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2018 | Buch

Introduction to the Natural Numbers Kapitel lesen Erstes Kapitel lesen

A Readable Introduction to Real Mathematics

verfasst von: Daniel Rosenthal, David Rosenthal, Peter Rosenthal

Verlag: Springer International Publishing

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Designed for an undergraduate course or for independent study, this text presents sophisticated mathematical ideas in an elementary and friendly fashion. The fundamental purpose of this book is to teach mathematical thinking while conveying the beauty and elegance of mathematics. The book contains a large number of exercises of varying difficulty, some of which are designed to help reinforce basic concepts and others of which will challenge virtually all readers. The sole prerequisite for reading this text is high school algebra. Topics covered include: * mathematical induction * modular arithmetic * the Fundamental Theorem of Arithmetic * Fermat's Little Theorem * RSA encryption * the Euclidean algorithm * rational and irrational numbers * complex numbers * cardinality * Euclidean plane geometry * constructibility (including a proof that an angle of 60 degrees cannot be trisected with a straightedge and compass)* infinite series * higher dimensional spaces.

This textbook is suitable for a wide variety of courses and for a broad range of students of mathematics and other subjects. Mathematically inclined senior high school students will also be able to read this book.

From the reviews of the first edition:

“It is carefully written in a precise but readable and engaging style… I thoroughly enjoyed reading this recent addition to the Springer Undergraduate Texts in Mathematics series and commend this clear, well-organised, unfussy text to its target audiences.” (Nick Lord, The Mathematical Gazette, Vol. 100 (547), 2016)

“The book is an introduction to real mathematics and is very readable. … The book is indeed a joy to read, and would be an excellent text for an ‘appreciation of mathematics’ course, among other possibilities.” (G.A. Heuer, Mathematical Reviews, February, 2015)

“Many a benighted book misguidedly addresses the need [to teach mathematical thinking] by framing reasoning, or narrowly, proof, not as pervasive modality but somehow as itself an autonomous mathematical subject. Fortunately, the present book gets it right.... [presenting] well-chosen, basic, conceptual mathematics, suitably accessible after a K-12 education, in a detailed, self-conscious way that emphasizes methodology alongside content and crucially leads to an ultimate clear payoff. … Summing Up: Recommended. Lower-division undergraduates and two-year technical program students; general readers.” (D.V. Feldman, Choice, Vol. 52 (6), February, 2015)

