history of mathematics - National STEM Centre

More documents

Recommendations

Info

Searching for the abstract Chapter 11 Towards a rigorous approach Introduction In this unit you will see an example of how, over time, a piece of mathematics changes. At first the mathematics gets created with little or no justification or proof that the mathematics is 'correct'. With time, or because its use changes, so the mathematics becomes more formally established. It gets 'proved' This may remind you of the process of investigating and proving which Book 4, Chapter 16, encouraged you to adopt. You make conjectures and work on them, often using examples, pictures and diagrams. Only later do you get round to proving your conjectures. This process of proof is sometimes called formalising. When mathematics is formalised, it can also become extended and generalised. In Chapter 10 you will follow the development of negative numbers and rules for combining them from their early use in China in the 3rd century BC through to 18th century Europe. In Chapter 11 you will discover how negative numbers became 'respectable' and how the rules for combining all numbers, not just negative ones, were put on a firm footing. With this status, number systems can be further developed into more elaborate algebraic systems, which share some properties with number systems. You will follow the ideas of Wessel and Clifford, who both proposed extensions to the number system. This unit is designed to take about 10 hours of your learning time. About half of this time will be outside the classroom. It is more important that you understand the general message of these two chapters, rather than becoming too involved in all the details of the activities, f | Work through the chapters in sequence. There are summaries and further practice exercises in Chapter 12. Mathematical knowledge assumed • you will need to know a little about vectors in both two and three dimensions • it is also helpful, but not essential, to have a little knowledge of complex numbers, but any information that you need is explained in the unit. 123
10.1 Visualising negative numbers 10.2 Subtracted numbers are more acceptable 10.5 A geometrical argument 10.7 The equivalence of the different cases 10.9 Negative numbers I and the number line 10.10 Objections to negative numbers'': 10.11 n A debate about ; negative numbers;* 124 'Two minuses make a plus' 10.3 A Chinese livestock problem 10.6 The Greek version 10.8 Why a lapse in developnea.fl. In this chapter, you will learn about the struggle that mathematicians had, first, with accepting negative numbers and, secondly, with the idea that 'minus x minus makes a plus'. 10.4 The Chinese rules Even though you may yourself be quite at ease using negative numbers, you will see in this chapter that the concept created controversy and objections from the very earliest times. It is only comparatively recently, in the last century or so, that mathematicians have accepted negative numbers as a proper part of mathematics. Even now, many people struggle to make sense of them. As you read about the objections to negative numbers, you may find yourself sympathising with those who raised the objections in the first place. You may also see that there is more to negative numbers than you might previously have realised. Activities 10.1 and 10.2, together with the optional activities 10.3 and 10.4, are about negative numbers and the ways that people thought about them. The remaining activities, including the optional activities, 10.6 and 10.8, show the way that people thought about carrying out arithmetic operations with negative numbers, and how they 'explained' their ideas. In Activity 10.11, you will be asked to take part in a debate about 'two minuses make a plus', so you should, if you can, work on it in a small group. All the activities are suitable for working in a small group. Activities 10.3,10.4,10.6 and 10.8 are optional. Work on Activity 10.11 in a small group if you can. Work through the activities in sequence. Negative numbers Activity 10.1 Visualising negative numbers 1 You already have different ways of thinking about and visualising negative
Page 2 and 3:
Nuffield Advanced Mathematics Histo
Page 4 and 5:
Contents Introduction 7 Unit 1 The
Page 6 and 7:
Introduction This book, about the h
Page 8 and 9:
The Babylonians Chapter 1 Introduct
Page 10 and 11:
Introduction to the Babylonians On
Page 12 and 13:
Figure 1.2 Stele inscribed with Ham
Page 14 and 15:
Token V Figure 1.5 7 Introduction t
Page 16 and 17:
There are no practice exercises for
Page 18 and 19:
A sexagesimal number system is a sy
