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There's something about a language of symbols and equations that most people find intimidating. Mathematician Colin Pask gently introduces you to the main ideas of mathematics and painlessly demonstrates how they are expressed in terms of symbols. He teaches the reader not only why symbols are used, and how and why equations are constructed, but also exactly what is achieved by doing that. Through simple yet intriguing examples in number theory, Pask generates confidence in thinking mathematically and reveals the pleasure of seeing how mathematical patterns evolve and are explored.
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MATH FOR THE FRIGHTENEDFacing Scary Symbols and Everything Else That Freaks You Out about Mathematics
By COLIN PASK
Prometheus BooksCopyright © 2011 Colin Pask
All right reserved.
Chapter OneWHERE TO BEGIN?
You might expect me to begin by defining what I mean by mathematics. The difficulty with that is the lack of a simple comprehensive definition. One approach is to say "mathematics is the science of patterns." There is much to be said in favor of that definition and I will give illustrations as we go along. However, I think the best approach is for me to show you some actual mathematics in action. At the very end, we can try to find some general conclusions.
Some readers might have liked me to begin with applications of mathematics, perhaps to motivate them to learn more about the subject. At this stage, I could tell you about some applications, but I could not show you in detail just how the mathematics is being used in them. I want you to really appreciate the vital role played by mathematics in science and other areas. That means understanding why the mathematical ideas and formalism I am about to describe are essential for science. So please be patient while I develop the appropriate mathematical tools. Then I can do the job properly with the applications following in later chapters.
I need to choose a vehicle for introducing you to the way concepts and ideas are developed in mathematics. Those of you who immediately associate lots of detailed calculations, symbols and diagrams with the word mathematics may be surprised by the prominence of concepts and ideas. But in some ways the totality of words plus the symbols and diagrams of mathematics are the equivalent of words in poetry. A poet uses words to express thoughts, stories, emotions, and so on. In fact, many links have been made between mathematics and poetry. G. H. Hardy, one of the great mathematicians of the first half of the twentieth century, famously wrote:
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. A painter makes patterns with shapes and colors, a poet with words.
My problem is that while you are comfortable with words, you may be less familiar with mathematical symbols and diagrams (and even frightened of them!). That means I must introduce those things to you, along with the concepts and ideas.
The vehicle I have chosen involves the squares of numbers and relationships between them. (Remember, the square of a number is just that number multiplied by itself. To be concise, we write 3 × 3 = 32 for three squared.) Why should we begin with the squares of numbers? A well-known and ancient example motivates this choice.
A FAMILIAR EXAMPLE: PYTHAGOREAN TRIPLES
The mathematical result most familiar to everyone is probably Pythagoras's theorem. It gives a relationship between the squares of the lengths of the sides of a right-angled triangle. Many people will immediately recite: the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. A simple example is given by the right-angled triangle with sides of lengths 3, 4, and 5:
32 + 42 = 52 9 + 16 = 25
Other possible side lengths, which are all whole numbers, or integers, are:
8, 15, 17 119, 120, 169 4601, 4800, 6649 12709, 13500, 18541.
In each case, adding the squares of the two smaller numbers gives the square of the larger one. Sets of three integers that could be the side lengths of a right-angled triangle are known as Pythagorean triples.
Valid Pythagorean triples were known in Babylonian, Chinese, Greek, and Indian civilizations dating back thousands of years. Of the four examples just given, the first was known in ancient China, and the others come from a Babylonian record dated around 1600 BCE. (See box 2.) Some of these data may have been used in surveying and building, and it has been suggested that the construction of ritual altars accounts for their widespread appearance in the ancient world.
Notice that while these number sets can be thought of in terms of right-angled triangles, it is doubtful that anyone is really interested in a triangle with sides of lengths 4601, 4800, and 6649. It is the relation between the three numbers that is of interest. Trying out examples of Pythagoras's theorem has led us to play with the squares of numbers, but now it is the way the squares of numbers can be related that fascinates us and the geometric interpretation fades out of sight.
Remember, all the numbers in a Pythagorean triple are whole numbers, or integers, and that is what makes them so neat and appealing. Of course, it is easy to choose two integers, 3 and 7, say, and then say that the square root of 32 plus 72 can be the length of the hypotenuse side of a right-angled triangle. But that side length turns out to be 7.615773 ..., and few of you probably find the set of numbers 3, 7, 7.615773 ... particularly interesting or appealing.
