . 132 7.4 The graph of x12 . . . 133 7.5 The graph of ex . . . 134 7.6 The graph of ln x . . . 135 7.7 Closeup of graph of ln x . . . 136 7.8 Graph of sin x . . . 137 7.9 Graph of sin x + 1 . . . 138 7.10 Graph of 2 sin x . . . 139 7.11 Graph of sin(x + 90) . . . 140
7.12 Closeup of graph of sin(2x) . . . 141
LIST OF FIGURES xx
9.1 Types of turning point . . . 159
9.2 The Newton-Rhapson method . . . 171
13.1 The graph of f (x, y) = xy2+ 1 . . . 215
13.2 The graph of f (x, y) = x3(sin xy + 3x + y + 3) . . . 216
13.3 A graph with four turning points . . . 222
14.1 Double integration over the simple region R. . . 230
14.2 Double integration over x then y. . . 231
14.3 Double integration over y then x. . . 232
16.1 An L and R curcuit. . . 261
16.2 An L, C, and R curcuit. . . 262
16.3 The unit step function u(t). . . 266
16.4 The displaced unit step function u(t − c). . . 266
16.5 Building functions that are on and off when we please. . . 267
16.6 A positive waveform built from steps. . . 269
16.7 A waveform built from steps. . . 270
16.8 A waveform built from delayed linear functions. . . 272
16.9 An impulse train built from Dirac deltas. . . 274
17.1 A continuous (analog) function . . . 279
17.2 Sampling the function . . . 280
17.3 The digital view . . . 280
20.1 The Normal Distribution . . . 310
Chapter 1
Preliminaries
1.1
Introduction
We shall start the course, by recapping many definitions and results that may already be well known. As mathematics is a cumulative subject, it is necessary however, to ensure that all the basics are in place before we can go on.
We assume the reader is familiar with the elementary arithmetic of num-bers positive, negative and zero. We also assume the reader is familiar with the decimal representation of numbers, and that they can evaluate simple expressions, including fractional arithmetic.
1.2
Notation
We now list some mathematical notation that we may use in the course, or may be encountered elsewhere, this is shown in table 1.1.
Another important bit of notation is “. . . ”, which is used as a sort of mathematician’s etcetera. For example
• 1, 2, 3, . . . , 10 is short hand for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
• 1, 2, 3, . . . is short hand for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc.
It’s probably worth noting that in algebra when we use letters to represent numbers, then
1.3 Arithmetic 2
= equal to 6= not equal to
< less than ≤ less than or equal to
> greater than ≥ greater than or equal to
≡ equivalent to ≈ approximately equal to
⇒ implies ∞ infinity
Σ sum of what follows Π product of what follows
Table 1.1: Basic notation
• 3a is a shorthand for 3 × a • a is a shorthand for 1 × a • −a is a shorthand for −1 × a
1.3
Arithmetic
Two often forgotten pieces of arithmetic are:
1.3.1
The law of signs
When we combine signs, either by multiplying two numbers, or by subtracting a negative number for example, we use table 1.2 to determine the sign of the outcome. Put simply, a “−” sign reverses our direction, and so two of them take us back to the “+” direction and so on.
+ + → +
− − → +
− + → −
+ − → −
1.4 Decimal Places & Significant Figures 3
1.3.2
Order of precedence
We are familiar with the fact that expressions inside brackets must be evalu-ated first, that is what the bracket signifies. However, without brackets there is still an inherent order in which operations must be done. Consider this simple calculation
2 + 3 × 4
opinion is usually split as to whether the answer is 20, or 14. The rea-son is that multiplication should be performed before addition, and so the 3 × 4 segment should be calculated first. Be aware that not all calculators understand this, test yours with this calculuation.
Calculations should be performed in the order shown in table 1.3.
B Brackets first - they override all priority
O Order (Powers, roots)
D Division
M Multiplication
A Addition
S Subtraction
Table 1.3: Order of precedence
We note that the table provides us with a handy reminder, BODMAS.
1.4
Decimal Places & Significant Figures
Often we are required to produce answers to a specific degree of accuracy. The most well known way to do this is with the number of decimal places.
1.4.1
Decimal Places
The number of decimal places is a measure of how to truncate answers to a given accuracy. If four decimal places are required then we look at the fifth decimal place and beyond, if it is 5 or greater, we round up the last decimal places that is written, otherwise we simply leave it alone. Let us take an
1.4 Decimal Places & Significant Figures 4
example. π is a mathematical constant that continues infinitely through all its decimal places, never repeating its pattern.
π = 3.141592653589793 . . .
rounded to five decimal places we obtain
π ≈ 3.14159
since the next digit is 2, wheras rounding to four decimal places give
π ≈ 3.1416
since the next digit is 9 which is clearly bigger than 5.
It is good to use enough decimal places to obtain an accurate answer, but one must always remember the context of the answer. There is little point in calculating that the length of a piece of metal should be 2.328745 cm if all we will have to measure it with is a ruler accurate to 1 mm.
1.4.2
Significant Figures
Sometimes decimal places are not the most appropriate way to define accu-racy. There is no specific number of decimal places that suit all situations. For example, if we quote the radius of the Earth in metres, then probably no number of decimal places are appropriate for most purposes, as the answer will not be that accurate, and there will be so many other figures before it, they are unlikely to be significant.
