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More About This Textbook
Overview
Anyone can appreciate the beauty, depth, and vitality of mathematics with the help of this highly readable text, specially developed from a college course designed to appeal to students in a variety of fields. Readers with little mathematical background are exposed to a broad range of subjects chosen from number theory, topology, set theory, geometry, algebra, and analysis.
Starting with a survey of questions on weight, the text discusses the primes, the fundamental theorem of arithmetic, rationals and irrationals, tiling, tiling and electricity, probability, infinite sets, and many other topics. Each subject illustrates a significant idea and lends itself easily to experiments and problems. Useful appendices offer an overview of the basic ideas of arithmetic, the rudiments of algebra, suggestions on teaching mathematics, and much more, including answers and comments for selected exercises.
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Mathematics
The ManMade Universe
By Sherman K. Stein
Dover Publications, Inc.
Copyright © 1999 Sherman K. SteinAll rights reserved.
ISBN: 9780486138992
CHAPTER 1
Questions on Weighing
This chapter will raise some important questions about numbers. While they may seem to be mere recreational puzzles that could be understood and investigated by anyone who can do simple arithmetic, in fact they concern a fundamental property in number theory. Not until Chapter 3 will the "why" behind the answers be considered. The goal of this chapter is to offer an opportunity to experiment with an openended problem and to see how a single mathematical idea can appear in a variety of disguises.
The questions, which concern weighing, will be introduced by a few examples. Say that we have a twopan scale of the type seen in chemistry labs and statues of "Justice":
[ILLUSTRATION OMITTED]
Furthermore, we have an unlimited supply of 5 and 7ounce measures. Now, supposing that potatoes weigh only whole numbers of ounces, rather than any amount as they actually do, let us ask which potatoes we would be able to weigh with our balance and our two types of measures.
For instance, using only the 5ounce measures, we can weigh 5, 10, 15, 20, 25, 30, 35, ... ounces. Or using only the 7ounce measures, we can weigh 7, 14, 21, 28, 35, ... ounces. Moreover, we could place one of each type together on a pan:
[ILLUSTRATION OMITTED]
Thus we can weigh 12 ounces, 12 = 5 + 7. Or we could put one of each type of weight alone on a pan:
[ILLUSTRATION OMITTED]
In this way we can weigh 2 ounces, since a potato of this weight, together with the 5ounce measure, balances the 7ounce measure.
Can we weigh a 3ounce potato? Yes, by placing two 5ounce measures on one pan and a 7ounce measure with the potato:
[ILLUSTRATION OMITTED]
The balancing records the equation 3 + 7 = 2 · 5.
Can we weigh a 4ounce potato? Yes. For instance, by placing two 5ounce measures with the potato and two 7ounce measures on the other pan:
[ILLUSTRATION OMITTED]
The corresponding equation is
4 + 2 · 5=2 · 7.
Or we could place three 7ounce measures with the potato, which then balance five 5ounce measures. The equation in this case is 4 + 3 · 7 = 5 · 5.
Can we weigh a 1ounce potato? Even this can be done, as the reader may prefer to work out for himself before reading the next sentence. Two 7ounce measures and the potato on one pan balance three 5ounce measures on the other pan. The reader may wish to pause and devise still other ways of measuring this 1ounce potato with 5 and 7ounce measures.
Once we know that we can weigh a 1ounce potato, then we know that we can weigh any number of ounces. For instance, we can weigh a 6ounce potato as follows. First recall that two 7ounce measures and a 1ounce potato balance three 5ounce measures:
[ILLUSTRATION OMITTED]
Repeating this arrangement of measures sixfold weighs a sixounce potato. That is, from the relation 1 + 2 · 7 = 3 · 5 5 it follows that 6(1 + 2 · 7) = 6(3 · 5) or
6 + 12 · 7 = 18 · 5.
Of course, this may not be the simplest method for weighing a 6ounce potato. Indeed, 6 + 3 · 5 5 = 3 · 7, so we could have managed by placing three 5ounce measures with the potato and three 7ounce measures on the other pan. But the reasoning at least assures us that if we can weigh a 1ounce potato with a supply of two types of measures, then we can weigh any whole number of ounces with those measures.
So much for the combination 5 and 7. Suppose we turn to another combination, 8 and 21. Even if we have only 8ounce and 21ounce measures available, we can measure a 1ounce potato, since
1 + 3 · 21 = 8 · 8.
Eight 8ounce measures on one pan will balance the potato and three 21ounce measures on the other pan.
The reader may now suspect that perhaps any pair of measures can weigh a 1ounce potato. But this is not so. If, for example, we have only 6ounce and 8ounce measures, then we could never hope to measure a 1ounce potato, or, for that matter, any odd number of ounces. (The reader should pause to think about why this is so.)
