Take a grand tour of the best of modern math, its most elegant solutions, most clever discoveries, most mind-bending propositions, and most impressive personalities. Writing with a light touch while showing the real mathematics, author Peter Schumer introduces you to the history of mathematics, number theory, combinatorics, geometry, graph theory, and "recreational mathematics." Requiring only high school math and a healthy curiosity, Mathematical Journeys helps you explore all those aspects of math that mathematicians themselves find most delightful. You’ll discover brilliant, sometimes quirky and humorous tidbits like how to compute the digits of pi, the Josephus problem, mathematical amusements such as Nim and Wythoff’s game, pizza slicing, and clever twists on rolling dice.
Take a grand tour of the best of modern math, its most elegant solutions, most clever discoveries, most mind-bending propositions, and most impressive personalities. Writing with a light touch while showing the real mathematics, author Peter Schumer introduces you to the history of mathematics, number theory, combinatorics, geometry, graph theory, and "recreational mathematics." Requiring only high school math and a healthy curiosity, Mathematical Journeys helps you explore all those aspects of math that mathematicians themselves find most delightful. You’ll discover brilliant, sometimes quirky and humorous tidbits like how to compute the digits of pi, the Josephus problem, mathematical amusements such as Nim and Wythoff’s game, pizza slicing, and clever twists on rolling dice.
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Overview
Take a grand tour of the best of modern math, its most elegant solutions, most clever discoveries, most mind-bending propositions, and most impressive personalities. Writing with a light touch while showing the real mathematics, author Peter Schumer introduces you to the history of mathematics, number theory, combinatorics, geometry, graph theory, and "recreational mathematics." Requiring only high school math and a healthy curiosity, Mathematical Journeys helps you explore all those aspects of math that mathematicians themselves find most delightful. You’ll discover brilliant, sometimes quirky and humorous tidbits like how to compute the digits of pi, the Josephus problem, mathematical amusements such as Nim and Wythoff’s game, pizza slicing, and clever twists on rolling dice.
Product Details
| ISBN-13: | 9780471220664 |
|---|---|
| Publisher: | Wiley |
| Publication date: | 02/11/2004 |
| Series: | Wiley-Interscience Publication |
| Pages: | 216 |
| Product dimensions: | 6.15(w) x 9.25(h) x 0.60(d) |
About the Author
Read an Excerpt
Mathematical Journeys
By Peter D. Schumer
John Wiley & Sons
Copyright © 2004 John Wiley & Sons, Inc.All right reserved.
ISBN: 0-471-22066-3
Chapter One
Let's Get Cooking: A Variety of Mathematical Ingredients
We begin our mathematical journey by introducing (or reminding you about) some basic objects of mathematics. These include prime numbers, triangular numbers, and geometric squares. We will also present a couple of neat proofs and discuss a very useful tool for carefully establishing results dealing with the natural numbers, namely mathematical induction. This chapter lays the foundation for many of the latter chapters and hence might be a bit more elementary. On the other hand, don't worry about being bored. Mathematics asks (and often answers) interesting fundamental questions. We'll get to some really compelling things right away!
Let's avoid a philosophical discourse on the construction of (or the a priori existence of) the natural numbers. I assume that the natural numbers exist (at least in our humanly defined mathematical world). They are the counting numbers 1, 2, 3, ... and they continue forever in the sense that if N is a natural number, then so is N+1. We denote the set of all natural numbers by N. The set of natural numbers can be subdivided. The number 1 is special in that it is the unique multiplicative identity. This is just a fancy way to say that anything times 1is itself.
Next come the primes. You no doubt recall that these are natural numbers that are only divisible by 1 and themselves. For example, 2 is prime. So are 17 and 41 since they cannot be factored further. So is 10,123,457,689, the smallest prime containing all ten decimal digits. But 6 = 2 · 3 and 91 = 7 · 13 are not prime. Numbers like 6 and 91 belong to the third category, namely the composites. The primes are the multiplicative building blocks for all the natural numbers. The composite numbers are those that require at least two building blocks.