An alternative often used is to specify a number of significant figures. This is essential the number of non-zero numbers that should be displayed. Suppose that we specify four significant figures. Then the speed of light in m/s is written as:
c = 2, 997, 992, 458 ≈ 2, 998, 000, 000 m/s
which can be written more simply again in standard form (see below). The issue here is that the other figures are less likely to have any real impact on the answer of a problem. Similarly the standard atomic mass of Uranium is
238.02891 g/mol ≈ 238.0 g/mol
since we only have four significant figures, we round after the zero. Note that writing the zero helps indicate the precision of the answer.
1.5 Standard Form 5
1.5
Standard Form
In science, large and small numbers are often represented by standard form. This takes the form
a.bcd × 10n
if we are using four significant figures. For example, we saw above that to four significant figures
c = 2, 998, 000, 000 m/s = 2, 998 × 1, 000, 000 m/s
or, working a bit more, we move the decimal place each time to the left, (which divides the left hand number by ten), and multiply by another ten on the right to compensate.
= 2.998 × 1, 000, 000, 000 m/s
now all that remains to do, is to write the number on the right as a power of ten. We count the zeros, there are nine, and so
c ≈ 2.998 × 109 m/s.
The same applies for small numbers. The light emitted by a Helium-Neon Laser has a wavelength of
λ = 0.000, 000, 632, 8 m
but this is clearly rather unwieldy to write down. This time we move the decimal place to the right until we get to after the first non-zero digit. Each time we do this we essentially multiply by 10, and so to compensate we have to divide by ten. This can be represented by increasingly large negative values of the power.1
So here, we need to move the decimal place seven times to the right, and so we will multiply by 10−7.
λ = 6.328 × 10−7 m.
1.5.1
Standard prefixes
There are a number of prefixes applied to large and small numbers to allow us to write them more meaningfully. You will have met many of them before. The prefixes for large numbers are shown in table 1.5.1.
1.5 Standard Form 6
Name Prefix In English Power of Ten
deca da tens 101 hecto h hundreds 102 kilo k thousands 103 Mega M millions 106 Giga G billions 109 Tera T trillions 1012 Peta P quadrillions 1015 Exa E quintillions 1018
Table 1.4: SI prefixes for large numbers
Note that in the past there was a difference between billions as used in British English and American English. The English billion was one million million, wheras the American billion is one thousand million. The latter has won out now, and most references to a billion are to the American one. Also, there is a slight disparity between “normal” quantities, and the bytes used in computing. Since computing is based on binary, and therefore powers of 2, a kilobyte (kB) is not 1000 bytes, but 1024 bytes.2So in computing, 1024
is used rather than thousands to build up such quantities. The prefixes for large numbers are shown in table 1.5.1.
Because of these prefixes, it is normal within engineering to adapt stan-dard scientific form to get the power of ten to be a multiple of three. Let us revisit our wavelength example:
λ = 6.328 × 10−7 m.
So we would prefer to tweak the power of ten here. We could do this
λ = .6328 × 10−6 m = 0.6238 µm,
but this is pretty ugly to have a fractional number. It would be more normal to write
λ ≈ 632.8 × 10−9 m = 623.8 nm,
1.5 Standard Form 7
Name Prefix In English Power of Ten
deci d tenths 10−1 centi c hundredths 10−2 milli m thousandths 10−3 micro µ millionths 10−6 nano n billionths 10−9 pico p trillionths 10−12 femto f quadrillionths 10−15 atto a quintillionths 10−18
Chapter 2
Number Systems
We remind ourselves of different sets of numbers that we will refer to later.
2.1
Natural numbers
The set of all numbers
1, 2, 3, . . .
is known as the set of positive integers, (or natural numbers, or whole numbers, or counting numbers and is denoted by N.
2.2
Prime numbers
A prime number is a positive integer which has exactly two factors1, namely itself and one.
Thus 2 is the first prime number, and the only even prime number. So the prime numbers are given by
2, 3, 5, 7, 11, 13, 17, 19, . . .
2.3 Integers 9
2.3
Integers
The set of all numbers given by
. . . , −3, −2, −1, 0, 1, 2, 3, . . .
is known as the set of integers and is denoted by the symbol Z.
2.4
Real numbers
The collection of all numbers in ordinary arithmetic, i.e. including fractions, integers, zero, positive and negative numbers etc. is called the set of real numbers and is denoted R. The set of real numbers can be visualised as a line, called the real number line or simply the real line. Each point on the lines represents a unique real number, and every number, including exotic examples such as π is represented by a unique point on the line.
Figure 2.1: The real number line
2.5
Rational numbers
The set of all numbers that can be written mn where m and n are integers, is known as the set of rational numbers and is noted by Q. (Note that n cannot by zero, as division by zero is not permitted).
For example, −23,2096165 , −2 = −21 , 0 = 01 are all rational numbers.
(Note that although division by zero is not peemitted, dividing zero by another number is, and as no other number can fit into zero at all, the result is zero).
It turns out that if you add, subtract, divide or multiply any two rational numbers together, you still get a rational number.
2.6 Irrational Numbers 10
There’s an easy way of working out whether a given number is rational or not. Simply expand it in it’s decimal form. Rational numbers always have a decimal expansion that ends, or repeats itself every so many digits.
For example
−2
3 = −0.6666666 . . . 3436.234523452345 . . .