We are now in a position to ask some basic questions. Suppose we have at our disposal an unlimited supply of measures of two types. How can we decide whether we can weigh a 1ounce potato with them? For instance, can we use 539ounce and 1619ounce measures to weigh a 1ounce potato? More generally we can ask: What potatoes can we weigh with an unlimited supply of two given types of measures? Keep in mind that all potatoes and measures weigh a whole number of ounces (0, 1, 2, 3, 4, ...). The whole numbers will usually be referred to as the natural numbers.
The questions really concern numbers, not potatoes. Let us gradually translate the second question into the language of numbers: Denote the weights of the two measures by A and B ounces respectively. In our first combination we had A = 5 and B = 7. The weight of the potato will be denoted by W ounces.
There are various methods of weighing the potato. One consists in putting several Aounce measures on the pan with the potato and several B ounce measures on the other pan. How many of each we use will depend on W, A, B, and our arithmetic. Say that we use X of the A ounce measures and Y of the B ounce measures:
[ILLUSTRATION OMITTED]
The corresponding equation is
W + XA = YB,
an equation that asserts merely that the scale in the figure balances. (We omit the multiplication sign between letters.)
For A = 5 and B = 7, let us see what X and Y are for various W. When W = 1, for instance, we have 1 + 4 · 5 = 3 · 7. Here X = 4 and Y = 3. Also 1 + 11 · 5 = 8 · 7, so that X = 11 and Y = 8 also perform the weighing when W=1.
Another method of weighing consists in placing only Bounce measures with the potato and only Aounce measures on the other pan. For A = 5, B = 7, the equation 1 + 2 · 7 = 3 · 5 illustrates this. We have W + XB = YA, where W = 1, X = 2, Y = 3.
A third method consists in placing the potato on one pan and the measuring weights on the other pan. For example, if W = 12 we have 12 = 5 + 7; if W = 27 we have 27 = 4 · 5 + 1 · 7. This method corresponds to an equation of the type W = XA + YB.
No other practical method exists, for there would be no point in placing measures of equal weight on both pans, since they could be removed without affecting the balance. Thus we need to consider only three types of equations:
W + XA = YB, W + XB = YA, W = XA + YB.
In all of these, X and Y are to be natural numbers, possibly including zero. The question about potatoes now becomes one about natural numbers: Let A and B be natural numbers. For which natural numbers W can we find natural numbers X and Y such that at least one of these equations holds:
W + XA = YB, W + XB = YA, W = XA + YB?
The potatoes are gone, but we are left with three equations to deal with. We can simplify matters further (reducing the three equations to one) by making use of the negative numbers, 1, 2, 3, 4, 5, ..., which lie to the left of 0 on the number line. (See Appendix A.) The numbers
..., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ...
are called integers. (Note that any natural number is an integer.)
With the aid of the negative numbers, we will reduce the first two equations, W + XA = YB and W + XB = YA, to the third type, W = XA + YB. Consider first, W + XA = YB. This can be rewritten as W = (X) A + YB, which is of the third type. (For instance, we may rewrite 1 + 4 · 5 = 3 · 7 as 1= (4)5 + 3 · 7.) Similarly, W + XB = YA, where X and Y are natural numbers, can be reduced to the third form by writing it as W = YA + (X)B.
The question about potatoes now reduces to this: Let A and B be natural numbers. Which natural numbers W can be expressed in the form W = MA + NB for certain integers M and N?
Let us see which values of M and N describe our earlier work for the combination A = 5 and B = 7. The following table records some cases we considered, beginning with 5, 10, ...; 7, 14,....
The row for "3", for instance, tells us
[ILLUSTRATION OMITTED]
Just as a potato may be weighed in more than one way, so M and N are not necessarily unique. For the case W = 6 there were the two weighings
6 + 12 · 7 = 18 · 5 (6 = 18 · 5 + (12)7)
and
3 · 5 + 6 = 3 · 7 7 (6 = (3)5 + 3 · 7).
A little arithmetic produces still more representations of 6; for instance,
6 = 4 · 5 + (2)7; 6 = 11 · 5 + (7)7; 6= (10)5 + 8 · 7.
Our first question, concerning the possibility of weighing a 1ounce potato, now reads: For which pairs of natural numbers A and B can we find integers M and N such that 1 = MA + NB?
Take the combination A = 24, B = 73. Since 3 · 24 = 72, we have 1 = (3)24 + 1 · 73. Thus for the combination 24 and 73 M and N exist. But what about the combination 24 and 75? Can they measure 1? What about 21 and 34? What about 89 and 233? And, we may wonder: If a pair of measures can't measure a 1ounce potato, what is the smallest positive weight they can measure?
