history of mathematics - National STEM Centre

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Consider the following different cases. a / intersects AB. b / intersects an extension of AB. c [Harder] / is parallel to AB. 12 Summaries and exercises 2 Given a line segment AB and a point P on AB, use Euclid's rules to construct on AB a point Q such that AQ = BP. 3 Explain how to construct a square equal in area to a given triangle. 4 More Greek mathematics Chapter summary • the major contributions and the background to the lives of • Euclid (Chapter 3 and Activities 4.1 to 4.3) • Archimedes (Activities 4.6 and 4.7) • Apollonius (Activities 4.8 and 4.9) • Hypatia (text at the end of Chapter 4) • how to interpret Euclidean theorems of geometric algebra in modern algebraic notation (Activity 4.1) • how to use the Euclidean algorithm to find the highest common factor (Activity 4.2) • how the ancient Greeks proved the infinity of primes (Activity 4.3) • the meaning of 'incommensurability' and the significance of its discovery to the ancient Greeks (Activity 4.4) • an example of how the Greeks continued to take an interest in numerical methods: Heron's algorithm for finding square roots (Activity 4.5) • how to use and interpret Archimedes's method for estimating n (Activity 4.6) • how Archimedes found the area under a parabolic arc (Activity 4.7) • how Apollonius defined and named the conic sections (Activities 4.8 and 4.9). Practice exercises 1 Here is a proposition from Euclid. If there are two straight lines, and one of them can be cut into any number of segments, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments. a Construct a diagram to represent the proposition. b Make an algebraic statement equivalent to the proposition. 2 Use the Euclidean algorithm to find the greatest common divisor of 2689 and 4001. 3 Prove proposition 14 from Book VIII of Euclid's Elements that, if a 2 measures b 2 , then a measures b, and, conversely, that if a measures b, then a 2 measures b 2 . 4 Write brief notes on Archimedes's contribution to mathematics. 153
754 12 Summaries and exercises 5 Arab mathematics Chapter summary • some inheritance problems from Islamic law (Activity 5.1). • how Arabic work on trigonometry built on the work of Ptolemy to compile trigonometric tables (Activities 5.2 and 5.3) • examples of problems from the East worked on by Arab mathematicians (Activity 5.4) • how Thabit ibn Qurra investigated amicable numbers (Activity 5.5) • some Chinese methods for solving equations which were available to the Arabs (Activities 5.6 and 5.7) • the origins of the word 'algebra' (text before Activity 5.8) • the six types of quadratic equation solved algorithmically, with geometric justification, by al-Khwarizmi (text and Activity 5.8) • some of the 19 types of cubic solved by Omar Khayyam (text and Activity 5.9) • methods for finding local maxima and minima to help find the solutions of a cubic (Activity 5.10). Practice exercises 1 A particular equation which was considered by al-Khwarizmi can be translated as follows. I one square and ten roots of the same equal thirty-nine. a Express this equation in a modern algebraic form. b Draw a square ABCD with side AB representing the 'root' of the equation. Extend AB to E and AD to F where BE = DF = 5, and complete the larger square on side AE. Extend the lines BC and DC to meet the larger square. Shade the areas representing the 'one square and ten roots of the same' in al-Khwarizmi's problem. c Hence find the positive solution to the equation. 2 Explain why al-Khwarizmi's classification of quadratic equations does not include the case or cases in which squares and roots and numbers equal zero. 3 Which of the Arabic numerals look most like those used in English texts today? What advantages and disadvantages can you see in the Arabic versions? 4 Use al-Khwarizmi's method to solve x 2 + I2x = 64. 5 Use Homer's method to solve x 2 + Ix = 60750. 6 Write brief notes on Omar Khayyam's contribution to mathematics. 7 Draw on your studies of Babylonian and Arab mathematics to write an account of two different approaches to problems which lead to quadratic equations. 6 The approach of Descartes Chapter summary • a review of the legacy of Greek geometrical methods (Activities 6.2 and 6.3)
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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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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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98 Descartes It is clear that the p
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Descartes ractice exercises lor thi
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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
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Page 175 and 176: 14 Answers 4 If he had a daughter t
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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
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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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