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30 Figure 12. 7 Transcription of Figure 2.8b Summaries and exercises 1 Introduction to the Babylonians Chapter summary • general background (whole chapter). Practice exercises There are no practice exercises for this chapter. 2 Babylonian mathematics Chapter summary • an introduction to the Babylonian number system (Activities 2.1 to 2.5) • Babylonian arithmetic (Activity 2.6) • the Babylonian method for finding square roots (Activities 2.7 to 2.9) • solution of quadratic equations (Activities 2.10 and 2.11) • Babylonian geometry (Activities 2.12 and 2.13). Practice exercises 1 Look at Figure 12.1 and answer the following questions. a Using your calculator or otherwise, convert 42; 25,35 and 1; 24,51,10 into decimal numbers. Give your answers to at least seven significant figures accuracy. b To which number is 1; 24,51,10 an approximation? c Describe a geometric algorithm to estimate square roots which the Babylonians may have used to calculate this approximation. d Write the algorithm to calculate -
752 12 Summaries and exercises Two texts which call for comment here represent versions of the same exercise, and they are significant, not only because of the abstract interest in numerical relations indicated by the enormous numbers involved, but also because they concern the problem of irregular numbers. Number 671 is written by a bungler who succeeded in making half a dozen writing errors in as many lines ... Jestin Number 50 se guru 7:l sila 3 7 Jestin Number 671 (rev) se sila 3 7 ! guru 7 lu: 1 su ba-ti lu: 1 su ba-ti lu-bi (obv) gurus 45,42,51 45,36,0 (written on three lines) se sila 3:3 su ?-tag 4 ? Translation of Number 50 The grain (is) 1 silo. 7sila (7 liters) each man received. Its men: 45,42,51. 3 sila of grain (remaining) A silo (guru) in this period contained 40,0 gur, each of which contained 8,0 sila. The correct answer seems to have been obtained by the following process: (1) 5,20,0,0 times 0; 8,34,17,8 = 45,42,51; 22,40 (2) 45,42,51 times 7 = 5,19,59,57 (3) 5,20,0,0-5,19,59,57 = 3 a Explain in your own words what problem the students have been asked to solve. b Explain where the number 5,20,0,0 comes from. c Explain step (1) in your own words. d Verify that 45,42,51 times 7 equals 5,19,59,57. e What mathematical error has the pupil apparently made in Jestin 671 ? 5 Give a geometric and an algebraic description of the Babylonian method of solving an equation of the form x +ax = b. 3 An introduction to Euclid Chapter summary • how to use the Greek rules to carry out constructions for drawing a perpendicular, bisecting an angle, bisecting a line segment, drawing a perpendicular to a line segment (Activities 3.1 and 3.2) • how to construct a square equal in area to any given polygon (Activity 3.3) • the significance of the three classical problems of ancient Greece (Activity 3.4) • some Euclidean proofs to work through (Activities 3.5 and 3.6) • how Euclid's approach established an axiomatic basis for geometry (Activities 3.7 and 3.8). Practice exercises 1 Given a line segment AB and a straight line /, use Euclid's rules to construct points C and D on / such that CD = AB.
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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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i——— Figure 6.12 Figure 6.13
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Descartes Practice exercises for th
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Figure 7.2 Descartes Activity 7.1 i
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N- i- L Descartes Thus I have In th
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A Descartes Activity 7.7 Reflecting
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94 Descartes 23 It is true that the
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96 Descartes Figure 7.11 shows a ge
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98 Descartes It is clear that the p
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Page 107 and 108: 102 Descartes Activity 8.1 Activity
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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
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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
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Page 153 and 154: Searching for the abstract How does
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Page 159 and 160: 754 12 Summaries and exercises 5 Ar
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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
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