history of mathematics - National STEM Centre

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7 / Towards a rigorous approach although these operations may not be labelled by + and x. Write down the rules for combining these objects. Translate these rules into a new language by consistently replacing the symbols used for your two operations by the symbols + and x. If your objects were represented by numbers, would these translated rules look the same as the rules for combining real numbers by addition and multiplication? If so, then your collection of objects satisfies the same axioms as the real numbers. If not, then the axioms for the two systems are different; in which case, how do they differ? Charting the progress and status of negative numbers through history highlights some interesting points concerning the development of any piece of mathematics. It has shown how many results are first generated by analogy, intuition and extrapolation, and do not necessarily develop a rigorous foundation until later. However, once the rigour has been established, extensions and generalisations can be produced in which the rules governing the combination of numbers by + and x are used to represent the combination of new objects. These rules then acquire the status of the rules governing an algebraic system, since they no longer just apply to the original numbers. And, potentially, this algebraic system allows the possibility of generalisation to an abstract structure, which exists in spaces which you cannot necessarily visualise. Reflecting on Chapter 11 What you should know • that mathematical results are conjectured using various methods: extrapolation, analogy, intuition; and that none of these methods is necessarily rigorous • that complex numbers can be described as an extension of the real number system, and that they may be modelled in the plane • that the properties of negative numbers do not follow the 'principle of permanence of equivalent forms' • that it is possible to model a positive number as a step along the number line and that multiplication of the number by -1 is analogous to reversing its direction • that it is possible to extend this model of the real numbers into the plane to produce a different algebraic system from the complex numbers (this is Clifford's model) • that the unit vectors can be modelled as basic steps at right angles to one another in both two and three dimensions, and that it is possible to define an operator which turns one basic step into the other • that it is possible to combine numbers, steps and turning operators, using an operation analogous to multiplication, but which has different properties from multiplication of numbers • that systems, which can be visualised and represented by geometrical objects, can be extended and generalised to more abstract situations, for example, to higher dimensions, which can no longer be represented pictorially but which are equally valid mathematically. 149
Searching for the abstract ; Practice exercises for this chapter are on page 157. 150 Preparing for your next review • You should be able to describe examples of results that have been conjectured through each of the methods of extrapolation, analogy and intuition. • You should be able to explain the properties of the imaginary number i by modelling it as a line of unit length inclined at 90° to the unit length line +1. • You should be able to describe and to justify examples of systems which have the same properties (axioms) as real numbers and those with different properties. • You should be ready to explain how the system of real numbers can be extended in several ways to describe new mathematical systems. • You should be able to explain, using diagrams, the basic elements of Clifford's model in two and three dimensions and to justify their properties. • You should be able to calculate the results of combining together various elements in Clifford's model using addition and multiplication. • You should be aware of the potential to generalise systems such as the real numbers or complex numbers. • Answer the following check questions. 1 By referring to the mathematics in Books 1-5, list some mathematical ideas and systems which you now see as extensions and/or generalisations of the real numbers. Give reasons for your answers by referring to the rules for combining the elements in your various examples. 2 What is the 'principle of permanence of equivalent forms' as proposed by George Peacock? Give an example which supports the principle when the positive numbers are extended to include the negative ones. 3 Give an example of a result in number systems which was originally proposed by extrapolation from a previously known result. 4 Write down two different geometrical interpretations of the imaginary number i and briefly say why each of them is consistent with i 2 — -1. 5 How do quaternions differ from rational numbers in their rule of multiplication? Write down the rules for multiplying the quaternion units /, J and K. 6 By drawing diagrams, explain why you need three different kinds of basic turning operators in three dimensions. How many would you need in five dimensions?
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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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A sexagesimal number system is a sy
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ff m yfr < n Figure 2.5 «« Activi
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Activity 2.6 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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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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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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Page 165 and 166: 13 Hints Activity 7.3, page 89 1 Wr
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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 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
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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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