Number Story From Counting to Cryptography
Numbers have fascinated people for centuries. They are familiar to everyone, forming a central pillar of our understanding of the world, yet the number system was not presented to us "gift-wrapped" but, rather, was developed over millennia. Today, despite
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Number Story From Counting to Cryptography PETER M. HIGGINS
COPERNICUS B O OKS An Imprint of Springer Science+Business Media
Peter M. Higgins, BA, BSc, PhD Department of Mathematical Sciences, University of Essex Wivenhoe Park, Colchester, UK
Published in the United States by Copernicus Books, An imprint of Springer Science+Business Media, LLC
Mathematics Subject Classification (2000): 11-01 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2007936363
ISBN 978-1-84800-000-1
e-ISBN 978-1-84800-001-8
Printed on acid-free paper. © Springer-Verlag London Limited 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. 9 8 7 6 5 4 3 2 1 Springer Science+Business Media springer.com
Preface chapter 1
chapter 2
chapter 3
chapter 4
chapter 5
ix
The First Numbers
1
How Should We Think About Numbers? The Structure of Numbers
5 8
Discovering Numbers
17
Counting and Its Consequences
23
Some Number Tricks
31
What Was the Domino? Casting Out Nines Divisibility Tests Magical Arrays Other Magic Number Arrays
34
Some Tricky Numbers
35 39 49 57 61 65
Catalan Numbers Fibonacci Numbers Stirling and Bell Numbers Hailstone Numbers The Primes Lucky Numbers
84
Some Useful Numbers
85
Percentages, Ratios, and Odds Scientific Notation Meaning of Means
88
67 72 75 77
85 90
v
chapter 6
chapter 7
chapter 8
chapter 9
On the Trail of New Numbers
101
Pluses and Minuses Fractions and Rationals
104
Glimpses of Infinity
117
The Hilbert Hotel Cantor’s Comparisons Structure of the Number Line Infinity Plus One
120
chapter 11
vi
122 128 133
Applications of Number: Chance
137
Some Examples Some Collectable Problems on Chance
141
The Complex History of the Imaginary Algebra and Its History Solution of the Cubic
chapter 10
105
148
165 168 174
From Imaginary to Complex
185
The Imaginary World Is Entered The Polar System Gaussian Integers Glimpses of Further Consequences
189
200
The Number Line under the Microscope
209
Return to Egypt Coin Problems, Sums, and Differences
2
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