There are endless questions to be asked about the natural numbers, especially the primes. The Fundamental Theorem of Arithmetic states that all natural numbers have a unique factorization as a product of primes (discounting order of the factors). Since 2,002 = 2 · 7 · 11 · 13, no other product of primes could equal 2,002. But how many primes are there? Are there any consisting of more than a million digits? How many are of the form [N.sup.2]+1, that is, exceed a square number by 1? How spread apart are they? Some of these questions were answered by the ancient Greek geometers more than 2000 years ago. Others still have not been answered fully. Let's get right to it and answer the first question, which is perhaps the most fundamental of all.
Theorem 1.1: There are infinitely many primes.
How does one go about demonstrating the veracity of the statement above? Certainly any list of primes, no matter how long, will not suffice to show that there are infinitely many primes. Perhaps showing that the primes satisfy some pattern would help. For example, if for each prime it was true that its double plus 1 was prime, then we'd have a process for producing as many primes as we wished. Let's give it a try by carrying out this process on the first prime: the sequence begins 2, 5, 11, 23, 47, ... -look promising? So far so good, but the next number in our sequence is 95, a composite number. This isn't going to be that easy. But perhaps a similar idea might work.
A theorem is a mathematical statement whose veracity has been rigorously demonstrated to the satisfaction of mathematicians. At least, that's my definition of a theorem. If it sounds a bit tenuous, it no doubt is. On very rare occasions, a result is erroneously considered to be a theorem for many years before a logical weakness or flaw is discovered in its purported proof. But such instances are extremely unusual. Although mathematics is a human endeavor, mathematicians treat their work very stringently and the standard for proof is high. So from here on you are a mathematician who is about to read the following proof presented by Euclid in The Elements (Book IX, Proposition 20), written about 300 B.C.E. (Before the Common Era). See if it convinces you.
Proof of Theorem 1.1: There are some primes, for example, 2 is prime. So now consider a nonempty set of primes, S = {[P.sub.1], [P.sub.2],..., [P.sub.n]}. The number N = P1·P2· ... ·Pn +1 is not divisible by P1 because division by [P.sub.1] leaves a remainder of 1. Similarly, N is not divisible by any of the primes in the set S. So N is either another prime not among those listed in S or N is composite, but made up of a product of primes none of which are in the set S. So the set S is not a complete list of all the primes. Since S was an arbitrary set of primes, no finite list of primes can be complete. Therefore, the number of primes is infinite.
The result is stunning and the proof is elegant. We have answered a simple but deep question about the prime numbers and hence about the natural numbers themselves. The natural numbers have an infinite number of multiplicative building blocks. We won't run dry! And notice that we've also answered our second question. Are there any primes consisting of more than a million digits? Yes, there are. In fact, there are infinitely many of them (since only finitely many primes could possibly have a million or fewer digits.) Naming such a prime may be quite a different matter, but later we will tell some of that story as well.
The primes begin 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. There are various sized gaps between successive primes. For example, the gaps from our list above are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4. Since two is the only even prime number, all gaps after the initial gap of 1 are even numbers. How big can these numbers get? Does 2 appear infinitely often? Do all even numbers appear at some point? The next theorem answers our first question.
Theorem 1.2: There are arbitrarily large gaps between successive primes.
What we require are arbitrarily long strings of consecutive composite numbers. Below we construct such a string. Before we begin, recall the definition of factorial. If n is a natural number, n! (read "n factorial") is the product of all the natural numbers up to and including n. For example, 5! = 1 · 2 · 3 · 4 · 5 = 120.
Proof of Theorem 1.2: Given a natural number N [greater than or equal to] 2, consider the sequence of N consecutive numbers (N + 1)! + 2, (N + 1)! + 3, ... , (N + 1)! + N + 1. Note that 2 divides (N + 1)! since 2 is one of the factors in the product that defines (N + 1)!. So 2 divides (N +1)! + 2 and hence (N + 1)! + 2 is composite. Similarly, 3 divides (N + 1)! + 3 and so (N + 1)! + 3 is composite as well. Analogously, all the N consecutive numbers from (N +1)! + 2 to (N + 1)! + N + 1 are composite. Since the number N is arbitrary, there are strings of consecutive composite numbers of any given length. Hence there are arbitrarily large gaps between successive primes.