As a society becomes more complex, it needs to develop methods for keeping track of trade, organizing workers and materials, planning and regulating necessary food supplies, setting taxes, sorting out inheritance problems, and so on. These all suggest the development of a mathematical approach, and many of the early records we have refer to such organizational matters and the types of problems they create.
In the Lansing Papyrus, an ancient Egyptian document written around 1200 BCE, a scribe reports a teacher encouraging students of mathematics:
See, I am instructing you so that you may become one who is trusted by the King, so that you may open treasuries and granaries, so that you may take delivery from the corn-bearing ship at the entrance to the granary, so that on feast days you may measure out the God's offerings.
Our first steps in mathematics involve things like counting, combining (adding), and sharing (dividing). An application might require the sum of two numbers,
119 + 120 = 239
Perhaps this sum is used to give the total number of workers to be fed, but we can also view it independently as a relation between the numbers 119, 120, and 239. However, the relation
1192 + 1202 = 1692
between the numbers 119, 120, and 169 seems to be of a quite different type. The result is not at all obvious, hardly the sort of thing one might naturally stumble upon. Triples involving large numbers were unlikely to have been used as builders' aids, for example. This suggests a deeper level of mathematical thinking and an appreciation of mathematical ideas and ways to exploit them. With such results there seems to be a step away from mathematics as just a tool for dealing with some of life's practical problems. It is the result about the numbers themselves that seems fascinating and worthy of exploration.
For that reason, the uncovering of evidence of the knowledge of Pythagorean triples may be used as a measure of the level of mathematical advancement in early civilizations. How many different Pythagorean triples were known? Is there evidence of a systematic approach to finding and tabulating them? These questions raise some significant mathematical points, which I will discuss in a later chapter.
We conclude that the Pythagorean triples may be given simply as interesting results about numbers and quite separate from the geometrical properties of right-angled triangles, which were the likely origin for their initial investigation.
Perhaps the most commonly discussed measure of mathematical progress in ancient civilizations concerns the calculation of the area and circumference of circles and ultimately the value of p, the ratio of a circle's circumference and its diameter. That calculation, too, eventually became a problem in number theory and algebra.
An astute reader may be puzzled at this point. Surely Pythagoras lived a long time after the Babylonians who gave those famous examples of Pythagorean triples? That is so, and it emphasizes the fact that what we call Pythagoras's theorem was probably known in several ancient civilizations. Importantly, what the Greek mathematicians did was to relate Pythagoras's theorem to certain more fundamental facts, as I will explain in a later chapter.
You may also be wondering how on earth those Babylonians came across triples involving such large numbers. Not just any old three numbers can be chosen to form a Pythagorean triple. Finding triples by just playing with a few numbers is not easy, even today with a powerful electronic calculator. There is still some debate, but it appears that the ancient Babylonians used clever reasoning and a mathematical formula to produce valid triples. While the working may not be too hard, it is still tricky and I will postpone showing it to you until chapter 8.
CHOOSING A STARTING POINT
Pythagorean triples have shown us how numbers may be related in complex ways through their squares. They give us a fine example of purely mathematical results that have long been of interest. But to make things easy, I am going to use an even simpler property of squares of numbers as our first example for in-depth study.
As a vehicle for introducing you to mathematical ideas and methods, I am going to use a simple result concerning squares of numbers. It was discussed by Leonardo of Pisa, better known today as Fibonacci. Ah, yes, I hear many of you say, Fibonacci numbers and the rabbit population growth problem. It is the same man, but I am not talking about rabbits. Fibonacci is an important figure in the history of mathematics more generally, not just for the rabbit problem.
Mathematics flourished in many ancient civilizations, but around two thousand years ago Greeks such as Thales, Euclid, Appolonius, Archimedes, and Diophantus began to develop more systematic descriptions of the subject and its methods and logical foundations. As the Greek civilizations declined, so did the mathematical activity, but the results and further developments were kept alive in various regions, particularly in the Arab world. Eventually an interest in mathematics began to emerge in Western Europe as old results and texts were rediscovered or translated. Recognition of this non-Eurocentric history of mathematics is mostly quite recent. (See box 3.)
