history of mathematics - National STEM Centre

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12 Summaries and exercises • how Viete's law of homogeneity restricted the development of algebra (Activities 6.4 and 6.5) • how Descartes's method represented a new approach to the use of algebraic notation (Activities 6.6 to 6.11) • how to apply Descartes's method to van Schooten's problem (Activity 6.12) • an algebraic analysis of the problem of trisecting an angle (Activity 6.13). Practice exercises The practice exercises for this chapter are combined with those in Chapter 8. 7 Constructing algebraic solutions Chapter summary • how to translate an algebraic analysis into a geometric construction (Activity 7.1) • how Descartes developed constructions requiring circles (Activities 7.2 to 7.6) • how Descartes's method can lead to solutions which are curves (Activities 7.7 to 7.9) • Descartes' s discussion of the nature of curves (Activity 7.10 and preceding text) • analysing curves produced using construction instruments (Activities 7.11 to 7.13) • Descartes's discussion of acceptable curves (Activity 7.14) • Descartes's geometrical construction of the trisection of an angle (Activity 7.15). Practice exercises The practice exercises for this chapter are combined with those in Chapter 8. 8 An overview of La Geometrie Chapter summary the importance of Descartes's work seen so far (Activity 8.1) how Descartes classified curves (Activities 8.2 to 8.4) how Descartes attempted to standardise equations (Activity 8.5) how Descartes treated negative solutions (Activity 8.6) how to use Descartes's rule of signs (Activities 8.7 and 8.8) how Descartes constructed a normal to a curve (Activity 8.9) the contribution of Descartes to analytical geometry (Activity 8.10, and surrounding text) • an example of the work of Fermat in analytical geometry (Activity 8.11). Practice exercises 1 Write brief notes on Descartes' s contribution to mathematics. 2 Construct Descartes's graphical solution to the cubic equation x 3 = -4.x +16, 755
156 12 Summaries and exercises using the circle (x-8) 2 +(y + j) =8 2 +(f) and the parabola x2 =y. Explain why the intersection of this circle and the parabola provides the solution. 3 Use the method of Descartes to solve the equation y 1 = -ay + b 2 9 The beginnings of calculus Chapter summary • how Euclid defined a tangent to a circle (Activity 9.1) • how Archimedes constructed a tangent to a spiral (Activity 9.2) • how Fermat found maxima and minima, and how he used this in his general method to construct tangents to a curve (Activities 9.3 and 9.4) • how Descartes's definition of a tangent, and how his method for constructing tangents differed from Fermat's definition and method (Activity 9.5) • the development of integral notation (Activities 9.6 and 9.7) • Leibniz: gradient and area (Activity 9.8) • Newton and differentiation (Activity 9.9). Practice exercises 1 Write brief notes on Leibniz's contribution to mathematics. 2 What three long-standing classes of problems were solved by the introduction of the calculus? Explain why calculus was important in the solution of these problems. 10 'Two minuses make a plus' Chapter summary how to visualise addition and multiplication on a number line (Activity 10.1) subtracted numbers (Activity 10.2) an example of subtracted numbers arising in ancient China (Activity 10.3) Chinese rules for combining negative numbers (Activity 10.4) a geometric approach to multiplying negative numbers using Indian, Arab and Greek sources (Activities 10.5 to 10.7) thinking about why there is a gap in development (Activity 10.8) some particular difficulties with negative numbers (Activities 10.9 and 10.10) a debate based on source material (Activity 10.11). Practice exercises The practice exercises for this chapter are combined with those in Chapter 11. 77 Towards a rigorous approach Chapter summary • thinking about the validity of proof by extrapolation (Activity 11.1 and 11.4) • using analogy to justify the rules for combining negative numbers (Activity 11.2)
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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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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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102 Descartes Activity 8.1 Activity
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Page 129 and 130: 10.1 Visualising negative numbers 1
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Page 163 and 164: 158 Hints Activity 2.3, page 15 2 F
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Page 167 and 168: outcome 1 step to the right Figure
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
Page 173 and 174: 74 Answers 2 Suppose that you can f
Page 175 and 176: 14 Answers 4 If he had a daughter t
Page 177 and 178: 14 Answers Activity 6.5, page 79 1
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
Page 185 and 186: 14 Answers of multiplication by neg
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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