history of mathematics - National STEM Centre

More documents

Recommendations

Info

Figure 6. U Figure 6.15 6 The approach of Descartes 1 Figure 6.14 shows a straight line AB, with a point C on it. Produce AB to the point D such that the rectangle with sides AD and DB, is equal (in area) to the square with side CD. Call the lengths AC, CB and BD, a, b and x. 1 Formulate an equation involving a, b and x. 2 Express x in terms of a and b. Trisecting an angle Activity 6.13 Trisecting an angle Figure 6.16 A classical problem from Greek mathematics was the problem of trisecting an angle. Let O be an angle, shown in Figure 6.15. Construct two lines that divide the angle at O into three equal parts. Descartes, who did not know that this problem cannot be solved with a pair of compasses and a straight edge, hoped to find a solution using his new method. It was only in 1837 that a relatively unknown Frenchman, Pierre-Laurent I Wantzel (1814-1848), proved that it was impossible to trisect an angle j using only a straight edge and a pair of compasses. | The algebraic analysis of this problem is given in Activity 6.13. The geometric construction is left until Chapter 7. Let O be the centre of a circle with radius 1. The points N, Q, T, P, R and M are as shown in Figure 6.16. With the help of the figure, you can re-formulate the problem: 'Let NP be a chord. Find the length of chord NQ'. 1 By considering angles standing on the chord PQ, prove that ZNOQ = ZQNR. 2 Prove that triangle NOQ is similar to triangle QNR, and triangle ROM is similar to triangle QNR. 3 Let NP = q and NQ = z. Prove that RM = q - 2z. 4 Use similar triangles to prove that z 3 = 3z - q. Reflecting on Chapter 6 What you should know • the major contributions made by Descartes and Viete to the development of mathematics 85
Descartes Practice exercises for this chapter are on ;0age J 5$. ..,....,,.,, ...,,,,„,, ...,.,,.„ 86 • the meaning of the law of homogeneity • how to explain Descartes's approach to the construction of the bisector of the line AB. Preparing for your next review • Reflect on the 'What you should know' list for this chapter. Be ready for a discussion on any of the points. • Answer the following check questions. 1 In A history of mathematics (Second edition), by Carl B Boyer and Uta C Merzbac, it states that Descartes substituted homogeneity in thought for homogeneity in form, a step that made his geometric algebra more flexible - so flexible indeed that today we read xx as "x squared" without ever seeing a square in our mind's eye. Explain the first part of this quotation. 2 Write a paragraph reflecting on the advantages of the law of homogeneity in terms of explaining algebra to beginners. In your answer make use of the diagrams in The Arabs unit for solving quadratic equations.
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 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
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
Page 67 and 68: 62 The Arabs Activity 5.6 Horner's
Page 69 and 70: 64 The Arabs Indian text. For over
Page 71 and 72: D B P N M K Figure 5,9a The Arabs D
Page 73 and 74: 68 The Arabs Figure 5.10 Activity 5
Page 75 and 76: The Arabs Practice exercises for I
Page 77 and 78: 72 6.1 Read about Descartes The app
Page 79 and 80: Descartes Activity 6.1 Head about D
Page 81 and 82: 76 Descartes Figure 6.5 Figure 6.5
Page 83 and 84: Descartes fViete (Vieta in Latin) w
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: i——— Figure 6.12 Figure 6.13
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
Page 97 and 98: A Descartes Activity 7.7 Reflecting
Page 99 and 100: 94 Descartes 23 It is true that the
Page 101 and 102: 96 Descartes Figure 7.11 shows a ge
Page 103 and 104: 98 Descartes It is clear that the p
Page 105 and 106: Descartes ractice exercises lor thi
Page 107 and 108: 102 Descartes Activity 8.1 Activity
Page 109 and 110: 104 Descartes In the next section y
Page 111 and 112: Descartes Today we call these numbe
Page 113 and 114: Descartes Activity 8.9 Normals Figu
Page 115 and 116: Descartes Practice exercises for th
Page 117 and 118: 772 The beginnings of calculus 9.1
Page 119 and 120: Figure 9.4 774 Calculus you can wri
Page 121 and 122: Calculus including Descartes, under
Page 123 and 124: 118 Calculus Activity 9.7 Cavalieri
Page 125 and 126: Figure 9. 7 / 120 Calculus dy 1 Wri
Page 127 and 128: Calculus JPractiee exercises this c
Page 129 and 130: 10.1 Visualising negative numbers 1
Page 131 and 132: 126 Searching for the abstract For
Page 133 and 134: Searching for the abstract Activity
Page 135 and 136: Searching for the abstract John Wal
Page 137 and 138: Searching for the abstract 31. Hith
Page 139 and 140: 134 Searching for the abstract If y
Page 141 and 142:
Searching for the abstract Practice
Page 143 and 144:
138 Searching for the abstract Extr
Page 145 and 146:
Searching for the abstract /esseTs
Page 147 and 148:
Searching for the abstract d Do you
Page 149 and 150:
144 Searching for the abstract math
Page 151 and 152:
146 Searching for the abstract As a
Page 153 and 154:
Searching for the abstract How does
Page 155 and 156:
Searching for the abstract ; Practi
Page 157 and 158:
752 12 Summaries and exercises Two
Page 159 and 160:
754 12 Summaries and exercises 5 Ar
Page 161 and 162:
156 12 Summaries and exercises usin
Page 163 and 164:
158 Hints Activity 2.3, page 15 2 F
Page 165 and 166:
13 Hints Activity 7.3, page 89 1 Wr
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
show all

history of mathematics - National STEM Centre

Create successful ePaper yourself

Delete template?

Save as template?