Ponder for a moment what Theorems 1.1 and Theorems 1.2 say. There is an unlimited supply of primes, but gaps between them can be as large as you like. Also note what the theorems do not claim. Theorem 1.2 does not say that there is an infinite string of consecutive composites. For if there were, then there would have to be a last prime, contradicting Theorem 1.1. Nor does Theorem 1.2 imply that our construction gives the first instance of a particularly sized gap. However, both theorems do tell us something significant about prime numbers.
On the one hand, there are lots and lots of primes. On the other hand, they're not so dense in the natural numbers that there aren't long patches without them. Compare this with the odd numbers. There are infinitely many odd numbers, but there's never a gap larger than two between them. How about the set of squares-1, 4, 9, 16, etc.? Certainly there are infinitely many of them. And the gaps are successive odd numbers that get arbitrarily large. So in some sense the primes are a bit more like the squares perhaps. Deeper theorems do distinguish between the density of squares and the density of primes among the natural numbers, but for now at least we have another concrete model where results analogous to Theorems 1.1 and 1.2 apply. (For those of you who are curious, there are more primes than squares in some strict analytical sense.)
What about our other questions? It has not been proven that every possible even gap appears somewhere along the endless list of primes, but number theorists tend to believe that it is so. In fact, A. de Polignac conjectured that every even number appears infinitely often as a gap between consecutive primes (1849). And although there are lots of examples of the smallest gap of two, even a proof that there are infinitely many such twin primes remains elusive. However, many gigantic examples have been discovered. Currently the record is the twin pair 33218925 · [2.sup.169,690] ± 1, found by Daniel Papp (2002). Each number is a 51,090-digit prime.
Now let's consider the triangular numbers, for each n defined as the sum of the first n natural numbers. Denote them by [t.sub.n]. For example, [t.sub.4] = 1+2+3+4 = 10. Geometrically, we can think of [t.sub.n] as the number of bowling pins with n rows laid out in the usual triangular array. Imagine some giant extraterrestrial species bowling with [t.sub.10] bowling pins. The 46-55 split is a real killer! A most natural question is, "What is [t.sub.n]?" Is there a nice formula which gives us [t.sub.n] for all n? Of course there is. Why would I bring it up? The formula is simple and provides us with the classic example in which to introduce the technique of proof by induction.
Theorem 1.3: For all n [greater than or equal to] 1, tn = n(n + 1)/2.
The method of mathematical induction is a simple one. It consists of verifying the assertion for the initial value (n = 1 in this case) and then showing that if the theorem is true for the case n, then it must be true for case n + 1. In this way, it's much like an infinite set of dominoes all lined up in a row. If we knock over the first one and carefully set up all the rest so that each one knocks over the succeeding domino, then eventually any particular domino will fall. This method of proof was widely used by the brilliant French philosopher and mathematician Blaise Pascal (1623-1662) and so this method of proof is often attributed to him. However, recently mathematical historians have discovered that the same technique was utilized by Levi ben Gerson (1288-1344), the Jewish medieval Biblical scholar, astronomer, philosopher, and mathematician. No doubt the clever idea of mathematical induction has been independently discovered by many scholars. Its elegance and simplicity belie its power and broad applicability.
Proof of Theorem 1.3: For n = 1 , [t.sub.1] = 1 = 1 (2)/2. So Theorem 1 is correct in this case. If the formula is correct for [t.sub.n], then let's show that it is correct for [t.sub.n+1] as well.
1 + 2 + ... + n + (n + 1) = (1 + 2 + ... + n) + (n + 1) = n(n + 1)/2 + (n + 1) by the inductive hypothesis = [n(n + 1) + 2 (n + 1)]/2 = (n + 1)(n + 2)/2,
which is the formula for [t.sub.n+1]. By mathematical induction, the formula holds for all n [greater than or equal to] 1.
Make sure to double-check the little bit of algebra in the proof above and be certain that you are comfortable with the logic behind mathematical induction. If so, you are well on your way toward thinking like a real mathematician. Now if you need to know the sum of the first 1,000 natural numbers, the solution is easy: [t.sub.1,000] = 1,000(1,001)/2 = 500,500. Watch out for that bowling ball!