There is another way of looking at these questions. Recall the representation 1 = 8 · 8 + (3)21. The basis of this is that 8  8 differs from 3 · 21 by just 1. We may have found this representation by listing 1 · 8 = 8, 2 · 8 = 16, 3 · 8 = 24, 4  8 = 32, 5  8 = 40, 6 · 8 = 48, 7 · 8 = 56, 8 · 8 = 64, ... and 1 · 21 = 21, 2 · 21 = 42, 3 ? 21 = 63, ... until we found an entry in one list that differed by 1 from an entry in the other list.
The numbers of the form "integer times 8" are called the multiples of 8. The multiples of 21, or of any integer, are defined similarly. This illustration shows some of the multiples of 8 and some of the multiples of 21:
[ILLUSTRATION OMITTED]
Thus the question "Can A and B measure 1?" can be phrased in terms of multiples: "Are there multiples of A and of B that differ by exactly 1?" Since the multiples of a number are regularly spaced on the number line, the question can also be interpreted geometrically: If we have an unmarked ruler A inches long, and another B inches long, can we measure a distance of one inch?
The reader has probably guessed the answers to these general questions. Chapter 3 will return to this topic. There the equation W = MA + NB will turn out to be the basic tool for establishing fundamental properties of the natural numbers.
Exercises
1. Is every natural number also an integer? Is 0 a positive integer? Is 0 a negative integer? Is every positive integer also a natural number?
2. Drawing pictures of the twopan scale, show three different methods of weighing a 1ounce potato with 4 and 7ounce measures.
3. Drawing pictures of the twopan scale, show three different methods of weighing a 1ounce potato with 8 and 13ounce measures.
4. What do you think is the smallest weight of a potato that can be measured with (a) 8 and 12ounce measures, (b) 8 and 11ounce measures, (c) 9 and 12ounce measures? Why do you think so?
5. A 4ounce potato and five 7ounce measures balance three 13ounce measures. Express this in the form (a) W + XA = YB,. where all letters stand for natural numbers, (b) W = MA + NB, where M and N are integers.
6. Are there integers M and N such that (a) 1 = M · 7 + N · 10, (b) 1 = M · 8 8 + N · 10, (c) 1= M · 13 + N · 22, (d) 1= M · 6 + N · 21? If your answer is "yes," give values of M and N; if "no," explain why.
7. a. Show how to weigh a 1ounce potato with 11ounce and 18ounce measures.
b. Use (a) to find a method of weighing a 3ounce potato with 11ounce and 18ounce measures.
8. Draw the balanced loading of the twopan scale that corresponds to each of these equations: (a) 3 + 6 · 7 = 5 · 9, (b) 5 = 3 · 11 + (4)7, (c) 23 = 2 · 4 + 3 · 5.
9. Can a 1ounce potato be weighed with 5 and 8ounce measures in such a way that (a) the potato is with the 5's, and the 8's are in the other pan; (b) the potato is with the 8's, and the 5's are in the other pan?
10. Can a collection of 5ounce measures on one pan ever balance a collection of 8ounce measures on the other pan?
11. The products 5 · 10 and 3 17 differ by 1. Draw four different balanced loadings of the scales for weighing a 1ounce potato that reflect this arithmetic fact. (The measures may be 5, 10, 3, or 17.)
12. Find integers M and N, if you think they exist, such that
a. I = M · 4 + N · 9,
b. 1 = M · 9 + N · 11,
c. 1 = M · 10 + N · 15,
d. 1 = M · 23 + N · 25.
13. a. Draw on a number line several multiples of 7 and several multiples of 12.
b. Do the multiples of 7 meet the multiples of 12 anywhere other than at 0?
c. What is the closest that a multiple of 7 can be to a multiple of 12 without actually coinciding with it?
14. Consider 5 and 12ounce measures. Show that it is possible without using more than four 12ounce measures to weigh (a) a 1ounce potato, (b) a 2ounce potato, (c) a 3ounce potato, (d) a 4ounce potato, (e) a 5ounce potato, (f) a 6ounce potato, (g) a 7ounce potato. Will this be true for heavier potatoes?
15. a. Find a multiple of 9 that differs from a multiple of 11 by 1.
b. Show them on the number line.
16. List (a) four multiples of 3, (b) five multiples of 2, (c) six multiples of 7.
17. a. Show that 1 can be weighed with 8's and 11's.
b. Give some small values of W that can be weighed with 16's and 22's.
c. Give some small values of W that can be weighed with 24's and 33's.
18. Find four pairs of integers M and N such that (a) 12 = M · 2 + N 5, (b) 5 = M · 2 + N · 5, (c) 13 = M · 2 + N · 5.
19. What potatoes can be weighed with 4 and 7ounce measures, if we never use more than three 7ounce measures?
20. Consider weighings by 3 and 7ounce measures. Draw a picture of balanced scales recording the equality 3 · 7 = 7 · 3. (a) Show that if a potato can be weighed by placing it on the pan with 3ounce measures it can also be weighed by placing it on the pan with 7ounce measures. (b) Is this true more generally than for just 3's and 7's?
