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Activity 2.5 Babylonian fractions 2 Babylonian mathematics numbers into a more readable form, by writing a semicolon to separate the integral and fractional part of a number, a comma to distinguish the several sexagesimal positions, and a zero where it is intended. 1 Write the sexagesimal numbers 2,0,48; 12 and 3;9,18 in decimal notation. 2 Convert the decimal numbers 10 000 and 60 J- into sexagesimal numbers. Not all fractions are as easy to convert. 3 Convert § and -^ into sexagesimal numbers. In Activity 2.3, question 5, the 'irregular' numbers 7 and 11 were omitted from the table of reciprocals because they are non-terminating in a sexagesimal system, similar to y in our decimal system. 4 In the decimal system, which fractions convert to terminating decimals and why? Examine the connection between the denominators of the fractions and the number ten. 5 a Which fractions, when converted to sexagesimals, have representations which terminate? b What may be the advantage of the sexagesimal system when dealing with fractions? 6 Write in sexagesimal notation those of the following fractions which have representations which terminate: 2^,6^ and 83 ^ . By considering non-terminating fractions, the Babylonians were faced by the problem of infinity. However, they did not examine it systematically. Not until the Greeks would this challenge be tackled. At one point, a Mesopotamian scribe seemed to give upper and lower bounds for the reciprocal of the irregular number 7, placing it between 0; 8,34,16,59 and 0; 8,34,18. 7 Verify the bounds above by giving the periodicity of j in sexagesimal notation when dealing with fractions. Except for the lack of a separator between the integral and fractional part of a number, and a symbol for zero, the Babylonian number system seems to be equivalent to the modern decimal system. It enabled the Babylonians to do arithmetic operations much as you calculate today, and with comparable facility. Some of these operations can be found on an Old Babylonian tablet whose transcription is depicted in Figure 2.7. It contains text, numbers and operators, and describes a method for solving a problem which is composed of two simultaneous equations - that is, a 'length' and a 'width' which you would express as x and y. 17
18 The Babylonians Figure 2.7 Later in this chapter, you will examine more closely how the Babylonians dealt with such problems and how they solved quadratic equations. But first, in Activity 2.6, you will concentrate on their arithmetic operations.
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 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
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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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Descartes Today we call these numbe
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Descartes Activity 8.9 Normals Figu
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Descartes Practice exercises for th
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772 The beginnings of calculus 9.1
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Figure 9.4 774 Calculus you can wri
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Calculus including Descartes, under
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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 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 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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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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