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Fractions, decimals and percentages 13 Assignment 1 This assignment covers the material contained in Chapters 1 and 2. The marks for each question are shown in brackets at the end of each question. 5. Simplify (a) 2 2 3 ÷ 3 1 3 ( 1 1 (b) ( 4 7 × 2 1 ) ÷ 3 + 1 ) + 2 7 5 24 4 (9) 1. Evaluate the following: (a) 2016 − (−593) (b) 73 + 35 ÷ (11 − 4) (c) 120 − 133 ÷ 19 + (2 × 17) 15 (d) 2 5 − 1 15 + 5 6 (e) 49.31 − 97.763 + 9.44 − 0.079 (14) 2. Determine, by long multiplication 37 × 42 (3) 3. Evaluate, by long division 4675 (3) 11 4. Find (a) the highest common factor, and (b) the lowest common multiple of the following numbers: 15 40 75 120 (6) 6. A piece of steel, 1.69 m long, is cut into three pieces in the ratio 2 to 5 to 6. Determine, in centimeters, the lengths of the three pieces. (4) 7. Evaluate 576.29 19.3 (a) correct to 4 significant figures (b) correct to 1 decimal place (4) 8. Express 7 9 correct to 2 decimal places (2) 46 9. Determine, correct to 1 decimal places, 57% of 17.64 g (2) 10. Express 54.7 mm as a percentage of 1.15 m, correct to 3 significant figures. (3)
3 Indices, standard form and <strong>engineering</strong> notation 3.1 Indices The lowest factors of 2000 are 2 × 2 × 2 × 2 × 5 × 5 × 5. These factors are written as 2 4 × 5 3 , where 2 and 5 are called bases and the numbers 4 and 3 are called indices. When an index is an integer it is called a power. Thus, 2 4 is called ‘two to the power of four’, and has a base of 2 and an index of 4. Similarly, 5 3 is called ‘five to the power of 3’ and has a base of 5 and an index of 3. Special names may be used when the indices are 2 and 3, these being called ‘squared’and ‘cubed’, respectively.Thus 7 2 is called ‘seven squared’ and 9 3 is called ‘nine cubed’. When no index is shown, the power is 1, i.e. 2 means 2 1 . Reciprocal The reciprocal of a number is when the index is −1 and its value is given by 1 divided by the base. Thus the reciprocal of 2 is 2 −1 and its value is 1 or 0.5. Similarly, the reciprocal of 5 is 5−1 2 which means 1 or 0.2 5 Square root The square root of a number is when the index is 1 2 , and the square root of 2 is written as 2 1/2 or √ 2. The value of a square root is the value of the base which when multiplied by itself gives the number. Since 3 × 3 = 9, then √ 9 = 3. However, (−3) × (−3) = 9, so √ 9 =−3. There are always two answers when finding the square root of a number and this is shown by putting both a + and a − sign in front of the answer to a square root problem. Thus √ 9 =±3 and 4 1/2 = √ 4 =±2, and so on. Laws of indices When simplifying calculations involving indices, certain <strong>basic</strong> rules or laws can be applied, called the laws of indices. These are given below. (i) When multiplying two or more numbers having the same base, the indices are added. Thus 3 2 × 3 4 = 3 2+4 = 3 6 (ii) When a number is divided by a number having the same base, the indices are subtracted. Thus 3 5 3 = 2 35−2 = 3 3 (iii) When a number which is raised to a power is raised to a further power, the indices are multiplied. Thus (3 5 ) 2 = 3 5×2 = 3 10 (iv) When a number has an index of 0, its value is 1. Thus 3 0 = 1 (v) A number raised to a negative power is the reciprocal of that number raised to a positive power. Thus 3 −4 = 1 3 4 1 Similarly, = 23 2−3 (vi) When a number is raised to a fractional power the denominator of the fraction is the root of the number and the numerator is the power. Thus 8 2/3 = 3 √ 82 = (2) 2 = 4 √ and 25 1/2 = 2 251 = √ 25 1 =±5 (Note that √ √ ≡ 2 ) 3.2 Worked problems on indices Problem 1. Evaluate: (a) 5 2 × 5 3 , (b) 3 2 × 3 4 × 3 and (c) 2 × 2 2 × 2 5
Page 2 and 3: Basic Engineering Mathematics
Page 4 and 5: Basic Engineering Mathematics Fourt
Page 6 and 7: Contents Preface xi 1. Basic arithm
Page 8 and 9: Contents vii 13.3 Graphical solutio
Page 10 and 11: Contents ix 27.4 Vector subtraction
Page 12 and 13: Preface Basic Engineering Mathemati
Page 14 and 15: 1 Basic arithmetic 1.1 Arithmetic o
Page 16 and 17: Basic arithmetic 3 12. −23148 −
Page 18 and 19: Basic arithmetic 5 Problem 18. 23
Page 20 and 21: Fractions, decimals and percentages
Page 28 and 29: Indices and standard form 15 From l
Page 30 and 31: Indices and standard form 17 16 2
Page 32 and 33: Indices and standard form 19 3.6 Fu
Page 34 and 35: 4 Calculations and evaluation of fo
Page 36 and 37: Calculations and evaluation of form
Page 44 and 45: Computer numbering systems 31 3. (a
Page 46 and 47: Computer numbering systems 33 11 11
Page 48 and 49: Computer numbering systems 35 Probl
Page 50 and 51: 6 Algebra 6.1 Basic operations Alge
Page 52 and 53: Algebra 39 ) 2a 2 − 2ab − b 2 2
Page 54 and 55: Algebra 41 Using the third and four
Page 56 and 57: Algebra 43 x 2 is a common factor o
Page 58 and 59: Algebra 45 10. p 2 − 3pq × 2p ÷
Page 60 and 61: 7 Simple equations 7.1 Expressions,
Page 62 and 63: Simple equations 49 7.3 Further wor
Page 64 and 65: Simple equations 51 Problem 18. A r
Page 66 and 67: Simple equations 53 Squaring both s
Page 68 and 69: Transposition of formulae 55 Rearra
Page 70 and 71: Transposition of formulae 57 Now tr
Page 72 and 73: Transposition of formulae 59 6. 7.