21. Find integers M and N such that 1 = M · 5 + N · 9 and (a) M is positive, N negative; (b) M is negative, N positive. (c) Draw the corresponding loadings of the scales.
22. Find integers M and N such that 13 = M · 4 + N · 5 and (a) M and N are positive; (b) M is positive, N negative; (c) M is negative, N positive. (d) Draw the corresponding loadings of the scales.
23. a. What amounts of postage can you make with 5cent stamps and 8cent stamps? 2.
b. What type of weighing problem would be equivalent to the postage problem raised in (a)?
c. What requirement does the postage problem impose on M and N if the postage, W, equals M · 5 + N · 8?
24. What amounts of postage can be made with 4cent and 7cent stamps?
25. What amounts of postage can be made with 10cent and 13cent stamps?
26. What amounts of postage can be made if you are allowed to use at most one 1cent stamp, at most one 2cent stamp, at most one 4cent stamp, at most one 8cent stamp, at most one 16cent stamp, and so on (each denomination being a product of 2's)?
(Continues...)
Table of Contents
1. Questions on weighing
Weighing with a twopan balance and two measures—Problems raised—Their algebraic phrasing
2. The primes
The Greek primemanufacturing machine—Gaps between primes—Average gap and 1/1 + 1/2 + 1/3 + . . . + 1/N—Twin primes
3. The Fundamental Theorem of Arithmetic
Special natural numbers—Every special number is prime—"Unique factorization" and "every prime is special" compared—Euclidean algorithm—Every prime number is special—The concealed theorem
4. Rationals and Irrationals
The Pythagorean Theorem—he square root of 2—Natural numbers whose square root is irrational—Rational numbers and repeating decimals
5. Tiling
The rationals and tiling a rectangle with equal squares—Tiles of various shapes—use of algebra—Filling a box with cubes
6. Tiling and electricity
Current—The role of the rationals—Applications to tiling—Isomorphic structures
7. The highway inspector and the salesman
A problem in topology—Routes passing once over each section of highway—Routes passing once through each town
8. Memory Wheels
A problem raised by an ancient word—Overlapping ntuplets—Solution—History and applications
9. The Representation of numbers
Representing natural numbers—The decimal system (base ten)—Base two—Base three—Representing numbers between 0 and 1—Arithmetic in base three—The Egyptian system—The decimal system and the metric system
10. Congruence
Two integers congruent modulo a natural number—Relation to earlier chapters—Congruence and remainders—Properties of congruence—Casting out nines—Theorems for later use
11. Strange algebras
Miniature algebras—Tables satisfying rules—Commutative and idempotent tables—Associativity and parentheses—Groups
12. Orthogonal tables
Problem of the 36 officers—Some experiments—A conjecture generalized—Its fate—Tournaments—Application to magic squares
13. Chance
Probability—Dice—The multiplication rule—The addition rule—The subtraction rule—Roulette—Expectation—Odds—Baseball—Risk in making decisions
14. The fifteen puzzle
The fifteen puzzle—A problem in switching cords—Even and odd arrangements—Explanation of the Fifteen puzzle—Clockwise and counterclockwise
15. Map coloring
The twocolor theorem—Two threecolor theorems—The fivecolor theorem—The fourcolor conjecture
16. Types of numbers
Equations—Roots—Arithmetic of polynomials—Algebraic and transcendental numbers—Root r and factor X—r—Complex numbers—Complex numbers applied to alternating current—The limits of number systems
17. Construction by straightedge and compass
Bisection of line segmentBisection of angleTrisection of line segment—Trisection of 90° angle—Construction of regular pentagon—Impossibility of constructing regular 9gon and trisecting 60° angle
18. Infinite sets
A conversation from the year 1638—Sets and onetoone correspondence—Contrast of the finite with the infinite—Three letters of Cantor—Cantor's Theorem—Existence of transcendentals
19. A general view
The branches of mathematics—Topology and set theory as geometries—The four "shadow" geometries—Combinatorics—Algebra—Analysis—Probability—Types of proof—Cohen's theorem—Truth and proof—Gödel's theorem
Appendix A. Review of arithmetic
A quick tour of the basic ideas of arithmetic
Appendix B. Writing mathematics
Some words of advice and caution
Appendix C. The rudiments of algebra
A review of algebra, which is reduced to eleven rules
Appendix D. Teaching mathematics
Suggestions to prospective and practicing teachers
Appendix E. The geometric and harmonic series
Their properties—Applications of geometric series to probability
Appendix F. Space of any dimension
Definition of space of any dimension
Appendix G. Update
Answers and comments for selected exercises
Index