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10 'Two minuses make a plus' And, to use an awkward comparison which M. Chabert's pronounced Grenoblois drawl made even more clumsy, let us suppose that the negative quantities are a man's debts; how, by multiplying a debt of 10,000 francs by 500 francs, can this man have, or hope to have, a fortune of 5,000,000 francs? Are M. Dupuy and M. Chabert hypocrits like the priests who come to say Mass at my grandfather's, and can my beloved mathematics be a fraud? Oh, how eagerly I would have listened then to one word about logic, or the art of finding out the truthl Reflecting on Chapter 10 What you should know • negative numbers were initially rejected by the Chinese, Greeks and Babylonians, among others, as not being practical • subtracted numbers on the other hand were acceptable to these people and, likewise, subtraction as an operation was considered acceptable • rules for addition and subtraction of positive and negative numbers appeared as early as the 3rd century BC in China • rules for multiplying subtracted numbers appeared later in the 3rd century AD in geometric form and in algebraic form in the 9th century (al-Khwarizmi) • rules for multiplying and dividing negative numbers were first proposed by Brahmagupta in the 7th century AD although they can be derived by extrapolation from al-Khwarizmi's results • much later in the 17th century in Europe, negative numbers started to be accepted as solutions of equations • negative numbers were modelled as directions and debts as a means of justifying the rules for multiplying and dividing negative numbers. However, these rules were not universally accepted because of apparent contradictions. Preparing for your next review • You should be prepared to explain how the operations of combining both positive and negative numbers by addition and multiplication can be modelled as movements along the number line. • You should be aware of the difference between negative and subtracted numbers and be able to discuss why the latter were more acceptable to early users of numbers. • You should be able to explain how algebraic formulae, involving multiplication of subtracted numbers, may be derived from areas of geometrical figures. • You should be able to discuss some of the objections to negative numbers that were raised throughout the centuries. • You should be able to explain some of the limitations in the derivation of rules of multiplication of numbers in the work of, say, Euler and Saunderson. 735
Searching for the abstract Practice exercises for this chapter are on 136 • Answer the following check questions. 1 What is the difference between a 'negative number' and a 'subtracted number'? 2 List three different ways in which negative numbers may be used in models of physical situations. 3 'If b represents a point on the number line, then b also represents the distance between the point representing 0 and that representing b.' Is this statement true, or false, or only true under certain circumstances? Give reasons for your answer.
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Nuffield Advanced Mathematics Histo
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Contents Introduction 7 Unit 1 The
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Introduction This book, about the h
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The Babylonians Chapter 1 Introduct
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Introduction to the Babylonians On
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Figure 1.2 Stele inscribed with Ham
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Token V Figure 1.5 7 Introduction t
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There are no practice exercises for
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A sexagesimal number system is a sy
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ff m yfr < n Figure 2.5 «« Activi
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Activity 2.5 Babylonian fractions 2
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Activity 2.6 Babylonian arithmetic
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Figure 2.10 Activity 2.9 2 Babyloni
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Type Quadratic Table 2.1 x +ax = b
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Activity 2.12 Geometry Figure 2.13a
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Practice exercises for this chapter
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3 An introduction to Euclid fthese
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The Greeks Activity 3.3 B Figure 3.
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34 The Greeks Activity 3.4 g Now us
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36 The Greeks inscribes another wit
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38 The Greeks Activity 3.7 therefor
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The Greeks Interpret postulate 3 in
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42 More Greek mathematics 4.1 Some
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The Greeks Q.E.D. stands for 'quod
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Figure 4.3 46 The Greeks • treat
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Figure 4.6 48 The Greeks together w
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The Greeks Activity 4.8 The cone Th
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The Greeks Practice exercises for t
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5 Arab mathematics You should think
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The Arabs Western Arabic, or Gobar,
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The Arabs Activity 5.2 Figure 5.3a
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7776 Arabs Activity 5.4 This proble
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62 The Arabs Activity 5.6 Horner's
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64 The Arabs Indian text. For over
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D B P N M K Figure 5,9a The Arabs D
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68 The Arabs Figure 5.10 Activity 5
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The Arabs Practice exercises for I
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72 6.1 Read about Descartes The app
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Descartes Activity 6.1 Head about D
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76 Descartes Figure 6.5 Figure 6.5
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Descartes fViete (Vieta in Latin) w
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Descartes The next two sections con
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82 Descartes Activity 6.9 a 3 - b 3
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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
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Page 117 and 118: 772 The beginnings of calculus 9.1
Page 119 and 120: Figure 9.4 774 Calculus you can wri
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Page 123 and 124: 118 Calculus Activity 9.7 Cavalieri
Page 125 and 126: Figure 9. 7 / 120 Calculus dy 1 Wri
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Page 129 and 130: 10.1 Visualising negative numbers 1
Page 131 and 132: 126 Searching for the abstract For
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Page 137 and 138: Searching for the abstract 31. Hith
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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
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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
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14 Answers A i- B 5 E %2 5)-25 = 39
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Index Ptolemy 49, 55, 58 Pythagoras
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