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction to the Natural Numbers
Abstract
The natural numbers are the numbers that we count with; that is, the numbers 1, 2, 3, 4, 5, 6 and so on. We describe the basic properties of the natural numbers. We explain why the product of two negative integers is positive. We prove that there is no largest prime number, and discuss two famous unsolved problems about prime numbers.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 2. Mathematical Induction
Abstract
Mathematical induction is a technique that is useful for proving many theorems. We describe this technique in detail, and give a number of applications of it. We also explain the well-ordering principle, and show that it implies the principle of mathematical induction.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 3. Modular Arithmetic
Abstract
Modular arithmetic is a way of studying divisibility properties of natural numbers. It provides techniques for easily answering questions such as whether 3 plus 2 to the power 3,000,005 is divisible by 7. It has a number of applications, such as proving that a natural number is divisible by 9 if and only if the sum of its digits is divisible by 9. More importantly, it includes theorems such as Fermat’s (see Chapter 5) which are useful in many contexts, including in developing a technique for sending secret messages (see Chapter 6).
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 4. The Fundamental Theorem of Arithmetic
Abstract
The Fundamental Theorem of Arithmetic is the assertion that every natural number greater than 1 can be uniquely (up to the order of its factors) factored into a product of prime numbers. We present a direct proof of this important theorem.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 5. Fermat’s Little Theorem and Wilson’s Theorem
Abstract
Fermat’s Little Theorem states that, for every prime number p, if p does not divide the natural number a, then a to the power p − 1 leaves a remainder of 1 upon division by p. This beautiful theorem has a number of important theoretical and practical applications, one of which is to the technique for sending secret messages that is described in Chapter 6. We present proofs of Fermat’s Little Theorem and also of Wilson’s Theorem, another beautiful formula in modular arithmetic.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 6. Sending and Receiving Secret Messages
Abstract
We describe the RSA method for sending secret messages. This remarkable method allows a person who wishes to receive messages to announce to the world how messages are to be sent and, nonetheless, be the only person who can understand the messages received. We illustrate the method in this chapter, and prove its validity in the following chapter.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 7. The Euclidean Algorithm and Applications
Abstract
We describe the Euclidean Algorithm, which provides a way of expressing the greatest common divisor of two natural numbers as a “linear combination” of the numbers. This algorithm has a number of important applications, including forming the basis for a different proof of the Fundamental Theorem of Arithmetic. It is also an important ingredient in the RSA procedure for sending secret messages. A proof of Euler’s generalization of Fermat’s Little Theorem is also included.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 8. Rational Numbers and Irrational Numbers
Abstract
The “rational numbers” are the fractions; we discuss their basic properties in this chapter. We show that there are “irrational numbers,” including the square root of two. The collection of all rational and all irrational numbers is called the set of real numbers. We develop techniques for determining whether or not given real numbers are rational. We prove, for example, that if the square root of a natural number is rational, then the square root must be an integer. The “Rational Roots Theorem” is also proven.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 9. The Complex Numbers
Abstract
The polynomial x 2 + 1 does not have any roots within the set of real numbers. A new number, called i, is introduced as a root of that polynomial. The complex numbers are all the numbers of the form a + bi where a and b are real numbers. It is a remarkable fact that every polynomial with real coefficients (or even with complex coefficients) that is not a constant function has a root within the set of complex numbers. In this chapter, we develop the basic properties of the complex numbers.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 10. Sizes of Infinite Sets
Abstract
How many natural numbers are there? How many even natural numbers are there? How many rational numbers are there? How many real numbers are there? How many points are there in the plane? How many sets of natural numbers are there? How many different circles are there in the plane? An answer to all these questions could simply be: there are an infinite number of them. But there are more precise answers that can be given; there are, in a sense that is explained and explored in this chapter, an infinite number of different size infinities.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 11. Fundamentals of Euclidean Plane Geometry
Abstract
We describe the fundamentals of Euclidean geometry of the plane. We develop the concepts of congruence and similarity of triangles, and, in particular, prove that corresponding sides of similar triangles are in proportion. We also present a proof of the Pythagorean Theorem.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 12. Constructibility
Abstract
A straightedge is a ruler-like device that has no measurements marked on it; it is used to construct lines through any two given points. We investigate the famous question of which geometric figures can be constructed using only a straightedge and compass. It is easily shown that angles such as those of 60 degrees, 45 degrees, and 30 degrees are all constructible. On the other hand, it is quite difficult to prove that an angle of 20 degrees is not constructible and that, therefore, an angle of 60 degrees cannot be trisected (that is, divided into three equal parts) using only a straightedge and compass. We present a complete proof of this famous and remarkable fact. This is accomplished by studying the properties of “constructible numbers,” which also produces corollaries concerning constructibility of other geometric figures.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 13. An Introduction to Infinite Series
Abstract
An infinite series is an expression of the form a 1 + a 2 + a 3 + ⋯, where each a i is a real number. We discuss the question of when an infinite series has a “sum” in a precise sense that we will explain. A series that does have such a sum is said to “converge.” The basic properties of convergence of infinite series are rigorously proven. In particular, it is shown that every “infinite decimal” converges.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Chapter 14. Some Higher Dimensional Spaces
Abstract
Four-dimensional Euclidean space is defined, as is n-dimensional Euclidean space for every natural number n. A few basic properties of n-dimensional spaces are explored. Some infinite-dimensional spaces are also defined.
Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Backmatter
Metadaten
Titel
A Readable Introduction to Real Mathematics
verfasst von
Daniel Rosenthal
David Rosenthal
Peter Rosenthal
Copyright-Jahr
2018
Electronic ISBN
978-3-030-00632-7
Print ISBN
978-3-030-00631-0
DOI
https://doi.org/10.1007/978-3-030-00632-7