34.68 are all rational.
2.6
Irrational Numbers
Of course, not all real numbers are rational, and in fact many numbers you will already have met are not. These numbers are called irrational numbers.
Examples are √2,√3, and in fact √p where p is prime.
When written in their decimal forms, irrationals are never ending and non-repeating. This means that irrational numbers can never be written down exactly.
Although in secondary schools we often write π = 227, suggesting that π is rational, this is only a simple (and not very accurate) approximation. In fact π is also irrational, as is Euler’s constant e.
Chapter 3
Basic Algebra
Algebra is perhaps the most important part of mathematics to master. If you do not, you will find problems in all the areas you study, all caused by the underlying weakness in your algebra skills. In many ways, algebra is representative of mathematics in that it deals with forming an easy problem out of a difficult one.
3.1
Rearranging Equations
Imagine an equation as a pair of balanced weighing scales. What will happen if we add 2kg on both sides? The scales will remain in balance. If we multiply the weights by three on both sides? The scales will remain in balance. In fact, even if we take the sine of both weights, the scales remain in balance. The leads to the fundamental result you must remember.
You can do anything to both sides of an equation and you will obtain an equivalent equation.
3.1.1
Example
We shall look at an extremely trivial example of this concept in use. In our learning of algebraic manipulation we are often told that we can take things across the equals sign and change the sign. Rarely are we told why this works. Let’s examine it.
3.1 Rearranging Equations 12
x + 4 = 9
Well, it is simple to see what value x has in this case. However, we wish to show how rearranging works in these very simple cases. We wish to find x, and this is really saying we want to manipulate the equation into the form:
x =??
Where ?? represents the answer. Therefore, we wish to have an x on it’s own, on one side of the equation, with everything else on the other side of the equals sign. To that end, we start to look at what is attached to x, how it is attached, and how we should remove it. In our example, 4 is attached to the x by the process of addition. Now, how do you get rid of a 4 that has been added? Of course, the answer is to subtract it, but we must not simply do this on one side, rather in accordance with 3.1 we must do it on both sides of the equation to maintain its validity.
So we obtain
x + 4 − 4 = 9 − 4
Now the +4 − 4 on the L.H.S. cancel, leaving zero, and this step wouldn’t be written normally. So we finally obtain
x = 9 − 4 = 5
If you observe that it appears that the +4 crossed the equals sign to become a −4 on the R.H.S.. However, now we know what has actually happened.
3.1.2
Order of Rearranging
Of course, in most examples, more than one thing is attached to the x, and usually by a combination of operations. It may be equally correct to rearrange by removing these in any order, but some ways will almost certainly be easier than others.
We have already noted in 1.3.2 that some operations are naturally done before others, and so when we see an expression such as:
3x2− 4 = 8 it really means
3.1 Rearranging Equations 13
((3(x2)) − 4) = 8
where the brackets serve simply to underline the order in which things are done. The effect is somewhat similar to an onion with the x in the very centre. We could peel the onion from the outside in for the most tidy approach. That is, we remove things in the reverse order to the way they were attached in the first place.
So in our simple example, the 4 is subtracted last, so remove it first, (adding 4 on both sides).
3x2− 4 + 4 = 8 + 4 ⇒ 3x2 = 12
Now the x still has two things attached, the three, which is multiplied on, and the 2 which is a power. Powers are done before multiplication, so we remove in the reverse order again. Therefore we divide by 3 on both sides.
3 3x
2 = 12
3 ⇒ x
2 = 4
Now we only have one thing “stuck” to the x, and that is the power of 2. To remove this we simply take the square root on both sides:
x = ±2. Recall that -2 squared is also 4.
3.1.3
Example
Rearrange the following expression for x.
4x + 6 = 2x − 3
Solution
We still wish to rearrange to get x =, but we must notice here that x occurs in two places. We could remove the 3 and 6 on the LHS to obtain
x = 2x − 9 4
(try it as an exercise), but this is not very helpful, as x is now defined in terms of itself, so we still don’t know it’s value.
3.1 Rearranging Equations 14
Instead we first gather all the x terms together, and we do this by per-forming the same operation on both sides. For example, we don’t want the 2x on the RHS, it is positive and so it present by addition. We subtract it on both sides.
4x + 6 − 2x = 2x − 3 − 2x ⇒ 2x + 6 = −3
So we now have a simpler equation, with x only on one side. We can proceed as before now to remove things from the x in the LHS. Subtract 6 on both sides.
2x + 6 − 6 = −3 − 6 ⇒ 2x = −9 Finally divide by 2 on both sides.
x = −9 2
3.1.4
Example
Rearrange the following expression for x:
3(2x + 3) − 6 = 0
Solution
In this case we find that x is encased in brackets. To get at x so we can rearrange for it we could multiply out the bracket and rearrange from there. This is left as an exercise for the reader.
Another way to deal with it is to think of the bracket as an “onion within an onion” to continue the analogy we began above. Begin by taking things off this bracket, rather than x directly. We add 6 both sides
3(2x + 3) − 6 + 6 = 0 + 6 ⇒ 3(2x + 3) = 6. Now divide by 3 on both sides to obtain
3(2x + 3)
3 =
6
3 ⇒ 2x + 3 = 2.
Note the brackets can “fall off” at this point naturally. Now we start with out new “onion”, subtracting 3 both sides.
