eBook
Available on Compatible NOOK devices, the free NOOK App and in My Digital Library.
Related collections and offers
Overview
Much in our daily lives is defined in numerical terms-from the moment we wake in the morning and look at the clock to dialing a phone or paying a bill. But what exactly is a number? When did man begin to count and record numbers? Who made the first calculating machine-and when? At what point did people first think of solving problems by equations? These and many other questions about numbers are answered in this engrossing, clearly written book.
Written for general readers by a teacher of mathematics, the jargon-free text traces the evolution of counting systems, examines important milestones, investigates numbers, words, and symbols used around the world, and identifies common roots. The dawn of numerals is also covered, as are fractions, addition, subtraction, multiplication, division, arithmetic symbols, the origins of infinite cardinal arithmetic, symbols for the unknown, the status of zero, numbers and religious belief, recreational math, algebra, the use of calculators — from the abacus to the computer — and a host of other topics.
This entertaining and authoritative book will not only provide general readers with a clearer understanding of numbers and counting systems but will also serve teachers as a useful resource. "The success of Flegg's lively exposition and the care he gives to his surprisingly exciting topic recommend this book to every library." — Choice.
Product Details
| ISBN-13: | 9780486166513 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/15/2013 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 304 |
| File size: | 10 MB |
Read an Excerpt
NUMBERS
Their History and Meaning
By Graham Flegg
Dover Publications, Inc.
Copyright © 1983 Graham FleggAll rights reserved.
ISBN: 978-0-486-16651-3
CHAPTER 1
Encountering Numbers
This life's first native source.
(Spenser)
O where is the spell that once hung on thy numbers?
Arise in thy beauty ...
(Irish song)
There is no way in which we can escape from numbers. Numbers are an integral part of everyday life and, as far as we can tell, have always been so. When we wake up in the morning, our first almost instinctive action is to look at our watch or alarm-clock. This immediately confronts us with virtually all the crucial aspects of numbers. We have to locate the numerals to which the hands are pointing or, if we have a digital timepiece, to read a set of numerals. We thereby come face-to-face with written number symbols. If we have an ordinary clock or watch, the numerals may be Roman, I, II, III, ... or Hindu-Arabic, 1, 2, 3, ...; if we have a digital timepiece they will be the latter. When we are asked what the time is, we have to reply in words – 'two o'clock' perhaps. We have now been forced to make use of a number-word, the word 'two' corresponding to the numeral 2. But the word and symbol relate to a specific context, time, and here we face one of the important phenomena of life – the fact that some things are continuous and some things are discrete. This distinction has its counterpart in numbers. We think of time and space as being continuous. Yet, although we may be confident that we know exactly what we mean, the concept of continuity is by no means easy to define. It is almost certain that any precise definition which we attempted to make would not stand up to serious mathematical scrutiny – that is, unless we are specialist mathematicians and have been initiated into the deep mysteries of those numbers which are known technically as 'real'.
When we come to measure time and space, we encounter another problem. In theory at least, if time and space are continuous then whatever we use to measure them must also be continuous if we are to measure them exactly. We can obviously measure them approximately with something that is discrete. Further, we can measure them as accurately as we choose; yet there are several reasons why we cannot measure them exactly. A digital clock, for example, may measure time to a small fraction of a second. Quartz measuring instruments can measure time to an almost unbelievable degree of accuracy. No instrument can, however, measure time exactly. We can understand this in a crude kind of way if we attempt to measure a table with a ruler or tape. We can measure it to the nearest inch – this is obviously only a very crude approximation. We can measure it to the nearest centimetre – this will be a little more accurate. We can measure it to the nearest millimetre, and, if we use a strong magnifying glass, we can perhaps measure it to the nearest tenth of a millimetre. Somewhere we have to stop – we reach the end of the capability of the measuring system adopted. In any case, there will have been little point in attempting to measure to fractions of a millimetre because our reading also depends on how accurately we have aligned the zero of the measure to the other side of the table. This is a problem which affects all measurement. Although measurement seems as if it ought to be a continuous phenomenon, in practice it is discrete. Again, we find this is reflected in numbers. Just as there are numbers which correspond to the continuous, there are also numbers corresponding to the discrete, the most obvious of these being the whole numbers, though the numbers which correspond most closely to the measuring situation are known as the 'rational' numbers – the ratios of whole numbers.
