history of mathematics - National STEM Centre

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Activity 10.7, page 129 13 Hints 1 Make a special choice of the numbers a and c which reduces the equation to one which just has two negative numbers multiplied together on the left-hand side. Having made this choice, substitute it in both sides of al-Khwarizmi's equation to obtain the rule for multiplying negative numbers together. 2 Follow the same sort of argument here as applied in question 1 except that you are not necessarily restricting yourself to choices of a and c. Remember that division by a number e is the same as the operation of multiplication with — . e Activity 10.8, page 129 1 Think of any single significant mathematical development during that period which might be occupying the minds of mathematicians. You could look through the other units in this book. Does that give you any ideas? Activity 10.9, page 130 1 Start by making a brief summary of the passage. Activity 10.10, page 131 1 a Here is the translation of Arnauld's extract. I do not understand why the square of -5 is the same as the square of + 5 , and that they are both 25. Neither do I know how to reconcile that with the basis of the multiplication of two numbers which requires that unity is to the one of the numbers what the other is to the product. This is both for whole numbers and for fractions. For 1 is to 3, what 4 is to 12. And 1 is to ^ what \ is to -^ . But I cannot reconcile that with the multiplication of two negatives. For can one say that +1 is to -4 , as - 5 is to +20? I do not see it. For +1 is more than -4 . And by contrast - 5 is less than +20. Whereas in all other propositions if the first term is larger than the second, the third must be larger than the fourth. Activity 10.11, page 133 1 a Summarise both extracts before you try to list the assumptions which each author makes. Here are notes on Euler's extract as an example: • no doubt that positive multiplied by positive is positive, but separately examines positive by negative and negative by negative; • negative numbers modelled as a debt; multiplying a debt by a positive number bigger than 1 gives a larger debt; thus -a times b and a times —b is —ab; • -a times —b must be either —ab or ab. It cannot be —ab since -a times b gives that and a times -b cannot give same result; so -a times -b must be ab . By looking through your summary can you see the assumptions which Euler and Saunderson make? b You will first need to summarise Arnauld's objections to negative numbers and the assumptions which he makes. You can get help to do this if you consult the hint and also the answers to Activity 10.10. 161
outcome 1 step to the right Figure 13.1 162 13 Hints 1 step upwards' To stage the debate, or to draw up a plan for your presentation or essay, you could consider the following format: • first, summarise Arnauld's extract in Activity 10.10; • secondly, summarise Euler's and Saunderson's extracts. In each case you should draw out the key points. Then, you could start on the debate by • presenting what Arnauld is likely to believe is the flaw in Euler's and Saunderson's arguments; • allowing Euler and Saunderson to answer back or counterattack. Remember not to use modern arguments to present the flaws in the various arguments. Try to argue from the points of view of Arnauld, Euler and Saunderson based on what you know of them from the extracts. You could consult your list of assumptions made by Euler and Saunderson; by looking at them can you find a 'chink in their armour' from Arnauld's point of view, or vice versa? Activity 77.7, page 138 1 How did you set about establishing the result for the product of two negative numbers? You should consider whether the process you used was a justifiable one given the original role of the numbers a, b, c and d. Activity 11.4, page 141 1 One counter-example is the answer given to question Ic in Activity 10.10 in connection with Arnauld's opposition to negative numbers. Activity 11.8, page 143 1 b The operations for combining the elements in Boole's algebra will not be labelled by + and x and the elements will not be labelled by numbers. Imagine replacing the symbols in the rules for combining Boole's elements by + and x in a suitable way and imagine the elements to be numbers. Would the rules then look the same as those for the number system? If not, how would they differ? Activity 11.10, page 145 1 To get started, here is part b as an example; you can then try the other parts. Take a step of one unit to the right (i) and add to it a step of one unit 'upwards', (j). This produces a step as shown in Figure 13.1 but not in either of the basic directions (to the right and upwards); it is inclined at an angle to the basic directions. What is the angle at which this new step is inclined and what is its length? Activity 11.12, page 146 2 b Think about something you take for granted with real numbers. When you multiply three numbers together can you multiply them in any order; does the order matter?
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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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Introduction to the Babylonians On
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ff m yfr < n Figure 2.5 «« Activi
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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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3 An introduction to Euclid fthese
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34 The Greeks Activity 3.4 g Now us
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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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7776 Arabs Activity 5.4 This proble
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Page 187 and 188: 14 Answers b The required number is
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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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