3.2 Function Notation 15
and finally divide by 2 both sides
2x 2 = − 1 2 ⇒ x = − 1 2
3.1.5
Example
Rearrange the following expression for x:
−3 = 10 x − 5. Solution
We have a more serious problem here, namely that x is on the bottom line. We begin by removing the −5 to clarify the equation, by adding 5 on both sides of course.
−3 + 5 = 10
x − 5 + 5 ⇒ 2 = 10
x
Now, there’s not much attached to x, but the x is still on the bottom line. That means the x has been divided into something (the 10 in this case). To cancel the division by x, we multiply x on both sides.
2x = 10x
x ⇒ 2x = 10
which simplifies our equation quite a lot. We can now divide by two on both sides to finish.
2x
2 =
10
2 ⇒ x = 5
3.2
Function Notation
Very often when we wish to analyse the behaviour of an expression, we make it a function. A function can be thought of as a box, into which goes a value and out of which comes a, usually different, value.
You will have seen functions written as formulae before, for example
3.2 Function Notation 16
is a function for calculating the area of a circle. A value goes in (the radius of the circle) and a value comes out (the area of the circle).
At times we may use notation such as f (x) to represent a function. For example
f (x) = 2x − 3
is a very simple function. For different values we put in (x), we will get different values out f (x). The notation f (x) simply means the value of the output of the function.
This notation is very useful when we want to consider specific values that we insert. For example, we write f (2) to mean “find the output value of the function f , when the input value for x is 2”. You can see that we have simply replaced the x by a 2.
In our example stated above we get
f (2) = 2(2) − 3 = 1 f (−3) = 2(−3) − 3 = −9
f (0) = 2(0) − 3 = −3 f (w) = 2(w) − 3
In the last example, we had to replace x by w, but we can’t work out anything further, so we stop there.
Very often we want to be able to undo the result of our function. For example, we need to be able to reverse multiplication with division in order to rearrange equations, or reverse a square with a square root.
To do this we use an inverse function. An inverse function can be thought of a complementary box to our original one, so that when we plug the output of the first function into it’s input we get the original value. For example, with our simple f (x) above, we inserted 2 and got 1. Our inverse will have to take 1 and give us 2.
To find the inverse function, it is usually easier to give f (x) a letter, like y. In our example we obtain
y = 2x − 3
Now we rearrange the equation for x, using the rules described above. We obtain
x = y + 3 2
3.3 Expansion of Brackets 17
this is left as an exercise for the reader.
We’re pretty much done, but it is usual to label our inverse of f (x) with the notation f−1(x) and have our function in terms of x, not y. So, we swap x and y to obtain
y = x + 3 2 and now use our inverse function formula
f−1(x) = x + 3 2
Recall that with our simple example for f (x), that
f (2) = 2(2) − 3 = 1.
If we now feed this output into the input of the inverse, we should get back to our starting position (2).
f−1(1) = 1 + 3 2 = 2.
Just as before we insert the value in the brackets into x throughout the expression for the inverse function, and you can see that indeed the inverse function here has taken us back to the start.
We will meet other examples of inverse functions throughout this module. It’s not always possible to do this, and not all functions have inverses unfortunately.
3.3
Expansion of Brackets
When we have to multiply something by a bracketed expression, we use the so called distributive law.
a(b + c) = ab + ac(a + b)c = ac + bc We can easily show that this can be extended.
a(b + c + d + · · · ) = ab + ac + ad + · · · There are some simple things worth remembering
• The abscence of a number before an expression is the same as multi-plying by 1.
3.3 Expansion of Brackets 18
• The sign preceding a number belongs to that number and must be included in the multiplication.
• In particular, a minus sign before a bracket means −1 multiplied by that bracket.
3.3.1
Examples
Here are some examples. 1.
−2(3x − 5y + z) = −6x + 10y − 2z 2.
3(2x − y) − (x + 2y) = 6x − 3y − x − 2y = 5x − 5y 3.
4x(y − z + 2(x − y)) = 4x(y − z + 2x − 2y) = 4x(2x − y − z) = 8x2− 4xy − 4xz 4.
2y(3x − 4z(x + z)) = 2y(3x − 4xz − 4z2) = 6xy − 4xyz − 8yz2
3.3.2
Brackets upon Brackets
When we encounter a bracketed expression multiplied by another bracketed expression we can apply the same technique, although it appears more com-plicated.
Consider
(a + b)(c + d)
For the moment, we shall call z = (a + b). Then our expression appears simpler.
z(c + d) = zc + zd Now we reinsert the true value of z.
= (a + b)c + (a + b)d = ac + bc + ad + bd
So we reduce the whole problem to two separate expansions of the type we have already met. There are a number of rules of thumb to make this technique rather simpler, but many depend on multiplying only two brackets
3.3 Expansion of Brackets 19
together, each of which with exactly two terms. We shall examine a general technique without this shortcoming.
Consider once more
(a + b)(c + d)
Pick any bracket, for the sake of demonstration, we shall pick the first. Now take the first term in it (which is a). We now multiply this term on each term of the other bracket in turn, adding all the results.
= ac + ad + · · ·
When we reach the end of the other bracket, we return to the first bracket and move onto the next term, which is now b and do the same again, adding to our existing terms.