The problems associated with measurement do not end here, however. If we construct a square whose side measures one unit, there is no way in which we can measure its diagonal exactly using just the rational numbers. No matter how we argue that by dividing up that one unit indefinitely into ever smaller and smaller fractions we ought in the end to be able to measure the diagonal exactly, any reasonably competent mathematician can prove that our argument is false. This does not affect us in practice, however, because we can still measure the diagonal as accurately as is necessary. Mathematicians respond to this situation, known since the times of the Pythagoreans, by classifying the square root of two as an irrational number. Yet, even if we add to the rational numbers all those of the same kind as the square root of two, we still do not have enough numbers to measure continuous quantities exactly, even in theory.
The most fundamental way in which numbers arise is through counting. This is the way in which numbers were first understood by man. When we count, we assign objects to as many of the positive whole numbers as are necessary in turn until we have exhausted whatever it is we are counting. We do this automatically. But this seemingly simple process conceals another important aspect of numbers – the fact that they are ordered. In order to count up to five (say), we not only need to be familiar with the first five number-words, we need to be familiar with them in their order. It is probable that it was this 'ordinal' aspect of counting, as it is called, that was the first aspect of numbers of which man became aware.
Many of the apparent problems which are associated with counting, measuring, and continuous aspects of number arise because of the difference between what we call the 'real world' and the conceptual world which we create in our own minds. This is highly relevant to the question 'what are numbers?' Numbers are the basis on which the whole structure of mathematics has been built, and the positive whole numbers are the basis of all other numbers – for this reason they are called 'natural' numbers. Provided that we accept these numbers as being 'naturally' given, all others can be constructed from them, though it needs some fairly sophisticated concepts to make the jump from the discrete to the continuous and thus construct the real numbers from the rational numbers. This does not however tell us what kind of things numbers are. We get a possible clue if we think about the disparity between measuring in theory and measuring in practice. When we say 'in practice', we mean that we are speaking of something which we can actually do physically. When we say 'in theory', we are speaking of something which we can do in our minds. The distinction is that between physical 'reality' and conceptual 'reality'.
If we take the number two, we can be aware of this number in at least four different ways – as a numeral, as a number-word, as a concept in our minds, and as a property possessed by every collection of two objects. Although for many practical purposes we do not need to worry about these different aspects of numbers, it is very important that we are aware of them in any study of the history of numbers. There is nothing in the physical world which is two. There are, however, a great many things in the physical world to which 'two' may be usefully applied. Numbers are thus essentially concepts, and mathematics is the study of these concepts and of the structures which can be built from them. The concept of numbers arises directly out of our experience of the physical world in the same sort of way as our concept of colours. Numbers are idealizations in the mind of particular experiences encountered in the world. The number two does not have an independent existence of its own except as a concept, neither does redness have an existence except as a concept. Perhaps we should have coined the word 'twoness' rather than the word 'two' – the analogy would then have been a little more obvious.
To some extent we are led away from the appreciation of the conceptual status of numbers by the symbols which represent them. We are so used to manipulating these symbols that they come to take on, as it were, a life of their own. Since the symbols clearly exist in the physical world, we tend to grant the same status of existence to the concepts which they represent. The fact that we encounter and use both numerals and number-words every day of our lives gives them a deceptive familiarity. They exert their own particular spell upon us, and the concepts which they represent are thereby translated from the realm of the mind into the physical world which surrounds us.
Numbers are, nevertheless, endemic to the natural world in some remarkable ways. If numbers are conceptualizations of man, we may well ask how it is that they appear to govern so many of the phenomena in nature. Some of the most beautiful shapes to be seen in nature are found on close examination to be governed by series of numbers. This indicates that numbers are in some way connected with aesthetics, a connection often exploited by artists and architects in ways which very much correspond with the harmonies of music. The centre of a daisy is composed of scores of tiny florets arranged in two opposite sets of spirals, 21 spirals in a clockwise direction and 34 in a counterclockwise direction. A pineapple has eight spirals of bumps going in the clockwise direction and thirteen in the counter-clockwise direction. Pine-cones are built up in a similar pattern; this time five spirals go clockwise and eight counter-clockwise. There is nothing very spectacular in all this – not until we begin to look for other connections between these numbers.