Let's try our hand on a somewhat more intricate example. Define the Fibonacci numbers as the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. Each entry in the sequence from the third entry on is the sum of the two preceding Fibonacci numbers. Hence [F.sub.1] = 1, [F.sub.2] = 1, and [F.sub.n] = [F.sub.n-1] + [F.sub.n-2] for n [greater than or equal to] 3. The Fibonacci sequence is a good example of a sequence defined recursively. These numbers are named after the Italian mathematician Leonardo of Pisa (ca. 1175-1250), who promoted the use of the Hindu-Arabic numeral system. (He was also known as Fibonacci, son of Bonaccio.) His book, Liber Abaci (1202), contained 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 from the second month on becomes productive?" The sequence thus produces the number of pairs we've listed above.
There seem to be an endless number of interesting arithmetic formulae based on this simple sequence. For example, let's pick any Fibonacci number, square it, subtract from that the product of its two neighboring Fibonacci numbers (above and below), and see what results. If we choose [F.sub.7] = 13, then we get [([F.sub.7]).sub.2] - ([F.sub.6]·[F.sub.8]) = [13.sup.2] - 8·21 = 1. If we choose [F.sub.8] = 21, then a similar computation gives us [21.sup.2] - 13 · 34 = -1. Experiment with some other starting values. Really-try some other examples. Once you've done a few calculations, then read on. (I'm trying to help get you in the mindset for mathematical discovery. This is the essence of mathematics.)
No doubt you have noticed that the answer is always 1 or -1. In fact, if we pick an odd-indexed Fibonacci number we get +1 and if we choose an even-indexed Fibonacci number we obtain -1. This leads us to the conjecture that [F.sup.2.sub.n] - [F.sub.n-1] · [F.sub.n+1] = [(-1).sup.n-1]]. In fact, this is a theorem that we now prove in two different ways, each making use of mathematical induction.
Theorem 1.4 (J.D. Cassini, 1680): For all n [greater than or equal to] 2, [F.sup.2.sub.n] - [F.sub.n-1] · [F.sub.n+1] = [(-1).sub.n-1].
First Proof of Theorem 1.4 (Direct): Our initial value is n = 2 : [F.sup.2.sub.2] - [F.sub.1] · [F.sub.3] = [1.sup.2] - 1 · 2 = [(-1).sup.1]. So the proposition is true in this case. Now let's assume that it holds for some unspecified value of n, that is, [F.sup.2.sub.n] - [F.sub.n-1] · [F.sub.n+1] = [(-1).sup.n-1]. We will show that the proposition necessarily follows in the next case, namely for n + 1.
[F.sup.2.sub.n+1] = [F.sub.n+1] ([F.sub.n] + [F.sub.n-1]) by the definition of [F.sub.n+1]
= [F.sub.n] + 1[F.sub.n] + [F.sub.n] - 1[F.sub.n+1] + ([F.sup.2.sub.n] - [F.sup.2.sub.n
Continues...
Excerpted from Mathematical Journeys by Peter D. Schumer Copyright © 2004 by John Wiley & Sons, Inc.. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Preface.Acknowledgments.
1. Let’s Get Cooking: A Variety of Mathematical Ingredients.
2. The Green Chicken Contest.
3. The Josephus Problem: Please Choose Me Last.
4. Nim and Wythoff’s Game: Or How to Get Others to Pay Your Bar Bill.
5. Mersenne Primes, Perfect Numbers, and Amicable Pairs.
6. The Harmonic Series . . . and Less.
7. Fermat Primes, the Chinese Remainder Theorem, and Lattice Points.
8. Tic-Tac-Toe, Magic Squares, and Latin Squares.
9. Mathematical Variations on Rolling Dice.
10. Pizza Slicing, Map Coloring, Pointillism, and Jack-in-the-Box.
11. Episodes in the Calculation of Pi.
12. A Sextet of Scintillating Problems.
13. Primality Testing Below a Quadrillion.
14. Erdös Number Zero.
15. Choosing Stamps to Mail a Letter, Let Me Count the Ways.
16. Pascal Potpourri.
Appendix: Comments and Solutions to Problems Worth Considering.
Bibliography.
Index.
What People are Saying About This
"This is such a fount of fascinating knowledge and problems that all professors and teachers who want to motivate and challenge their talented students should consult it." (Choice, June 2004, Vol. 41 No. 10)