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7 / Towards a rigorous approach De Morgan was very keen to describe the number system in an abstract way, and to keep numbers themselves out of an axiomatic description of the number system. There was a major implication of this approach. All systems which obey the axioms also obey any rules derived from the axioms. This is true of the real and complex numbers, both of which satisfy the axioms. However, Hamilton was inventing an extension (which he called quaternions) of complex numbers. You will see later that quaternions do not satisfy the same rules as numbers. Activity 11.7 Do all systems have the same axioms? Activity 11.8 Boolean algebra 1 Check that De Morgan's view is correct at least as far as the real and complex numbers are concerned. That is, check that the axioms as specified in Activity 11.5, question la, for combining elements under + and x are the same for both the real and the complex number systems. 2 Also, given any real number a, the axioms require you to be able to specify another number b such that a + b = 0.lfa^0, the axioms also require you to be able to specify another number c such that a x c = 1. If a is the complex number 2 + i, find b and c. George Boole (1815-1864) had no formal mathematical education. He became a primary school teacher and during that time taught himself more mathematics and became interested in a debate over logic which was taking place. As a result, in 1847 he published a short work on logic which was recognised by De Morgan as being revolutionary. Nevertheless it did not receive much recognition at the time although it probably enabled Boole to get a position teaching at Queen's College in Cork. Seven years later he produced the classic Investigation into the laws of thought, which established the algebra of logic, or the algebra of sets, now known as Boolean algebra. In other words, Boole set up the rules by which the elements in his sets, or in his logic, might be combined. 1 a Find out as much as you can about Boole and Boolean algebra and make notes as you do your research. b As you do your research, decide for yourself whether the elements of Boole's algebra satisfy the same rules of combination as the real numbers. c Use your notes to prepare and make a presentation to your fellow students on Boole and Boolean algebra. As part of your presentation, include your answer to part b. Remember that your fellow students may not know anything about axioms, so make your presentation in a way which is easy for them to understand. William Kingdon Clifford (1845-1879) was a graduate of Trinity College, Cambridge. He was renowned for his teaching, and for his ability to visualise 143
144 Searching for the abstract mathematical ideas and bring them to life. Clifford almost always uses models in which algebraic quantities can be visualised geometrically or diagrammatically. For the system of real numbers, Clifford takes a step to the right of length 1 to be the basic element from which everything else is built. Multiplying this basic step by a positive number, another step, gives a longer step. Other operations on this step are those labelled by + and -, which represent the operations of 'retaining' or 'reversing' the direction of the step. Here is Clifford's own explanation. Activity 11.9 Clifford's model In the equation 2 x (+3) = +6 the last term on each side is a step, the first is an operator and the equation means by doubling I can turn a step 3 to the right into a step 6 to the right. Now what operator is required to turn the step -3 to the left into the step +6 to the right? First we reverse the step by an operator which we will call r, |K~3)} = +3: thus it becomes +3. Now double it, and the whole operation is written, 2r(-3) = +6, so the required operator is 2r , which means reverse and then multiply by 2. But we may change the order of the process, viz. double and then reverse and we get the same result r2(-3) = +6. ... Let k3 mean triple without reversing. And let us suppose any step taken, tripled, reversed, doubled and reversed again. The two reversals will clearly destroy each other and give 'no reversal' or k, and we shall have our step sextupled without reversal. This may be written in an equation like the last. And in the same way we have two others in which the direction is reversed: I r2(-3) = +6 r2(r3) = k6 r2(+3) = -6 r2(k3) = r6: and we are led to assign a new meaning to the symbols + and -; we may use them instead of k and r respectively. 1 Clifford interprets all real numbers and the operations on them by taking as the basic building blocks a step of length 1 to the right and the operations of +, - and x. a Describe how you would obtain from basic building blocks a step of length 12 to the left. b Describe the process of taking a step of 2 to the left, tripling it, reversing its direction and then doubling it. Write the process as an algebraic expression.
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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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72 6.1 Read about Descartes The app
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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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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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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
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Page 113 and 114: Descartes Activity 8.9 Normals Figu
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Page 123 and 124: 118 Calculus Activity 9.7 Cavalieri
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Page 129 and 130: 10.1 Visualising negative numbers 1
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Page 137 and 138: Searching for the abstract 31. Hith
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Page 147: Searching for the abstract d Do you
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Page 153 and 154: Searching for the abstract How does
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Page 157 and 158: 752 12 Summaries and exercises Two
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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
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