history of mathematics - National STEM Centre

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5 Arab mathematics Brahmagupta first found a solution x = 1 and y = 3. He then carried out the process represented in Table 5.1. Smaller root Larger root 1 6 1 35 Table 5. 7 Activity 5.5 Amicable numbers Multiply cross-wise and add to find a new value of x, that is, x = 6. The value of y is then found, that is, y = 17. ^ 1 Repeating the process gives 17J x = 35, y = 99. 3 1
62 The Arabs Activity 5.6 Horner's method overlooked by all the famous 'amicable number chasers' - to be discovered in 1866 by an Italian schoolboy! Fortunately, there are other approaches. The German mathematician Euler (1707-1783) found more than 60 pairs, using methods he developed himself - methods that still form the basis of present-day search techniques. Algorithms for finding solutions to equations The following two activities illustrate two ideas which originated in China and which were used by Arab mathematicians and handed on to the West. The first is an example of a numerical solution of higher-order equations by a Chinese method, which culminated in the work of Chin Chiu Shao in 1247 entitled Chiu-chang suan- shu, which appeared in its simpler versions in Arab mathematics about the same time. Nowadays this method is known as 'Horner's method'. In 1819, an English schoolteacher and mathematician, W G Horner, published a numerical method of finding approximate values of the roots of equations of the type f(x) = a0xh + alx"~ ] + ... + an_ { x + an =0. The procedure that Horner re-discovered is identical to the computational i scheme used by the Chinese over 500 years earlier. Here is an example in which this method is used to solve x 2 + 252x - 5292 = 0. First, find that there is a root between 19 and 20. Then make the substitution y = x - 19 to obtain (y + 19) 2 + 252(y + 19) - 5292 = 0, or y 2 + 290y -143 = 0, which you know has a solution between 0 and 1. Now say that 143 y = ——-— = 0.491, so that* = 19.491. (1 + 290) To find the cube root of 12 978 (that is, to solve the equation jc 3 -12 978 = 0), first note that there is a solution between 20 and 30, so you substitute _y = x - 20 to obtain the equation y 3 + 6Qy 2 + \200y- 4978 = 0, which has a solution between 0 and 10. Now check that this equation has a solution between 3 and 4, so write z = .y-3 and obtain the equation z 3 + 69z 2 + 1587z-811 = 0. Then you can say 81 1 that z = ————— = 0.49, so that x = 23.49. 1+69+1587 Work in a group for this activity. Be prepared to make a short oral report on question 3 at your next review. 1 Use Horner's method to find the square root of 71 824. 2 Use Horner's method to solve the equations a x 2 -23x- 560 = 0 b x 6 + 6000 x = 191 246 976.
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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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Page 18 and 19: A sexagesimal number system is a sy
Page 20 and 21: ff m yfr < n Figure 2.5 «« Activi
Page 22 and 23: 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
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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
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Page 77 and 78: 72 6.1 Read about Descartes The app
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Page 81 and 82: 76 Descartes Figure 6.5 Figure 6.5
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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
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Page 99 and 100: 94 Descartes 23 It is true that the
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Page 103 and 104: 98 Descartes It is clear that the p
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Page 111 and 112: Descartes Today we call these numbe
Page 113 and 114: Descartes Activity 8.9 Normals Figu
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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 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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Searching for the abstract Practice
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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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