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PM752C: Approaching Infinity INFINITY Prepared for the course team by John Mason Centre for Mathematics Education Project MATHEMATICS UPDATE Course Team Gaynor Arrowsmith, Project Officer, Open University Lynne Burrell, Academic Editor, Open University Leone Burton, External Assessor, Thames Polytechnic Joy Davis, Liaison Adviser, Open University Peter Gates, Author, Open University Pete Griffin, Author, Open University Nick James, Liaison Adviser, Open University Barbara Jaworski, Author, Open University John Mason, Author and Project Leader, Open University Ruth Woolf, Project Secretary, Open University Acknowledgments Project MATHEMATICS UPDATE was funded by a grant from the Department of Education and Science. We are most grateful for comments from Ruth Eagle, Arthur Hanley, Michelle Selinger, Eileen Billington, Gillian Hatch and many others who may not have realised at the time that they were working on parts of this pack. The Open University, Walton Hall, Milton Keynes MK7 6AA. First published 1988. Copyright O 1988 The Open University. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, without written permission from the publisher. Printed in Great Britain by The Open University. Further information on this and other Open University courses may be obtained from the Learning Materials Service Office, Centre for Continuing Education, The Open University, PO Box 188, Milton Keynes MK7 6DH. ISBN 0 335 15918 4 CONTENTS INTRODUCTION AIMS WAYS OF WORKING ENCOUNTERS WITH THE INFINITE Decimals and bicimals; Counting; Geometry; Division by zero; Bisection; Completed acts COUNTING AND ORDERING Counting; Ordering GEOMETRICAL INFINITIES Infinite circles; Geometric series; Zigzags; To the limit; Centre of gravity ZERO ARGUING WITH INFINITY There exists for all; Limits PHILOSOPHICAL THE LAST WORD GLOSSARY 0 INTRODUCTION The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite . David Hilbert 1862 - 1943 How many numbers are there? How many fractions? How many points on a line? Can something be 'infinitely small' but not zero? What is the value of 0.999 . ? What does . mean? 111 In what sense is1 +p +a + . equal to 2? Does each point on a line correspond to a number, and vice versa? Does the infinite expression 1 - 1 + 1 - 1 + 1 - . have a value? These and similar questions are explored in this pack. Rarely will straight answers be given, because the power of these questions lies in the struggle to attempt to resolve them to your own satisfaction, and to that of your colleagues. The activities presented here are intended to help you make contact with significant and sometimes difficult questions about the infinite in mathematics. Don't be satisfied with easy answers! Nothing is mathematical, unless it is shot through with infinity. Caleb Gattegno l967 Working on the activities suggested in this pack will provide an opportunity to consider Gattegno's assertion, and to explore ways in which the infinite can be exposed and developed in almost any mathematical topic. Everyone has a sense of the infinite, and adolescents are particularly attracted to thinking about it, perhaps because it is connected with their growing awareness of themselves as people caught up in a huge intricate web of social, physical and intellectual relationships. Questions such as What is at the edge of the universe - or can it have an edge? 0 Must there have been a beginning, or could there be an infinite past? Will life continue for ever? Will any part of me carry on indefinitely? are of concern to most people at some time in their lives. They are not addressed directly by mathematics, but the ways in which mathematicians approach the idea of infinity finds resonance in the thoughts and concerns of many pupils. The very earliest of written records contain references to questions about infinity: Where nothing else is seen, or heard, or known, there is the infinite. Where something else is seen, or heard, or known, there is the finite. The Infinite is immortal; but the finite is mortal. Chandogya Upanishad 800 - 400 BC Infinity is a fathomless gulf, into which all things vanish. Marcus Aurelius 121 - 180 AD When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named. Thomas Hobbes 1588 - 1679 Infinity is a dark illimitable ocean, without bound. John Milton l608 - l674 I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics. The infinite is but a facon de parler, meaning a limit to which certain ratios may approach as closely as desired, when others are permitted to increase indefinitely. Car1 Friederich Gauss 1777 - 1855 The oldest records of human beings suggest that the incomprehensible, the transcendent experience, the unfathomable is of consuming interest. Associated with spiritual experiences, the term