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Did you grow up thinking math is boring? It’s time to reconsider. This book will teach you everything you ever wondered about numbers—and more.
How and why did human beings first start using numbers at the dawn of history? Would numbers exist if we Homo sapiens weren’t around to discover them? What’s so special about weird numbers like pi and the Fibonacci sequence? What about rational, irrational, real, and imaginary numbers? Why do we need them?
Two veteran math educators explain it all in ways even the most math phobic will find appealing and understandable.
You’ll never look at those squiggles on your calculator the same again.
|Product dimensions:||9.00(w) x 6.00(h) x 1.10(d)|
About the Author
Bernd Thaller is associate professor for applied mathematics at the Institute for Mathematics and Scientific Computing at the University of Graz in Austria. He is the author or coauthor of four books in mathematics.
Read an Excerpt
Their Tales, Types, and Treasures
By Alfred S. Posamentier, Bernd Thaller
Prometheus BooksCopyright © 2015 Alfred S. Posamentier and Bernd Thaller
All rights reserved.
Chapter 1: Numbers and Counting, 11,
Chapter 2: Numbers and Psychology, 51,
Chapter 3: Numbers in History, 77,
Chapter 4: Discovering Properties of Numbers, 105,
Chapter 5: Counting for Poets, 129,
Chapter 6: Number Explorations, 161,
Chapter 7: Placement of Numbers, 183,
Chapter 8: Special Numbers, 221,
Chapter 9: Number Relationships, 245,
Chapter 10: Numbers and Proportions, 287,
Chapter 11: Numbers and Philosophy, 329,
Appendix: Tables, 367,
NUMBERS AND COUNTING
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." 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.
Depending on your personal background, some of these examples, and perhaps some others, will come to your mind when you think of nine (see figure 1.1). And similar or even larger amounts of rich associations come with many other numbers, giving them individuality and meaning. These numbers, forming parts of every individual's mental network, are not that impersonal after all. "Numbers have souls, and you can't help but get involved with them in a personal way," writes Paul Auster (1947-) in his novel The Music of Chance And when he emphasizes this point, the statement even gets a slightly absurd touch:
After a while you begin to feel that each number has a personality of its own. A twelve is very different from a thirteen, for example. Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, a shady character who won't think twice about breaking the law to get what he wants. Eleven is tough, an outdoorsman who likes tramping through the woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what he's told; nine is deep and mystical, a Buddha of contemplation.
1.2. WHAT IS A NUMBER?
As this is probably difficult to answer, let us ask a different question: "Can you give an example of a number?" Probably, the answer will be something like 5, or five. But then, what about V or | or 3 + 2 or cinque?
Clearly, the symbol 5 is not a number — it is just a symbol. It is a common mistake to take a symbolic representation for the "real thing." But this mistake is very understandable because our everyday language does not distinguish between them and calls everything a number. But as long as we talk about the "meaning of numbers," we have to be precise: A symbol, like 5, serves to designate a number, but it is not the number itself. Indeed, the number five can be represented by quite different symbols — for example, by the Roman symbol V or the Chinese [TEXT NOT REPRODUCIBLE IN ASCII]. The number five can even exist without any written symbol at all — it was probably used by Homo sapiens long before the invention of writing and expressed by showing the fingers of one hand.
In the same way, the spoken word five (a combination of sounds) and the written word five (a combination of letters) are just representations of the number five. The number itself is an abstract idea, and it can be expressed in many different ways and by other words. For example, the word for five in French is cinque, in German it is funf and in Japanese it is go. In any case, all these different representations — symbol, word, sound, or even a dot pattern like [??] — should evoke the same idea of the number five. In linguistics, a word designating a number, like five, or twenty-four (no matter whether it is spoken or written), is called a numeral or a number word.
So far, we have not really explained what a number is; rather, we have said what it is not: It is not a symbol or a number word, which are just names. We are going to distinguish between the abstract idea number and the words or symbols used to designate numbers. The abstract idea is unique and invariable; symbols and words are a mere matter of convention and hence quite arbitrary. Moreover, there is a difference between the idea of a number and its different (although related) applications. The number described by the symbol 5 could be used, for example, to describe the fifth place in a sequence (as an ordinal number) or the number of objects in a collection (as a cardinal number) or the length of a flagpole in yards (as a measuring number).
In this chapter, we want to describe the "thing behind the symbol," the genesis, true meaning, and scope of the abstract idea number, which belongs to the greatest inventions of humankind.
In order to approach this concept, we shall first concentrate on the most basic aspect of numbers: their ultimate and original raison d'etre. A first reason for the existence of numbers is that they can be used for counting.
The numbers that can be used for counting are denoted by 1, 2, 3, 4, 5, and so on, and they are called natural numbers. Sometimes, zero is also included in the list of natural numbers, in order to be able to express the absence of things. On the other hand, for the Greek philosophers of antiquity, counting started with two objects, hence one was not regarded as a number. But no matter where we let them begin, the natural numbers are the basis for the understanding and mathematical construction of other types of numbers, such as negative numbers, rational numbers, and even real numbers — the numbers used for measuring quantities. German mathematician Leopold Kronecker (1823–1891) has best described the fundamental role of the natural numbers in his often-quoted dictum: "God made natural numbers; all else is the work of man."