= ac + ad + bc + bd + · · ·
Now we return to the first bracket, and move to the next term. We find we have actually exhausted our supply of terms, and so our expansion is really complete.
(a + b)(c + d) = ac + ad + bc + bd
To multiply several brackets together at once we should multiply two only at a time. For example
(a + b)(c + d)(e + f ).
We begin my multiplying one pair together, let us say the first two, to obtain:
= (ac + ad + bc + bd)(e + f ).
We may then complete the expansion, it is left to the reader as an exercise to confirm that the full expansion will be:
= ace + ade + bce + bde + acf + adf + bcf + bdf.
3.3.3
Examples
Here are some examples. 1.
3.4 Factorization 20
2.
(x + y)2 = (x + y)(x + y) = x2+ xy + xy + y2 = x2+ 2xy + y2 (See binomial expansions later).
3.
(x + y)(x − y) = x2− xy + xy − y2 = x2− y2 (This is called the difference of two squares).
4.
(3x − 2y)(x + 3) = 3x2 + 9x − 2xy − 6y 5.
(2x − y)(x + 2y) = 2x2+ 4xy − xy − 2y2 = 2x2+ 3xy − 2y2
3.4
Factorization
Factorization is the opposite of expansion, we often prefer to condense and simplify expressions rather than expand them. Indeed, even in the expansion examples above we simplified the expressions along the way to make life easier for ourselves.
Factorization is recognizing that an expression like this
4x + 2y
could be written as
4x + 2y = 2(2x + y)
simply because we can clearly see that expanding the result gives us the original. A factor common to all terms is observed - in this case the number 2 clearly divides into all the terms. The factor is divided into the expression and written outside the result which is bracketed.
The factor may often be some algebra, and not just a number. In the expression
x2− 3x
we see that x divides into both terms. We can thus write
x2 − 3x = x(x − 3).
The ability to spot factors does not come easily, but with a great deal of practice.
3.5 Laws of Indices 21
3.4.1
Examples
Let us look at some examples. Remember that in each case, expanding the end result should give us our original expression, and this allows you to check and follow the logic.
1. 3x + 12y2− 6z = 3(x + 4y2− 2z) 2. x3+ 3x2+ 4x = x(x2+ 3x + 4) 3. x3+ 3x2 = x2(x + 3) 4. 2x2+ 4xy + 8x2z = 2x(x + 2y + 4xz)
We can also factorise expressions into two or more brackets multiplied together, but this is more difficult and we shall examine it later.
3.5
Laws of Indices
The term index is a formal term for a power, such as squaring, cubing etc, and the plural of index is indices.
There are some simple laws of indices, which are shown in table 3.1.
1 xa× xb = xa+b 2 xa÷ xb = xa−b 3 (xa)b = xab 4 x0 = 1 5 x−b = x1b 6 x1b = b √ x 7 xab = b √ xa
3.5 Laws of Indices 22
3.5.1
Example “proofs”
We shall attempt to show how a selection of these results “work”, but such demonstrations are for understanding and are not examinable.
Let us consider the first law, with a concrete example:
x3× x2
We don’t know what the number x is, but all that is important is that the “base” values of each number are the same.
We recall that powers mean a string of the same thing multiplied together, so that: x3× x2 = x × x × x | {z } x3 × x × x | {z } x2 .
Clearly there is no difference between the “×” inside the braced section and between them. In otherwords, this is just
x3× x2 = x × x × x | {z } x3 × x × x | {z } x2 = x × x × x × x × x | {z } x5 = x5
a string of five xs multiplied together, exactly the definition of x5, and the 3 and 2 add to make 5.
We shall show how one other result works, using the more general a and b.
Consider
(xa)b. By definition, this is just
xa× xa× · · · × xa
| {z }
b times
with b of these xaterms. (Recall, x6just means 6 of the x terms multiplied
together).
Now each xaterms is itself a collection of a xs multiplied together. So we
can expand further
b groups z }| { x × · · · × x | {z } a times × x × · · · × x | {z } a times × · · · × x × · · · × x | {z } a times .
Each collection of x terms with a brace below is an expanded xa, so
3.6 Laws of Surds 23
braces, each containing a xs. Therefore we have a long string of xs multiplied together, which are a × b in number, exactly the definition of xa×b.
It is worth going through this argument in a concrete case, for example (x2)3 to help follow the logic.
3.5.2
Examples
Here are some examples 1. 25× 23 = 25+3= 28(= 256) 2. 28÷ 23 = 28−3 = 25(= 32) 3. (32)2 = 32×2= 34(= 81) 4. 1614 = 4 √ 16(= 2) 5. 15−3 = 1 153(= 1 3375) 6. 27−23 = 1 2723 = √3 1 272 = 1 9
3.6
Laws of Surds
A surd is technically a square root which has an irrational value, but we often talk about surds whenever we manipulate square roots. We have two main results for manipulating square roots, as shown in table 3.2
1 √a√b = √ab 2 √ a √ b = pa b
3.7 Quadratic Equations 24
3.6.1
Examples
The second law of surds is most often used to work out the square roots of fractions. 1. r 1 4 = √ 1 √ 4 = 1 2 2. r 4 7 = √ 4 √ 7 = 2 √ 7
The first law was often used to split large square roots into smaller ones, by attempting to divide the original number by a perfect square.