Page 74 and 75: Simultaneous equations 61 Hence y =
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Simultaneous equations 63 It is oft
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Simultaneous equations 65 Thus the
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Simultaneous equations 67 Substitut
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10 Quadratic equations 10.1 Introdu
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Quadratic equations 71 In Problems
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Quadratic equations 73 Summarizing:
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Quadratic equations 75 Neglecting t
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11 Inequalities 11.1 Introduction t
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Inequalities 79 i.e. 11.4 Inequalit
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Inequalities 81 Solving quadratic i
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12 Straight line graphs 12.1 Introd
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Straight line graphs 85 Problem 2.
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Straight line graphs 87 (b) Rearran
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Straight line graphs 89 Degrees Fah
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Straight line graphs 91 y 147 140 A
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Straight line graphs 93 Stress (pas
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Graphical solution of equations 95
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14 Logarithms 14.1 Introduction to
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Logarithms 105 i.e. −5 = 4x i.e.
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15 Exponential functions 15.1 The e
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Exponential functions 109 If in the
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Exponential functions 111 Problem 1
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Exponential functions 113 ( ) 5.14
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Exponential functions 115 (b) Trans
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16 Reduction of non-linear laws to
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Reduction of non-linear laws to lin
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Graphs with logarithmic scales 125
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18 Geometry and triangles 18.1 Angu
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Geometry and triangles 133 (d) 227
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Geometry and triangles 135 (a) Equi
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Geometry and triangles 137 Hence XZ
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Geometry and triangles 139 Now try
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Geometry and triangles 141 Assignme
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Introduction to trigonometry 143 Fr
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Introduction to trigonometry 145 Si
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Introduction to trigonometry 147 If
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Introduction to trigonometry 149 To
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20 Trigonometric waveforms 20.1 Gra
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Trigonometric waveforms 153 between
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Trigonometric waveforms 155 45° 60
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Trigonometric waveforms 157 y 4 0 y
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Trigonometric waveforms 159 v rads/
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Trigonometric waveforms 161 Assignm
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Cartesian and polar co-ordinates 16
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Cartesian and polar co-ordinates 16
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Areas of plane figures 167 W X Tabl
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Areas of plane figures 169 (b) Area
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Areas of plane figures 171 14. Calc
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Areas of plane figures 173 is 12 50
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The circle 175 Q B Now try the foll
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The circle 177 From equation (2), a
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The circle 179 Thus, for example, t
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Volumes of common solids 181 Proble
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Volumes of common solids 183 Proble
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Volumes of common solids 185 Fig. 2
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Volumes of common solids 187 From F
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Volumes of common solids 189 Volume
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25 Irregular areas and volumes and
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Irregular areas and volumes and mea
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Triangles and some practical applic
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27 Vectors 27.1 Introduction Some p
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Vectors 209 r 10° b Having obtaine
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Vectors 211 b Fig. 27.11 o Fig. 27.
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Vectors 213 N Thus the velocity of
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Adding of waveforms 215 y 6.1 6 4 2
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Adding of waveforms 217 The horizon
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Number sequences 219 The first four
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Number sequences 221 5. Determine t
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Number sequences 223 Hence 1 ( 1 9
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Number sequences 225 Now try the fo
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Presentation of statistical data 22
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31 Measures of central tendency and
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Measures of central tendency and di
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Measures of central tendency and di
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32 Probability 32.1 Introduction to
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Probability 243 (a) The probability
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Probability 245 Two brass washers a
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33 Introduction to differentiation
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Introduction to differentiation 249
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34 Introduction to integration 34.1
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Introduction to integration 259 ∫
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Introduction to integration 261 Pro
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Introduction to integration 263 A t
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Introduction to integration 265 Ass
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List of formulae 267 (ii) Parallelo
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List of formulae 269 Cartesian and
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Answers to exercises 271 Exercise 7
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Answers to exercises 273 7. ab 6 c
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Answers to exercises 275 5. 0.013 3
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Index Abscissa, 83 Acute angle, 132
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Index 287 Interior angles, 132, 134
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