The Enjoyment of Math (Princeton Science Library) - Softcover

Hans Rademacher; Otto Toeplitz

Softcover

ISBN 10: 0691023514 ISBN 13: 9780691023519

Verlag: Princeton University Press, 1966

Inhaltsangabe

What is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through.

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Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting.

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The Enjoyment of Math

By Hans Rademacher, Otto Toeplitz, Herbert Zuckerman

PRINCETON UNIVERSITY PRESS

Copyright © 1957 Princeton University Press
All rights reserved.

ISBN: 978-0-691-02351-9

Contents

Preface, v, 
Introduction, 5, 
1. The Sequence of Prime Numbers, 9, 
2. Traversing Nets of Curves, 13, 
3. Some Maximum Problems, 17, 
4. Incommensurable Segments and Irrational Numbers, 22, 
5. A Minimum Property of the Pedal Triangle, 27, 
6. A Second Proof of the Same Minimum Property, 30, 
7. The Theory of Sets, 34, 
8. Some Combinatorial Problems, 43, 
9. On Waring's Problem, 52, 
10. On Closed Self-Intersecting Curves, 61, 
11. Is the Factorization of a Number into Prime Factors Unique?, 66, 
12. The Four-Color Problem, 73, 
13. The Regular Polyhedrons, 82, 
14. Pythagorean Numbers and Fermat's Theorem, 88, 
15. The Theorem of the Arithmetic and Geometric Means, 95, 
16. The Spanning Circle of a Finite Set of Points, 103, 
17. Approximating Irrational Numbers by Means of Rational Numbers, 111, 
18. Producing Rectilinear Motion by Means of Linkages, 119, 
19. Perfect Numbers, 129, 
20. Euler's Proof of the Infinitude of the Prime Numbers, 135, 
21. Fundamental Principles of Maximum Problems, 139, 
22. The Figure of Greatest Area with a Given Perimeter, 142, 
23. Periodic Decimal Fractions, 147, 
24. A Characteristic Property of the Circle, 160, 
25. Ciuves of Constant Breadth, 163, 
26. The Indispensability of the Compass for the Constructions of Elementary Geometry, 177, 
27. A Property of the Number 30, 187, 
28. An Improved Inequality, 192, 
Notes and Remarks, 197, 

CHAPTER 1

The Sequence of Prime Numbers

6 is equal to 2 times 3, but 7 cannot be similarly written as a product of factors. Therefore 7 is called a prime or prime number. A prime is a positive whole number which cannot be written as the product of two smaller factors. 5 and 3 are primes but 4 and 12 are not since we have 4 = 2 x 2 and 12 = 3 x 4. Numbers which can be factored like 4 and 12 are called composite. The number 1 is not composite but, because it behaves so differently from other numbers, it is not usually considered a prime either; consequently 2 is the first prime, and the first few primes are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ··· .

A glance reveals that this sequence does not follow any simple law and, in fact, the structure of the sequence of primes turns out to be extremely complicated.

A number can be factored a step at a time until it is reduced to a product of primes. Thus 6 = 2 · 3 is immediately expressed as a product of two primes, while 30 = 5 · 6 and 6 = 2 · 3 gives 30 = 2 · 3 · 5, a product of three primes. Similarly, 24 has four prime factors (24 = 3 · 8 = 3 · 2 · 4 = 3 · 2 · 2 · 2), of which three happen to be the same prime, 2. In the case of a prime such as 5 one can only write 5 = 5, a product of a single prime. By means of this step-by-step factoring, any positive whole number except 1 can be written as a product of primes. Because of this, the prime numbers can be thought of as the building blocks of the sequence of all positive whole numbers.

In the ninth book of Euclid's Elements the question of whether the sequence of primes eventually ends is raised and answered. It is shown that the sequence has no end, that after each prime yet another and larger prime can be found.

Euclid's proof is very ingenious yet quite simple. The numbers 3, 6, 9, 12, 15, 18, ··· are all multiples of 3. No other numbers can be divided evenly by 3. The next larger numbers 4, 7, 10, 13, 16, 19, ···, which are multiples of 3 increased by 1, are certainly not divisible by 3; for example, 19 = 6 · 3 + 1, 22 = 7 · 3 + 1, etc. In the same way the multiples of 5 increased by 1 are not divisible by 5 (21 = 4 · 5 + 1, etc.). The same thing is true for 7, for 11, and so on.

Now Euclid writes down the numbers

2 · 3 + 1 = 7

2 · 3 · 5 + 1 = 31

2 · 3 · 5 · 7 + 1 = 211

2 · 3 · 5 · 7 · 11 + 1 = 2311

2 · 3 · 5 · 7 · 11 · 13 + 1 = 30,031, etc.

The first two primes, the first three primes, and so forth, are multiplied together and each product is increased by 1. None of these numbers is divisible by any of the primes used to form it. Since 31 is a multiple of 2 increased by 1, it is not divisible by 2. It is a multiple of 3 increased by 1 and hence is not divisible by 3. It is a multiple of 5 increased by 1, hence not divisible by 5. 31 happens to be a prime and it is certainly larger than 5. 211 and 2311 are also new primes, but 30031 is not a prime. However, 30031 is not divisible by 2, 3, 5, 7, 11, or 13, and hence its prime factors are greater than 13. As a matter of fact, a little figuring shows that 30031 = 59 · 509, and these prime factors are greater than 13.

The same argument may be applied as far as one wants to go. Let p be any prime and form the product of all primes from 2 to p; increase this product by 1 and write

2 · 3 · 5 · 7 · 11 ··· p + 1 = N.

None of the primes 2, 3, 5, ··· p divides N, so either N is a prime (certainly much greater than p) or all the prime factors of N are different from 2, 3, 5, ··· p, and hence greater than p. In either case, a new prime greater than p has been found. No matter how large p is there is always another larger prime.

This part of Euclid is quite remarkable, and it would be hard to name its most admirable feature. The problem itself is only of theoretical interest. It can be proposed, for its own sake, only by a person who has a certain inner feeling for mathematical thought. This feeling for mathematics and appreciation of the beauty of mathematics was very evident in the ancient Greeks, and they have handed it down to later civilizations. Also, this problem is one that most people would completely overlook. Even when it is brought to our attention it appears to be trivial and superfluous, and its real difficulties are not immediately apparent. Finally, we must admire the ingenious and simple way in which Euclid proves the theorem. The most natural way to try to prove the theorem is not Euclid's. It would be more natural to try to find the next prime number following any given prime. This has been attempted but has always ended in failure because of the extreme irregularity of the formation of the primes.

Euclid's proof circumvents the lack of a law of formation for the sequence of primes by looking for some prime beyond instead of for the next prime after p. For example, his proof gives 2311, not 13, as a prime past 11, and it gives 59 as one past 13. Frequently there are a great many primes between the one considered and the one given by the proof. This is not a sign of the weakness of the proof, but rather it is evidence of the ingenuity of the Greeks in that they did not try to do more than was required.

As an illustration of the complexity of the sequence of primes, we shall show that there are large gaps in the series. We shall show, for example, that we can find 1000 consecutive numbers, all of which are composite. The method is closely related to that of...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 1966
• Sprache: Englisch
• ISBN 10: 0691023514
• ISBN 13: 9780691023519
• Einband: Taschenbuch
• Anzahl der Seiten: 205
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: gpsr@libri.de; gpsr@libri.de; ; Europaallee 1; Bad Hersfeld; 36244; Deutschland