1. √
12 =√4 × 3 =√4 ×√3 = 2√3
2. √
80 =√16 × 5 =√16 ×√5 = 4√5
A very important application of this to come later is that of complex numbers.
3.7
Quadratic Equations
A quadratic equation in x is an equation of the form
ax2+ bx + c = 0, a 6= 0
where a, b and c are constants (that is, they do not change value as x does.
It’s vital that a is not zero, or the equation collapses into that of a straight line. However, it is quite allowable to have b or c (or both) zero.
3.7.1
Examples
Here are some examples of equations which are, and which are not quadratic equations, shown in table 3.3. Equations 1,2 and 3 are genuine quadratics, even though terms are missing in 2 and 3. Equation 4 is the equation of a straight line, or if you like a = 0 which is not permitted. Equation 5 contains an x3 term, and so is a cubic equation and not a quadratic whose highest term must be x2.
3.7 Quadratic Equations 25
Equation Quadratic? a b c
1 2x2+ 3x − 4 = 0 YES 2 3 -4
2 x2+ 2x = 0 YES 1 2 0
3 x2+ 3 = 0 YES 1 0 3
4 4x + 2 = 0 NO n/a n/a n/a
5 x3+ 3x − 2 = 0 NO n/a n/a n/a
Table 3.3: Examples of quadratic equations
3.7.2
Graphical interpretation
To examine the solutions of this equation, it is helpful to consider the graph of the function given by
y = ax2+ bx + c.
When this graph is plotted it gives a characteristic “U” shaped curve, called parabola. It has many interesting properties, but the most important is that it is symmetrical about a vertical line through the maximum, or minimum point. Our equation corresponds to the above function when y = 0, or to put it another way, the solutions of our equation occur when the curve cuts the x-axis (where y is zero).
Figure 3.1: The quadratic equation
This gives us our first problem, which is that we cannot even be sure that the equation has solutions. If this seems strange or confusing, recall that we never claimed that all equations could be solved. Our problem falls into three categories.
3.7 Quadratic Equations 26
1. The curve cuts the x-axis twice;
2. The curve cuts the x-axis once only (just touching, no more);
3. The curve does not cut the x-axis at all.
This situation is shown in figure 3.1. In this figure all the curves have been drawn as “maximum” parabolas, many quadratics show “minimum” parabolas but the cases are just the same. The y-axis has been ommitted as it is not relevant to the problem at hand which is determining the number of solutions of the corresponding equation.
3.7.3
Factorization
One approach to try and solve a quadratic is the so called factorization or sum and product method of solution.
We imagine that we can write our equation in the form
(x − α)(x − β) = 0.
Why have we written it this way? Well, for a product of two things to be zero, one or other, or both must be zero. Therefore in this case, if the left-hand bracket is zero we see that x = α, and if the right-hand bracket is zero we see that x = β. Therefore α and β are our two solutions.
If we expand these brackets out, we obtain
x2− αx − βx + αβ = x2− (α + β) + αβ = 0
Now we try to compare this to our quadratic equation, but before we do so, we “standardize” our equation, by dividing it by a:
ax2+ bx + c = 0 ⇒ x2+ b ax +
c a = 0.
We now compare the coefficients of the x terms in each equation (assum-ing they are in fact two forms of the same equation). (A coefficient is a number multiplied upon a term).
Comparing we obtain
x2 term 1 = 1 This is why we “standardize”;
x term ba = −(α + β)
3.7 Quadratic Equations 27
The two significant equations are this
−b
a = α + β; c a = αβ which tell us that the sum of the solutions is −b
a and that the product
of the solutions is ca. If we can guess two numbers with these properties, we have solved the quadratic equation.
This method has the advantages that the theory is simple and straight forward, but has two crippling disadvantages. Firstly, we have to guess two numbers, and even in simple cases this may be very difficult, in a hard example next to impossible. Secondly, and more importantly, before we start we have no way of knowing how many solutions the equation has. If it has two, this theory works well, if it has one, then α and β have the same value (we say the solution is repeated). However, if there are no solutions, it will be impossible to guess our two numbers, as they do not exist, but this may not be obvious from the start.
3.7.4
Quadratic solution formula
Fortunately, we do have a more flexible method of solution, but its proof is difficult.
The proof is given here, but is not required.
First of all consider the problem, we cannot simply take a square root to remove the square, as the other x term gets in the way. Consequently, we try to absorb the x term and the x2 term into one term. This is a difficult
procedure known as completing the square. Consider
(x + y)2 = (x + y)(x + y) = x2+ 2xy + y2 We wish to let this be our x2 + b
ax terms. For this to be true, we must
let y = 2ab . Note that this does not correspond to exactly what we want:
x + b 2a 2 = x2+ b ax + b2 4a2
which is our quadratic, except that the c
a term is missing, and the last
term above is extra, so we add and subtract these two terms respectively.
x2+ b ax + c a = 0 ⇒ x + b 2a 2 + c a = b2 4a2
3.7 Quadratic Equations 28 ⇒ x + b 2a 2 = b 2 4a2 − c a = b2− 4ac 4a2 ⇒ x + b 2a = ± r b2− 4ac 4a2
which using the laws of surds (see 3.5), yields our final result.
x = −b ± √
b2− 4ac
2a
3.7.5
The discriminant
As well as giving a fool-proof method of solving a quadratic, the solution formula has a small section which tells us some useful information. The section
b2− 4ac
which was the contents of the square root is known as the discriminant of the equation, as it tells us how many solutions the equation has.