If we start with the number one, and create a sequence of numbers built up in such a way that each number is the sum of the previous two numbers, we obtain:
1, 1, 2, 3, 5, 8, 13, 21, 34....
This sequence includes all the pairs of numbers which we have just noted to exist in various ways in nature. The special form of this sequence, known as the 'Fibonacci sequence' after its discoverer (Leonardo of Pisa, son of Bonaccio) has been known since the twelfth century. It arises directly from the following problem:
How many pairs of rabbits can be produced from a single pair in a year if every month each pair begets a new pair which, in turn, becomes productive from the second month onwards?
Again, this may not seem particularly interesting or extraordinary. This sequence is, however, by no means confined to daisies, fruit and a problem about rabbits. It occurs again and again in nature. It occurs, for example, in the way in which successive leaves grow around the stems of many plants and trees, and it is only one of many such instances in which sequences of numbers, apparently invented by man, are found to have existed in nature since the times before man appeared on the earth. This raises some interesting problems for debate. For example, do we invent mathematics or discover mathematics? Are numbers in some sense at least 'given' so that there comes a point at which any attempt to find still more basic definitions is useless? To what extent are the numbers which we find so extensively in nature, and the relations between these numbers, directing the way in which nature evolves? These and other similar questions have been debated by mathematicians, philosophers, and even theologians over the centuries. The question has been asked: 'is God a number?' Certainly, there has often been a close relationship between numbers and religious beliefs. Some theologians have denied that numbers are creations at all, and have suggested that numbers control both the Deity and His created universe. We shall not debate these questions, interesting though they certainly are. The inclusion of the word 'meaning' in the sub-title of the book is not meant to refer to abstract philosophical questions about numbers but rather to their practical significance in mathematics and in life generally. In the following chapters we are concerned to present as many of the basic facts about numbers as are readily accessible to the layman, and to do so in a historical context.
Most people probably agree that numbers play an important role in everyday life and are crucial to man's economic, scientific and technological development. It is not as fully appreciated that they play just as crucial a role in the 'humanities', even though this was well-known in Ancient Greece. It comes, of course, as no surprise to the mathematicians. No one who is truly versed in the art of numbers and the structures which can be built from them can fail to be aware that they have a particular kind of beauty which is all their own. The popular image of the pure mathematician as one who is divorced from both the harsh realities and the aesthetic beauties of the world lies far from the truth. In fact, the great majority of mathematicians have a serious interest in aesthetic studies of one kind or another. This is not because such studies are a relaxing contrast to mathematics, but because beauty of shape and sound is felt to be a reflection of the beauty and order to be found in numbers.
It has been claimed that mathematics is the 'queen of the sciences'. This carries with it the suggestion that mathematics has feminine characteristics, yet it was first made at a time when practical activities were thought of as being essentially masculine and the arts feminine. We might also note that justice and mercy were thought to have respectively masculine and feminine characteristics. In a sense, mathematics – and numbers in particular – effect a marriage between the complementary masculine and feminine aspects of life. Numbers reveal the unity which underlies all of life as we experience it. There is an increasing awareness of this today, despite the widespread popular prejudice against mathematics. We are perhaps gradually returning to the viewpoint of the Greek philosopher mathematicians whose belief was that 'all things are number', re-echoed in the claim of the nineteenth-century French philosopher Auguste Comte that 'there is no enquiry which cannot finally be reduced to a question of numbers.'
CHAPTER 2Counting with Numbers
The King was in his counting house
Counting out his money.
(Nursery rhyme)
He counted them at break of day –
And when the sun set where were they?