Fibonacci was one of the first people to understand the importance of the mathematics that had been preserved and added to in non-Western centers of learning. It has been said that he was the first significant mathematician of this era. Fibonacci's work stood almost unrivaled for three hundred years.
Fibonacci was born around 1180 in the commercial city state of Pisa. His famous and important book, Liber Abaci (The Book of Calculation), was first published in 1202. In it he introduced to the West our modern system of numerals and showed how to do an enormous number of calculations of different kinds, many of great interest in commerce (as well as the famous rabbit problem!).
Fibonacci also wrote Liber Quadratorum (The Book of Squares). This book, first published in 1225, is a systematic treatment of the mathematics of numbers and their squares. Liber Quadratorum sets out mathematical results with no thought of applications. It takes the form of twenty-four propositions comprising basic results and ideas about squares that Fibonacci explains, justifies, and exploits as he develops more and more new results.
It is proposition two from Liber Quadratorum that I have chosen to use in the next chapter to begin our journey into mathematics.
SUMMARY AND IMPORTANT MESSAGES
(These sections are at the end of each chapter. They review the key points made in the chapter and the general messages that readers should have picked up.)
Ancient mathematicians were aware of special relationships among certain sets of numbers, and their results have come down to us today in various ways. Results about the squares of numbers still fascinate us, and the extent of some of the examples makes us question how they were first derived. Fibonacci wrote a whole book dealing with the squares of numbers and their properties, and it suggests a suitable starting point for our journey into mathematics.
BOX 2. BABYLONIAN MATHEMATICS: PLIMPTON 322
The ancient Babylonians recorded many things on clay tablets. One surviving example is called Plimpton 322, named after George Plimpton, who collected it in 1923. It is thought to have been made around 1700 BCE in the ancient city of Larsa in Iraq. Plimpton 322 is 12.7 cm by 8.8 cm in size, and one edge shows that part has broken off.
Plimpton 322 is of great mathematical significance. However, that is not immediately obvious for three reasons. First, it is written using Babylonian symbols rather than the 1, 2, 3, 4 ... we are used to today. Second, the Babylonians used base 60—they counted in 60s rather than in 10s as we do. Third, it needs a little mathematical detective work to decide just what is recorded. Eleanor Robson has given a plausible recent version.
The tablet has a heading and fifteen lines recording Pythagorean triples, with a few obvious transcribing errors. Call the sides of the equivalent right-angled triangle a, b, and c, and take c as the hypotenuse and b as the longest of the two other sides, then Plimpton 322 leads to the table below. The last column gives a clue to the organization of the table.
BOX 3. THE HISTORICAL SPREAD OF MATHEMATICS
A trivially simple description of the history of mathematics has often been used in the past. With a little recognition of ancient civilizations, the start of "real mathematics" was placed in the ancient Greek civilization with prominent figures such as Pythagoras and Euclid. Then there was a dormant period (the Dark Ages), and finally mathematics began to flourish in Europe as the Renaissance began. The real story is much more complex, as is the shown in the figure. In particular, we can see how the earliest mathematics was preserved and added to by scholars in the Arab world when Baghdad was a great center for learning.
Excerpted from MATH FOR THE FRIGHTENED by COLIN PASK Copyright © 2011 by Colin Pask. Excerpted by permission of Prometheus Books. 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
Preface—Please Read before We Start....................11
The Overall Plan....................15
1. Where to Begin?....................17
2. Fibonacci's Proposition Two....................25
3. Equations: Information and Interpretation....................35
4. The Proof....................45
5. Thinking Around the Proof....................57
6. But Why Is It True? Seeing It Another Way....................69
7. Building Up the Mathematics....................83
8. Back to Pythagorean Triples....................101
9. Logical Steps and Linear Equations....................115
10. Parameters, Quadratic Equations, and Beyond....................127
11. Any Number....................145
13. First Applications....................181
14. Mathematics and the Invisible World....................199
15. CAT Scans....................219
16. Social Planning and Mathematical Surprises....................227
17. Geometry and Euclid....................253
18. An Alternative Approach: Geometry Meets Algebra....................271
20. Ways of Thinking and How We Do Mathematics....................309
21. Final Thoughts....................335
22. Answers for Your Examples....................349
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