b2− 4ac > 0 Two real solutions Case 1 above
b2− 4ac = 0 One real solution Case 2 above b2− 4ac < 0 No real solutions Case 3 above
3.7.6
Examples
Here are some examples of quadratic equations and their solutions. 1. x2 + 2x − 3 = 0 2. x2 − 4x + 4 = 0 3. x2+ x + 1 = 0 Solutions
1. In this case a = 1, b = 2, c = −3. We plug this into the solution formula to obtain
3.7 Quadratic Equations 29
x = −2 ±p2
2− 4(1)(−3)
2
which when calculated out yields answers of 1 and −3. 2. In this case a = 1, b = −4, c = 4. The solution formula yields
x = +4 ±p(−4)
2− 4(1)(4)
2
which when calculated out yields answers of 2 and 2. The repeated so-lution is nothing unusual, and indicated that this quadratic has only one solution.
3. In this case a = 1, b = 1, c = 1. We shall follow this case in more detail.
x = −1 ±p1 2− 4(1)(1) 2 = −1 ± √ 1 − 4 2 = −1 ± √ −3 2
Note that we have a negative square root here. We cannot calculate this, suppose that the answer was a positive number, then when squared we get a positive number, not −3. We also get a positive number when we square a negative number, and we get zero when we square zero. Thus no number can be √−3.
There are no solutions to this quadratic equation.
3.7.7
Special cases
Observe that if b = 0, or if c = 0 in the quadratic, the equation can be solved more directly. We show how for completeness, although the formula may still be used in these cases.
b = 0
We have here
ax2+ c = 0 ⇒ ax2 = −c ⇒ x2 = −c a Therefore
3.8 Notation 30
x = ± r
−c a.
Note that −ac must be positive, and this will be the case only if a and c have different signs, otherwise there are no solutions. Note also the ± in the formula, solving directly often leads to forgetting the solution coming from the negative branch of the square root.
c = 0
In this case we have
ax2+ bx = 0 ⇒ x(ax + b) = 0 from which we obtain, that either
x = 0
which is one solution, or
ax + b = 0 ⇒ x = −b a
3.8
Notation
We now introduce two more items of notation.
3.8.1
Modulus or absolute value
A very useful idea in mathematics is the notion of absolute value of a real number. If x is a real number, we define the modulus of x, or mod x, denoted |x| as follows.
|x| = larger of x and −x
3.8 Notation 31
Examples
Here are some examples
|2| = larger of 2 and −2 = 2; | − 2| = larger of −2 and 2 = 2; |0| = larger of 0 and −0 = 0.
In other words, the modulus function simply strips off any leading minus sign on the number.
Alternative definitions
There are some alternative, but equivalent definitions of |x|:
|x| = x if x ≥ 0 −x if x < 0 and |x| =√x2
where we adopt the convention that the square root takes the positive branch only, unless we include ±, which is commonly accepted.
3.8.2
Sigma notation
Suppose that f (k) is some expression involving k (a function of k formally speaking). For example, f (k) could be 2k or 2k + 1 etc. Then, the notation
n
X
k=m
f (k)
is a shorthand for the expression
f (m) + f (m + 1) + · · · + f (n)
In other words, we insert the value at the bottom of the sigma into the f expression, then insert that value plus one, plus two, etc., until we reach the value on top of the sigma, and add all the results together.
We will use the symbol ∞ to denote the lack of an endpoint in the sum-mation.
The ranges above and below the sigma are sometimes ommitted when it is clear what is being summed.
3.9 Exponential and Logarithmic functions 32
Examples
Here are some examples of sigma notation 1. 4 X k=0 2k = 20 + 21+ 22 + 23+ 24 = 31 2. ∞ X k=4 k = 4 + 5 + 6 + 7 + 8 + · · · 3. P5 k=2(−1) k+1 1 3k = (−1)3 1 32 + (−1)4 313 + (−1)5 314 + (−1)6 315 = − 1 32 + 1 33 − 1 34 + 1 35
3.8.3
Factorials
The notation n!, where n is an integer greater or equal to 0, is spoken “n factorial” and is a shorthand for
n × (n − 1) × (n − 2) × · · · × 2 × 1.
So for example
5! = 5 × 4 × 3 × 2 × 1 = 120.
It should be clear that this time of expressions gets very large, very quickly, and indeed 69! is so large that most calculators are unable to repre-sent it.
We note that by convention we accept 1! = 1 and 0! = 1.
3.9
Exponential and Logarithmic functions
We now consider another, slightly more complex type of function.
3.9.1
Exponential functions
In quadratics, we had x terms with specific constant powers, such as x2. A very powerful function is formed when x is in the exponent (yet another name for “power” or “index”).
3.9 Exponential and Logarithmic functions 33
y = kx
where k is some positive constant. These functions are important due to their extraordinary ability to climb or decrease, as we shall see later when we examine their graphs.
3.9.2
Logarithmic functions
For a positive number n, the logarithm or log of n to base b, written logbn is the power to which b must be raised to give n. To put this in mathematics:
logbn = x ⇔ bx = n
The antilogarithm or antilog of n to the base b is bn.
There are two main “types” of logarithms in use today:
• log10, often written simply as log; • loge, often written1 as ln.