We typically make our first conscious acquaintance with natural numbers when we learn to count. Whether or not one likes mathematics, the ability to count has become second nature to us. As soon as we acquire this ability, we forget about this tedious learning process. Counting, then, appears to be a simple exercise, and we are usually not aware of its inherent complexities. But, in fact, counting is a rather delicate process, and it takes some maturity in abstract reasoning to describe it in more detail.
Can you estimate the number of pebbles in figure 1.2? If you want to know exactly, you will have to count them. By observing ourselves when counting, we find that this task consists of several steps:
1. We start with an arbitrary object in the collection and say "one."
2. We mark this object as "already counted" (at least in our mind, in order to avoid counting it twice).
3. We select a new object (either by pointing with a finger or simply by looking at it).
4. We say the next "number word" (using number words always in the same strict order).
5. We go back to step 2 and repeat until there are no more uncounted objects. The last number word obtained in that way describes the number of objects.
Counting is a process of associating number words with objects in a collection. One of the more difficult tasks involved here is that one has to divide the collection of objects into those that have already been counted and those that still remain to be counted. This is fairly easy if we can put the objects in a row, but it could be impossible if the objects were moving and kept changing places.
When counting nonpermanent objects or events — for example, the chimes of a clock striking the hour — we typically say a number word as the event occurs. When the events are separated by long time intervals, we normally have to create a permanent record of that event — for example, tally marks on a sheet of paper — and finally determine the number of events from the record.
1.4. THE COUNTING PRINCIPLES
The act of counting is governed by five principles. They describe the conditions and prerequisites that make counting possible. We call them the "BOCIA" principles — from the words Bijection, Ordinality, Cardinality, Invariance, and Abstraction. They were proposed by Rochel Gelman (1942-) and C. R. Gallistel (1941–) within the field of cognitive psychology, where they can be used to describe and classify typical counting errors of children. Every child who learns to count masters these principles intuitively, through practice and experience, by trial and error.
In this section, we give a brief description of each of these principles. In the following sections, we elaborate on these principles in more detail and show how they are related to some fundamental mathematical observations. An awareness of the inherent complexities of the counting procedure will also help us to better understand the psychological dimension of the number concept in chapter 2, the intricacies of the historical development described in chapter 3, and the philosophical problems with the foundations of mathematics in chapter 11.
1. Bijection principle (one-to-one principle):
When we count the objects of a collection, we associate these objects with number words. We do this in a one-to-one manner — that is, we pair each object with a unique "counting tag." See figure 1.3.
In practice, counting is often done by pointing a finger at each object while reciting the well-known sequence "one, two, three, ...," and so on. When we do so, we have to be careful about the following two points:
We have to point to each object once and only once. (In that way, no element is left without a counting tag and no element receives two.)
We must use each of the number words only once. (In that way, two different elements of the collection cannot receive the same counting tag.)
This results in a unique one-to-one correspondence, a "pairing" between the objects and a set of counting tags, as illustrated in figure 1.3. In mathematics, a one-to-one correspondence or association is called a bijection, hence the name of this principle.
2. Ordinal principle (stable-order principle):
When we count, we do this in some order. At least in our minds, we first arrange the objects to be counted in a certain (but arbitrary) order, before we apply the counting tags to each object in turn, as shown in figure 1.4. The set of counting tags is also ordered. Typically, the name or label for the first counted object is one, then follows two, three, Land so on. The order of the counting tags must not be changed when counting is repeated or when another collection is counted.
Whenever we count something, we have to use the same set of ordered counting tags. Even when the collection is apparently in disorder, we have to decide upon the order in which the objects receive their counting tags, as symbolized by the arrows in figure 1.4. In that way, the counting tags, which always follow the same order, describe or even create the order of the objects within the collection: One of the objects will be the first — where counting starts — then each object has a unique successor, until we reach the last one — where counting ends. In mathematics, numbers used to label things in a row are called ordinal numbers, hence the name of that principle.
In order to apply this principle, we have to know the sequence of number words by heart. One must be able to recite the number words in their correct order. The commonly used sequence of number words is constructed in a very systematic way, with a strict built-in ordering and without limit. Once the system is understood, one can produce as many number words as needed, and one can always name the next after any given number word in the sequence. Our number words thus provide a useful reservoir of ordered counting tags that is never exhausted, no matter how large the collection we want to count.
3. Cardinal principle:
When we start counting with one, then the last number word reached after having counted all elements of the collection has a very special meaning: It not only is the counting tag of the last counted item but also describes a property of the collection as a whole. The last counting tag is the result of counting. In everyday life we would call this the "number of objects in the collection." The property of a collection that is described by the last number word is sometimes called its numerosity. Mathematicians call it cardinality, hence the name of that principle. The cardinality of the collection of disks in figure 1.4 is 19, or nineteen.
Excerpted from Numbers by Alfred S. Posamentier, Bernd Thaller. Copyright © 2015 Alfred S. Posamentier and Bernd Thaller. Excerpted by permission of Prometheus Books.
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Table of Contents
Chapter 1 Numbers and Counting 11
Chapter 2 Numbers and Psychology 51
Chapter 3 Numbers in History 77
Chapter 4 Discovering Properties of Numbers 105
Chapter 5 Counting for Poets 129
Chapter 6 Number Explorations 161
Chapter 7 Placement of Numbers 183
Chapter 8 Special Numbers 221
Chapter 9 Number Relationships 245
Chapter 10 Numbers and Proportions 287
Chapter 11 Numbers and Philosophy 329
Appendix: Tables 367