Skip to main content

Über dieses Buch

The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu­ ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources. Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate.



Chapter 1. Heuristics

Strategy or tactics in problem-solving is called heuristics. In this chapter we will be concerned with the heuristics of solving mathematical problems. Those who have thought about heuristics have described a number of basic ideas that are typically useful. The five classics on problem-solving by George Polya are masterpieces devoted entirely to the practical study of heuristics in mathematics. Among the ideas developed in these books, we shall focus on the following:
Search for a pattern.
Draw a figure.
Formulate an equivalent problem.
Modify the problem.
Choose effective notation.
Exploit symmetry.
Divide into cases.
Work backward.
Argue by contradiction.
Pursue parity.
Consider extreme cases.
Loren C. Larson

Chapter 2. Two Important Principles: Induction and Pigeonhole

Mathematical propositions come in two forms: universal propositions which state that something is true for all values of x in some specified set, and existential propositions which state that something is true for some value of x in some specified set. The former type are expressible in the form “For all x (in a set S), P(x)”; the latter type are expressible in the form “There exists an x (in the set S) such that P(x),” where P(x) is a statement about x. In this chapter we will consider two important techniques for dealing with these two kinds of statements: (i) the principle of mathematical induction, for universal propositions, and (ii) the pigeonhole principle, for existential propositions.
Loren C. Larson

Chapter 3. Arithmetic

In this chapter we consider problem-solving methods that are important in solving arithmetic problems. Perhaps the most basic technique is based on the fundamental theorem of arithmetic, which states that every integer can be written uniquely as a product of primes. The theoretical background necessary for the proof of this key theorem requires a discussion of the notion of divisibility. Therefore, we will begin the chapter by considering problems about greatest common divisors and least common multiples. Important to this understanding are the division algorithm and the Euclidean algorithm.
Loren C. Larson

Chapter 4. Algebra

Algebra is one of the oldest branches of mathematics, and it continues to be one of the most active areas of mathematical research. The subject is still rich in new ideas, and it shows no signs of soon becoming exhausted or barren.
Loren C. Larson

Chapter 5. Summation of Series

In this chapter we turn our attention to some of the most basic summation formulas. The list is quite short (e.g., the binomial theorem, arithmetic- and geometric-series formulas, elementary power-series formulas) but we shall see that a few standard techniques (e.g. telescoping, differentiation, integration) make them extremely versatile and powerful.
Loren C. Larson

Chapter 6. Intermediate Real Analysis

In this chapter we will review, by way of problems, the hierarchy of definitions and results concerning continuous, differentiable, and integrable functions. We will build on the reader’s understanding of limits to review the most important definitions (continuity in Section 6.1, differentiability in Section 6.3, and integrability in Section 6.8). We will also call attention to the most important properties of these classes of functions. It is useful to know, for example, that if a problem involves a continuous function, then we might be able to apply the intermediate-value theorem or the extreme-value theorem; or again, if the problem involves a differentiable function, we might expect to apply the mean-value theorem. Examples of these applications are included in this chapter, as well as applications of L’Hôpital’s rule and the fundamental theorem of calculus.
Loren C. Larson

Chapter 7. Inequalities

Inequalities are useful in virtually all areas of mathematics, and inequality problems are among the most beautiful. Among all the possible inequalities that we might consider, we shall concentrate on just two: the arithmetic-mean-geometric-mean inequality in Section 7.2 and the Cauchy—Schwarz inequality in Section 7.3. In addition, we shall consider various algebraic and geometric techniques in Section 7.1, and analytic techniques in Sections 7.4 and 7.5. In the final section, Section 7.6, we shall see how inequalities can be used to evaluate limits.
Loren C. Larson

Chapter 8. Geometry

In this chapter we will look at some of the most common techniques for solving problems in Euclidean geometry. In addition to the classical synthetic methods of Euclid, we will see how algebra, trigonometry, analysis, vector algebra, and complex numbers can be useful tools in the study of geometry.
Loren C. Larson


Weitere Informationen