Page 20 and 21:
ff m yfr < n Figure 2.5 «« Activi
Page 22 and 23:
Activity 2.5 Babylonian fractions 2
Page 24 and 25:
Activity 2.6 Babylonian arithmetic
Page 26 and 27:
Figure 2.10 Activity 2.9 2 Babyloni
Page 28 and 29:
Type Quadratic Table 2.1 x +ax = b
Page 30 and 31:
Activity 2.12 Geometry Figure 2.13a
Page 32:
Practice exercises for this chapter
Page 35 and 36:
3 An introduction to Euclid fthese
Page 37 and 38:
The Greeks Activity 3.3 B Figure 3.
Page 39 and 40:
34 The Greeks Activity 3.4 g Now us
Page 41 and 42:
36 The Greeks inscribes another wit
Page 43 and 44:
38 The Greeks Activity 3.7 therefor
Page 45 and 46:
The Greeks Interpret postulate 3 in
Page 47 and 48:
42 More Greek mathematics 4.1 Some
Page 49 and 50:
The Greeks Q.E.D. stands for 'quod
Page 51 and 52:
Figure 4.3 46 The Greeks • treat
Page 53 and 54:
Figure 4.6 48 The Greeks together w
Page 55 and 56:
The Greeks Activity 4.8 The cone Th
Page 57 and 58:
The Greeks Practice exercises for t
Page 59 and 60:
5 Arab mathematics You should think
Page 61 and 62:
The Arabs Western Arabic, or Gobar,
Page 63 and 64:
The Arabs Activity 5.2 Figure 5.3a
Page 65 and 66:
7776 Arabs Activity 5.4 This proble
Page 67 and 68:
62 The Arabs Activity 5.6 Horner's
Page 69 and 70:
64 The Arabs Indian text. For over
Page 71 and 72:
D B P N M K Figure 5,9a The Arabs D
Page 73 and 74:
68 The Arabs Figure 5.10 Activity 5
Page 75 and 76:
The Arabs Practice exercises for I
Page 77 and 78: 72 6.1 Read about Descartes The app
Page 79 and 80: Descartes Activity 6.1 Head about D
Page 81 and 82: 76 Descartes Figure 6.5 Figure 6.5
Page 83 and 84: Descartes fViete (Vieta in Latin) w
Page 85 and 86: Descartes The next two sections con
Page 87 and 88: 82 Descartes Activity 6.9 a 3 - b 3
Page 89 and 90: i——— Figure 6.12 Figure 6.13
Page 91 and 92: Descartes Practice exercises for th
Page 93 and 94: Figure 7.2 Descartes Activity 7.1 i
Page 95 and 96: N- i- L Descartes Thus I have In th
Page 97 and 98: A Descartes Activity 7.7 Reflecting
Page 99 and 100: 94 Descartes 23 It is true that the
Page 101 and 102: 96 Descartes Figure 7.11 shows a ge
Page 103 and 104: 98 Descartes It is clear that the p
Page 105 and 106: Descartes ractice exercises lor thi
Page 107 and 108: 102 Descartes Activity 8.1 Activity
Page 109 and 110: 104 Descartes In the next section y
Page 111 and 112: Descartes Today we call these numbe
Page 113 and 114: Descartes Activity 8.9 Normals Figu
Page 115 and 116: Descartes Practice exercises for th
Page 117 and 118: 772 The beginnings of calculus 9.1
Page 119 and 120: Figure 9.4 774 Calculus you can wri
Page 121 and 122: Calculus including Descartes, under
Page 123 and 124: 118 Calculus Activity 9.7 Cavalieri
Page 125 and 126: Figure 9. 7 / 120 Calculus dy 1 Wri
Page 127: Calculus JPractiee exercises this c
Page 131 and 132: 126 Searching for the abstract For
Page 133 and 134: Searching for the abstract Activity
Page 135 and 136: Searching for the abstract John Wal
Page 137 and 138: Searching for the abstract 31. Hith
Page 139 and 140: 134 Searching for the abstract If y
Page 141 and 142: Searching for the abstract Practice
Page 143 and 144: 138 Searching for the abstract Extr
Page 145 and 146: Searching for the abstract /esseTs
Page 147 and 148: Searching for the abstract d Do you
Page 149 and 150: 144 Searching for the abstract math
Page 151 and 152: 146 Searching for the abstract As a
Page 153 and 154: Searching for the abstract How does
Page 155 and 156: Searching for the abstract ; Practi
Page 157 and 158: 752 12 Summaries and exercises Two
Page 159 and 160: 754 12 Summaries and exercises 5 Ar
Page 161 and 162: 156 12 Summaries and exercises usin
Page 163 and 164: 158 Hints Activity 2.3, page 15 2 F
Page 165 and 166: 13 Hints Activity 7.3, page 89 1 Wr
Page 167 and 168: outcome 1 step to the right Figure
Page 169 and 170: 14 Answers 3 0;6,40 and 0;2,13,20 4
Page 171 and 172: 14 Answers 3 If AD intersects the l
Page 173 and 174: 74 Answers 2 Suppose that you can f
Page 175 and 176: 14 Answers 4 If he had a daughter t
Page 177 and 178: 14 Answers Activity 6.5, page 79 1
Page 179 and 180:
14 Answers 6 The equation z 2 + az
Page 181 and 182:
14 Answers b This meets the ellipse
Page 183 and 184:
14 Answers To get an interpretation
Page 185 and 186:
14 Answers of multiplication by neg
Page 187 and 188:
14 Answers b The required number is
Page 189 and 190:
14 Answers operator has the effect
Page 191 and 192:
14 Answers A i- B 5 E %2 5)-25 = 39
Page 193 and 194:
Index Ptolemy 49, 55, 58 Pythagoras
show all

history of mathematics - National STEM Centre

Create successful ePaper yourself

Delete template?

Save as template?