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5 Arab mathematics solutions the cubic has. He therefore classified the cubics differently, into those types with at least one positive solution and those types that may and may not have a positive solution depending on the exact coefficients. The following activity illustrates his approach. Activity 5.10 Graphical deductions The Renaissance is the name given to the period from the 14th to 16th centuries in Europe when there was a revival of cultural thinking. 1 Al-Tusi notes that for an equation x- 3 + c = ax 2 expressed in the form x (a — x) = c, whether the equation has a positive solution depends on whether the expression of the left-hand side reaches c or not. He then states that, for any value of x lying between 0 and a, x 2 (a — x) < (•} a) (i a) • Why is this true? 2 Show that a local maximum occurs for x in the expression x 2 (a - x) when JCG = -| a. There is no indication of how al-Tusi found this value. Suggest how he might have found it. 3 He then proceeds to make the following inferences: • if ^-a 3 < c no positive solution exists • if -If a 3 = c, there is only one positive solution, x — ja • if -fa a 3 > c, two positive solutions x l and x2 exist where 0 < x ]
The Arabs Practice exercises for I§| this chapter are on ;|l|f page 154, 70 Reflecting on Chapter 5 What you should know • some examples of inheritance problems • the contributions made by al-Khwarizmi and Omar Khayyam • that Arab mathematics is in part a synthesis of Greek mathematics and mathematics from the East • how decimal numbers were represented in the Arab number system • how sine tables were constructed • Thabit's method for amicable numbers • examples of how Arab mathematicians developed solutions of equations from geometry • the derivation of the word 'algebra' • al-Khwarizmi's classification of quadratic equations • al-Khwarizmi's method of solving quadratics geometrically • Omar Khayyam's method of solving cubic equations. Preparing for your next review • Reflect on the 'What you should know' list for this chapter. Be ready for a discussion on any of the points. • Answer the following check questions. 1 Show how you would apply al-Khwarizmi's method to solving equations such as 2x 2 + Wx = 48 or -j x + 5x = 28. See whether you can prove the method geometrically. 2 Summarise in not more than one page, the work of al-Khwarizmi and Omar Khayyam. Use the text in this unit, and any other sources available to you. 3 The Arab contribution to mathematics has often been perceived as one of guardian of Greek mathematics while the West was in the turmoil of the dark ages. Summarise your own understanding of the Arab contribution with this in mind. 4 Solve another version of the hundred fowls problem as given below. A rooster is worth 5 coins, a hen 3 coins and 3 chicks 1 coin. With 100 coins we buy 100 fowls. How many roosters, hens and chicks are there?
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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
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: 68 The Arabs Figure 5.10 Activity 5
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
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Figure 9. 7 / 120 Calculus dy 1 Wri
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Calculus JPractiee exercises this c
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10.1 Visualising negative numbers 1
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126 Searching for the abstract For
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Searching for the abstract Activity
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Searching for the abstract John Wal
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Searching for the abstract 31. Hith
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134 Searching for the abstract If y
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138 Searching for the abstract Extr
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Searching for the abstract /esseTs
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Searching for the abstract d Do you
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144 Searching for the abstract math
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146 Searching for the abstract As a
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Searching for the abstract How does
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Searching for the abstract ; Practi
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752 12 Summaries and exercises Two
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754 12 Summaries and exercises 5 Ar
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156 12 Summaries and exercises usin
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158 Hints Activity 2.3, page 15 2 F
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13 Hints Activity 7.3, page 89 1 Wr
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outcome 1 step to the right Figure
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14 Answers 3 0;6,40 and 0;2,13,20 4
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14 Answers 3 If AD intersects the l
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74 Answers 2 Suppose that you can f
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14 Answers 4 If he had a daughter t
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14 Answers Activity 6.5, page 79 1
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14 Answers 6 The equation z 2 + az
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14 Answers b This meets the ellipse
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14 Answers To get an interpretation
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14 Answers of multiplication by neg
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14 Answers b The required number is
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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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