infinity is only defined negatively, by what it is not rather than what it is, since literally it means not-finite. From the paradoxes of Zeno, who questioned simple perceptions of motion, to the insights of Georg Cantor, who made the bold step of completing infinite acts, mathematicians can be seen as attempting to approach and control infinity. It all begins, and perhaps ends, with counting. What now is taught in every primary school (one-to-one matching) was original, liberating and frightening to Georg Cantor and even more so to his contemporaries. On the voyage described in these materials, we have the chance to live through some of the mental struggles of generations .of mathematicians. AIMS To indicate various ways in which mathematicians employ and deal with the infinite. To provide situations which will engender confidence in working with the infinite. To draw attention to two aspects of infinity - the infinite as process unfolding in time and the infinite as a completed act. To suggest that infinity is never far away from any mathematical question. For a more detailed account of many of the ideas presented here, the following books are highly recommended. Ray Hemmings, Dick Tahta, Images of Infinity, Leapfrogs insight series, Leapfrogs, 1984. Rudy Rucker, Infinity and the Mind, Penguin, 1986. Eli Maor, To Infinity and Beyond, Birkhauser 1987. WAYS OF WORKING This pack consists of sections of related activities, each section focussing on an important aspect of infinity in mathematics. The first section introduces the six major themes, which are followed up in greater detail in the subsequent sections. It is not intended that you work systematically subsection by subsection, or even that you tackle every activity in a subsection. Rather, find some attractive activities, explore in and around them, addiscuss what you find with colleagues. Words in italics are either for emphasis, or are technical terms which are explained in the GLOSSARY. Where more precise definitions and theorems are wanted, recourse to standard texts will provide them, and more. Your most important activity will be your attempt to tell yourself a coherent (mathematical) story after working on one or more of,the suggested activities, and to see how different activities relate to each other. You cannot deem yourself to have finished until you have completed Section 7 THE LAST WORD. i It is important when working on an activity, not to get lost simply in the 'doing'. Every so often there are suggestions for reflection - for pausing, drawing back, and making sense of what has been noticed. Drawing together several ideas in the mind, and trying to hold them there, can help to integrate them into a whole. There are also some suggestions for pausing and considering similarities and differences between the activities. It is hoped that such pausing will become natural and automatic. It is impossible to mark in the text every useful pausing point. Reflection is partly a solitary activity, as you try to encompass a number of disparate ideas at once in your mind. It is also a communal activity, as you try to express to others the sense you have made, in the form of connections, similarities and differences. Your attempts to express ideas which are only half formed can certainly help you, and you can gain from the ideas and perspectives of others. To prepare for a productive discussion with colleagues, it is usually useful to try to write down a few sentences that capture what you think the various activities have been about, with any questions or uncertainties that you are aware of. Working on an activity, it is very possible that you will get stuck. What do you do then? The BOOKMARK: What to do when you're stuck contains some suggestions. Use it. Use any colleagues you have access to, to discuss your 'stuckness'. It may be that you are not clear what the activity involves. If so, then investigate for yourself, formulating your own questions. It may be that you are not clear what the activity is trying to get at. If that is the case, then try to make up some sort of story, write it down, and compare notes with colleagues. After working for a while on a mathematical activity, there is often a strong desire to know whether you are 'on the right track'. However, these notes contain no answers as such, for many reasons. The most salient reason is that generally there is no particular correct answer. There are many things worth noticing, and these are likely to emerge from reflection and discussion. It is essential that you develop your own criteria for deciding when you have 'done enough for now'. If you have some nagging doubt that there is more to discover, then perhaps it is worth going back and investigating further.
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