Examples
Logs are often used to condense a very large range of numbers to a more managable one.
Examine how in the following table the logs of a large range of values from one thousandth, to one thousand are contracted to a range from -3 to 3. Verifying that there logarithms are correct is left as a simple exercise to the reader.
Value 0.001 0.01 0.1 1 10 100 1000
Log10 -3 -2 -1 0 1 2 3
The scales of pH (chemical scale of acidity) and the decibel range, the Richter scale of earthquake intensity are all logarithmic,2 base 10.
Therefore a pH of 6 is 10 times more acidic than the neutral 7, and 5 is 100 times more acidic than 7, etc.
1The Scottish mathematician John Napier (1550 - 1617) did a great deal of work
on methods of computation and published his Mirifici logarithmorum canonis descripto (Discription of the marvellous rule of logarithms) in 1614. In his honour logs base e are often called Naperian logarithms. Logs were first used to help perform large calculations.
3.9 Exponential and Logarithmic functions 34
Warnings
Just as we have problems with the square roots of negative numbers, so we cannot take the logarithms of negative numbers, or of zero.
Laws of Logarithms
Logarithms are so useful because they exhibit the following properties, known as the laws of logarithms. These are closely related to the laws of indices and are shown in table 3.4.
1 logn(x × y) = logn(x) + logn(y) 2 logn(x ÷ y) = logn(x) − logn(y) 3 logn(xy) = y logn(x)
4 lognn = 1
5 logn1 = 0
Table 3.4: Laws of Logarithms
These laws easily account for their use in computation. To multiply two large numbers, one would find their logarithms from tables, add these two values, and then by the first law, this is the logarithm of the product, so one merely antilogs this number to obtain the answer.
The other major application ot these is in solving equations where the variable is in the index, i.e. some sort of exponential expression.
3.9.3
Logarithms to solve equations
When we obtain equations like this
ax = b
it is sometimes possible to guess the power to which we must raise a to obtain b. However, more often than not, a and b are not simple integers, and therefore almost impossible to guess x. In this case, take logarithms on both sides of the equation.
3.9 Exponential and Logarithmic functions 35
⇒ lognax = log nb
⇒ x logna = lognb (using laws of logs 3) ⇒ x = lognb
logna
3.9.4
Examples
Solve the following equations for x. 1. 3x= 27 2. 3x= 30 3. 22x− 5(2x) + 6 = 0 Solutions
1. Although it is clear here that the answer is x = 3 we use logs to illustrate the point. It really doesn’t matter what base of logs we use, provided we are consistent. Take logs both sides
log 3x= log 27 ⇒ x log 3 = log 27
using the third law of logs (see 3.9.2). We now can perform ordinary rearranging.
x = log 27 log 3 = 3
2. This is only slightly more complicated, although it is not possible to guess the answer easily.
log 3x= log 30 ⇒ x log 3 = log 30
x = log 30
log 3 = 3.0959 to four decimal places.
3. This problem looks very complicated. Taking logs immediately will get us into trouble. Partly because we have no law to split logs over addition (note this carefully, a common mistake), and that the log of zero is a problem (it’s something like −∞).
We might observe that the equation roughly resembles a quadratic (see 3.7, and this is the key. Let us assign u = 2x, then clearly
3.9 Exponential and Logarithmic functions 36
u = 2x ⇒ u2 = (2x)2 = 22x
by the laws of indices (see 3.5). Placing these into the equation yields
u2 − 5u + 6 = 0
which is now a plain quadratic. Solving by whatever method yields u = 2 and u = 3 as solutions. Having dealt with the quadratic, we now deal with the exponential problem, recall that u = 2x.
Consider the u = 2 solution, this means 2x = 2 so that clearly x = 1,
with no logarithms required.
The u = 3 solution proceeds as follows
2x = 3
and following the procedure above we obtain
x = log 3
log 2 = 1.5850
to four decimal places. So we have two solutions x = 1, x = 1.5850.
3.9.5
Anti-logging
Logarithms are used to undo expressions in which the x we wish to obtain is in the power. Similarly, exponential functions may be used to undo logarithms. Recall our definition.
logbn = x ⇔ bx = n this means that
blogbn = n
To put it another way, we can remove a logarithm by taking the base number to the power of the logarithm. Let’s use some numbers to illustrate the point.
Example
We know that
3.9 Exponential and Logarithmic functions 37
so suppose we were asked to solve the equation
log10x = −3
we know that x = 1000, but we might not notice this in a more difficult problem, or one we had not previously seen. To remove the log, we take the base number (in this case we are using base 10) and raise it to both sides:
10log10x = 10−3.
Now we know that the log and the anti-log (the process of raising the base to this power) cancel each other out on the left, so we obtain
x = 10−3 = 1000 and our equation is solved.
3.9.6
Examples
Solve the following equations for x: 1.
log x2 = 2 2.
4 ln x = 70
Solutions
1. In the absence of a specific base, we assume base 10, as noted above. So we may antilog both sides directly here, raising 10 to both sides.
10log x2 = 102 On the LHS, the anti-log and log cancel, leaving
x2 = 102 = 100 ⇒ x = ±10
2. In this case, the base of the logarithm is e. We could immediately take anti-logs on both sides, but the 4 in front of the ln x makes it messy. It is easier to rearrange to ln x first (onion within onion as before).
ln x = 70 4