(Lord Byron)
Counting is an everyday activity of man, and has been since before the dawn of history. It is an activity which is inseparable from speech. To be able to count, we must know a sequence of number-words and be able to relate these in their proper order to whatever is being counted. This does not mean, however, that an abstract understanding of numbers is needed. The intellectual step taking us from counting to numbers in the abstract is a comparatively sophisticated one which came late in man's history. Counting was the first of a long succession of practical and intellectual steps which has led to the mathematics of today. It lies at the root of all that we have learned about numbers from the simplest arithmetic to the complex calculations which have enabled man to set foot on the Moon and to devise the means of his own total annihilation.
Rudimentary Beginnings
There are certain rudimentary senses which man shares with many other creatures. These include an awareness of size and shape and some sort of an appreciation of quantity. Without them, counting would be impossible.
Man's sense of size enables him to distinguish between one object and another. It enables him to appreciate, for example, the special kind of difference between a pebble and a boulder or between a mountain and a hillock. This sense of size has always been a necessary part of his reaction to the world about him. A basic sense of size precedes any development of the concept of numbers. Numbers do not become involved even implicitly until the need arises to consider the result of putting, actually or mentally, several objects in order according to their sizes. Numbers also become unavoidable as soon as man needs to measure, however crudely, and to express differences in the sizes of various objects in quantitative terms.
Man's sense of quantity gives him the ability to see that there is a particular kind of difference between one collection of objects and another. It enables him to see that, for example, his neighbour has more cattle than he, and to do this long before he is able to associate it with numbers.
In the first instance these two senses will have been applied only crudely. This means that there will have been little difficulty in appreciating size differences between objects of very different kinds. There is, for example, little distinction between the awareness that a mountain is larger than a hillock and the awareness that it is larger than a stone. But when the sense of size becomes more refined and objects more nearly equal in size have to be compared, differences in shape become crucial. Eventually, with objects of the same kind and shape, it becomes ever more difficult to be sure about relative size, and we arrive at the need to carry out some kind of comparative measurement.
In a similar way, with the sense of quantity, crude comparisons between collections of large numbers of objects and those of much smaller numbers of objects present no difficulty. This is true even when the objects themselves are of different kinds. As the sense of quantity is applied in a more refined way, comparison becomes increasingly difficult. Eventually the need for more advanced abilities cannot be avoided.
(Continues...)
Excerpted from NUMBERS by Graham Flegg. Copyright © 1983 Graham Flegg. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
PrefaceChapter 1. Encountering Numbers
Chapter 2. Counting with Numbers
Rudimentary Beginnings; One-to-One Correspondence and Order; The Beginning of Numbers; Patterns and Groups; Finger-Counting;
Counting by Twos; Counting by Fives; Counting by Twenties; The History of Counting; The Age of our Decimal Counting; Counting with Knots
Chapter 3. Writing down Numbers
Tallies; The Dawn of Numerals; The Development of Numeral Systems; The Evolution of Place-value; Alphabetical Numerals; The Hindu-Arabic Numerals; Fractions; The Derivation of Number-Symbols
Chapter 4. Calculating with Numbers
Addition; Substraction; Multiplication; Division; Checks on Calculation; Ratio and Proportion; Prime Numbers and Factors; The Calculation of Roots; Logarithms; Arithmetic Symbols
Chapter 5. Computing with Numbers
Early Counting Boards; Counting Boards in the Middle Ages; Reckoning on the Lines; The Modern Abacus; The Slide Rule; Calculating Machines; Binary Arithmetic; Computers
Chapter 6. Unknown Numbers
The Linear Unknown; The Square of the Unknown; Higher Powers of the Unknown; More than One Unknown; Indeterminate Unknowns; Symbols for the Unknown
Chapter 7. Recreational Numbers
Think of a Number; Magic Squares; Number Shapes; Numbers with Special Properties
Chapter 8. Thinking about Numbers
Rational and Irrational Numbers; The Real Numbers; The Status of Zero; Complex Numbers; Defining the Natural Numbers; Numbers and Religious Belief
Chapter 9. Teaching and Learning Numbers
Words and Symbols; Elementary Calculation; Algebra; Numbers and Life
Bibliographical Note; Index