Numbers: Their Tales, Types, and Treasures

335 Pages·2015·9.63 MB·English

by  Alfred S. Posamentier

ALSO BY ALFRED S. POSAMENTIER AND INGMAR
LEHMANN
The Fabulous Fibonacci Numbers
Pi: A Biography of the World's Most Mysterious Number
Mathematical Curiosities
Magnificent Mistakes in Mathematics
The Secrets of Triangles
Mathematical Amazements and Surprises
The Glorious Golden Ratio
ALSO BY ALFRED S. POSAMENTIER
The Pythagorean Theorem
Math Charmers

Published 2015 by Prometheus Books
Numbers: Their Tales, Types, and Treasures. Copyright © 2015 by Alfred S. Posamentier and Bernd
Thaller. 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, digital, electronic, mechanical, photocopying, recording, or
otherwise, or conveyed via the Internet or a website without prior written permission of the publisher,
except in the case of brief quotations embodied in critical articles and reviews.
Cover image © Can Stock Photo Inc./Sylverats
Cover design by Jacqueline Nasso Cooke
Unless otherwise indicated, all interior images are by the authors.
Inquiries should be addressed to
Prometheus Books
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The Library of Congress has cataloged the printed edition as follows: Posamentier, Alfred S.
Numbers : their tales, types, and treasures / by Alfred S. Posamentier & Bernd Thaller.
pages cm
Includes bibliographical references and index.
ISBN 978-1-63388-030-6 (pbk.) — ISBN 978-1-63388-031-3 (ebook) 1. Number concept. 2.
Counting. 3. Arithmetic—Foundations. I. Thaller, Bernd, 1956-II. Title.
QA141.15.P67 2015
513.5—dc23
2015011662
Printed in the United States of America

Acknowledgments
Chapter 1: Numbers and Counting
Chapter 2: Numbers and Psychology
Chapter 3: Numbers in History
Chapter 4: Discovering Properties of Numbers
Chapter 5: Counting for Poets
Chapter 6: Number Explorations
Chapter 7: Placement of Numbers
Chapter 8: Special Numbers
Chapter 9: Number Relationships
Chapter 10: Numbers and Proportions
Chapter 11: Numbers and Philosophy
Appendix: Tables
Notes
Index
We would like to thank Norbert Holzer, an expert in the preparation of
elementary school teachers in Graz, Austria, and a specialist for dyscalculia and
its diagnosis, who provided us fine insights about how children develop their
ability to count. We are also grateful to Dr. Peter Schöpf, retired mathematics
professor of the Karl-Franzens University in Graz, for his keen insight into the
history and philosophy of mathematics. We also thank Peter Poole for his timely
support with a few topics in the book.
Many thanks to Catherine Roberts-Abel for very capably managing the
production of this book, and to Jade Zora Scibilia for the truly outstanding
editing throughout the various phases of production, with the assistance of Sheila
Stewart. Steven L. Mitchell, editor in chief, deserves praise for enabling us to
approach the general readership to expose the gems that lie among the
commonly known concept of numbers.
1.1.A MENTAL NETWORK
We can't live without numbers. We encounter them every hour of every day.
Numbers have shaped the way we think about the world. They penetrate every
aspect of our life. Our whole society is organized with the help of numbers; it
depends on numbers in many respects, and it has been that way since the dawn
of civilization. Numbers rule our life.
We need numbers for counting, for measuring, and for doing calculations.
We have numbers to describe dates and times and to tell the price of goods and
services. We use numbers when we buy our meals or count our days. Numbers
can be manipulated to improve statistics or to cheat in games. We are identified
by Social Security numbers, license numbers, credit card numbers, and
telephone numbers. Numbers describe sports records, baseball scores, and
batting averages. Science, economy, and business are all about numbers, and we
find numbers even in music, for example, in rhythm and harmony. To some,
numbers are a never-ending source of joy and fascination, while others feel that
numbers are depressing, impersonal, often incomprehensible, and without soul.
Undoubtedly, people who lack fundamental skills with numbers will face
diminished life chances, difficulties finding a job, and other serious impairments
in everyday life, similar to people who can't read.
The immense importance of numbers should make us pause a bit and think
about their nature and their origin. What are numbers? Where do they come
from? Who was the first to use them? Indeed, there is more to these questions
than meets the eye. In order to find answers, we will embark on a journey that
visits the realms of psychology, ethnology, history, and philosophy. In the
course of this journey, we will learn about ourselves, our mind, and our number
sense; we will think about reality and mathematics; and we will encounter
fascinating ideas and surprising facts.
Indeed, what is a number? At first, this may seem like a rather odd and
unnecessary question. The symbols 1, 2, 3, 4, and so on appear so utterly
familiar; their meanings seem so obvious that an explanation can only create
confusion. Numbers belong to our shared knowledge about the world. We all
recognize a number when we see one. It is notoriously difficult to explain
something that everybody knows already, in particular if one hasn't thought of it
before.
Marvin Minsky, in his book The Society of Mind, also muses about the nature
of numbers and asks why it would be so difficult to explain meaning to others:
“Because what something ‘means’ depends on every different person's state of
mind.”1 The hope that through an explanation or precise definition, “different
people could understand things in exactly the same ways” cannot be fulfilled,
“because in order for two minds to agree perfectly, at every level of detail, they'd
have to be identical.” Nevertheless, “the closest we can come to agreeing on
meanings is in mathematics, when we talk of things like ‘Three’ and ‘Five.’ But
even something as impersonal as ‘Five’ never stands isolated in a person's mind
but becomes part of a huge network.”
In everyday life, there are many occasions contributing to the growth of the
mental network of knowledge and meaning that is associated with a number.
Numbers are often encountered in situations that have little to do with
mathematics. Think for a moment of a number like four, and you will certainly
come up with a lot of situations where this number plays a role (such as, the four
wheels of a car, the four wisdom teeth, the four seasons, and so on). Even a less
obvious example, like the number nine, produces a lot of associations in various
contexts—there are Dante's nine circles of hell, Tolkien's nine rings of power,
and the nine worlds of Yggdrasil in Norse mythology. Beethoven composed nine
symphonies; a Chinese dragon has nine forms; Europeans like nine-pin bowling
games; in the Caribbean Sea we find nine-armed sea stars; in Jewish culture, the
Chanukah menorah is a nine-branched candelabrum; a baseball team has nine
players on the field, and a complete game has nine innings. An old saying goes
that a cat has nine lives; another, that nine tailors make a man; and when we are
very happy, we are on cloud nine. Ramadan is the ninth month in the Islamic
calendar; normal office hours start at nine in the morning; human pregnancy
usually lasts nine months. Dressing nicely is often referred to as being dressed to
the nines. Nine is a good number in Chinese mythology, but an unlucky number
in Japanese culture, where its pronunciation reminds one of the word for agony
or pain. And when we take the whole lot, we take the whole nine yards.