history of mathematics - National STEM Centre

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12 Summaries and exercises • extending the number line to include complex numbers (Activity 11.3) • looking at an axiomatic approach (Activities 11.5 to 11.7) • Boolean algebra as an example of an axiomatic system (Activity 11.8) • exploring Clifford algebra (Activities 11.9 to 11.14) • generalising beyond numbers (Activity 11.15). Practice exercises 1 In his Arithmetica infinitorum, John Wallis gives his view that a number larger than infinity can be found. I a ratio greater than infinity such as a positive number may be supposed to have to a negative number a Investigate the sequence of positive ratios — — — —. This is a sequence of the form — with n increasing. In numerical terms is the sequence descending or n ascending? b Now investigate the sequence of ratios — where n is any positive number. What n can you say about the sequence as n tends towards zero? Draw a graph using n as the horizontal axis. c By extending the sequence to negative values of n, again with n decreasing, can you explain why John Wallis may have put forward the view expressed? d What is wrong with Wallis's argument? 2 In the book Mathematical thought from ancient to modern times, Kline assigns to Augustus de Morgan the following: The imaginary expression -^(-a) and the negative expression -b have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. De Morgan illustrated this by means of a problem. A father is 56; his son is 29. When will the father be twice as old as the son? a Solve de Morgan's problem by setting up an equation to find out how many years from now the father will be twice as old as the son. b In what way does your solution support de Morgan's claim? c How do you interpret your solution and what assumptions are you making about numbers? d How can de Morgan's original question be phrased so that the solution no longer appears absurd? 3 Discuss critically various contradictory beliefs which have been held about negative numbers throughout the centuries and throughout the world. 757
158 Hints Activity 2.3, page 15 2 First exclude the two 'exceptional' numbers in the right-hand column. 3 What is a half in decimal notation, and in Babylonian? Activity 2.5, page 17 3 Writing a fraction in sexagesimal notation is comparable with the following decimal conversion. 10 10 Activity 2.11, page 24 = ~To — + i_ io~ = — To + -^ii " = A +4 = 0 . 25 10 io 2 rn 2 in in 2 2 x - y = 1 from line 1; -%(x — y) = \xl from lines 3 to 5. Activity 3.2, page 32 4 b A median of a triangle is a straight line from one vertex of a triangle to the mid-point of the opposite side. Activity 3.3, page 34 1 g Draw an arc of a circle, with centre H and radius HI. Extend GH to intersect this circle outside GH at N. Then GN is a line segment divided at H into lengths a and b. Now you need to construct a right-angled triangle with hypotenuse GN. Use the fact that all vertices C of the right-angled triangles ABC, of which AB is the hypotenuse, form a semi-circle with diameter AB. Activity 4.7, page 49 1 b Think about the coefficient of x in the parabola. c Think about the difference between calculating QM given the ^-coordinates of A and B, and finding the lengths UV and HK given the ^-coordinates of P, M, R. Activity 4.9, page 50 1 b Notice that the section MPN is parallel to the base BCE of the cone, and so is a circle. What does this tell you about the triangle MPN? How does this fact help you to use the idea of a mean proportional?
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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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Descartes ractice exercises lor thi
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102 Descartes Activity 8.1 Activity
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104 Descartes In the next section y
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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 and 128: Calculus JPractiee exercises this c
Page 129 and 130: 10.1 Visualising negative numbers 1
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
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Page 141 and 142: Searching for the abstract Practice
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Page 147 and 148: Searching for the abstract d Do you
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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
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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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