history of mathematics - National STEM Centre

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5 Arab mathematics If possible work in a group for Activity 5.8, You may be asked to make a short report based on your work in this activity. Historical background The period of Arab intellectual ascendancy can be thought of as beginning with the expansion of Islam in the first half of the 7th century. After a period of over a century of war and expansion a new capital of the Arab civilisation was established at Baghdad, which gradually became a centre of mathematical activity. A 'House of Wisdom' was established, and mathematical and other scholars from the whole Arab civilisation, as well as India and Greece, were welcomed. The Arabs quickly absorbed the learning of their neighbours. A multiplicity of influences therefore went into the creation of Arab mathematics. First, there was Babylonian mathematics and astronomy dating back over 2500 years. This mathematics was passed down either through the medium of a religious sect known as Sabeans, descendants of Babylonian star-worshippers who showed a particular aptitude for astronomy and numerology, or through a number of practical algorithms for solving linear and non-linear equations in one or more unknowns, similar to the old Babylonian methods described in Chapter 2. Second were the Greek written sources, regarded by many historians of mathematics as of paramount importance in the development of Arab mathematics. Central to the Greek tradition, as it was taken over by the Arab world, were Euclid's geometrical text, Elements, and Ptolemy's astronomical text, Almagest. Third, and less certain, were the Indian written sources. The line of influence is not so clear as that of the Greek mathematicians. But, it is clear that the Indians did much work on trigonometry and astronomy, which entered Arab mathematics through the Arabs' interest in astronomy. For example, an astronomical work from India, one of the Siddhantas, was translated into Arabic in about AD 775. Other knowledge came from a variety of sources. There existed a vast treasure trove of commercial practices, measurement techniques, number reckoning, recreational problems and other examples of 'oral mathematics' which resulted from large groups of merchants, diplomats, missionaries and travellers meeting one another along the caravan routes and the sea routes from China to Cadiz in Spain. The computational procedures or algorithms, arising from this source, often seemed more like recipes than well-thought-out mathematical methods. But a number of Arab mathematicians, inspired by these recipes, proceeded to devise their own techniques for solving problems. In doing so, they sometimes translated their recipes into a more suitable, theoretical form. Numerals One of the most far-reaching inventions of Indian mathematics was a positional, or place value, decimal numeral system. With symbols or digits to represent numbers from zero to nine, 0, 1, 2, ..., 9, it became possible to represent any integer both 55
The Arabs Western Arabic, or Gobar, c. 960 |You can find out more f labour, the origins, | {development and the I'global spread of numerals |in, for example, The crest lor" the peacock, by G G |«Joseph. *"" ' 56 uniquely and economically. Indian mathematicians created an arithmetic in which calculations could be carried out by people of average ability. The story of the development and the spread of Indian numerals, often known as the Hindu-Arabic numerals, is fascinating; the role of the Arabs in this development was crucial. Figure 5.1 Indian Brahmi numerals, c. 250 BC Indian Gwalior numerals, c. AD 500 67X9 VA 9• 1 * 3*5 e'T $ 9 « 123/^567890 European apices, c. 1000 European (Durer), c. 1500 Eastern Arabic, c 800 You can see from Figure 5.1 how our system of numerals evolved. The development started from certain inscriptions on the Ashoka pillars (stone pillars built by Ashoka, 272-232 BC, one of the most famous emperors of ancient India) and has now reached a time when computers form numbers using pixels. Both the Babylonians and the Chinese had numeral systems versatile enough to represent and manipulate fractions. But neither had a device, or symbol, for separating integers from fractions. The credit for such a symbol must go to the Arabs. For example, in giving the value of n, the ratio of the circumference of a circle to its diameter, to a higher degree of accuracy than before, as sah-hah 3 141 592 653 59 where the Arabic word 'sah-hah' above the 3 was in effect the decimal point. Inheritance problems Mathematics was recruited to the service of Islamic law. These laws were fairly straightforward; when a woman died her husband received one-quarter of her estate, and the rest was divided among the children in such a way that a son received twice
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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
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
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Page 18 and 19: A sexagesimal number system is a sy
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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
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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
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Page 59: 5 Arab mathematics You should think
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Page 65 and 66: 7776 Arabs Activity 5.4 This proble
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Page 97 and 98: A Descartes Activity 7.7 Reflecting
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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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138 Searching for the abstract Extr
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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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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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