Journey into Mathematics

An Introduction to Proofs

Joseph J. Rotman

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486151687
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Students learn how to read and write proofs by actually reading and writing them, asserts author Joseph J. Rotman, adding that merely reading about mathematics is no substitute for doing mathematics. In addition to teaching how to interpret and construct proofs, Professor Rotman's introductory text imparts other valuable mathematical tools and illustrates the intrinsic beauty and interest of mathematics.
Journey into Mathematics offers a coherent story, with intriguing historical and etymological asides. The three-part treatment begins with the mechanics of writing proofs, including some very elementary mathematics--induction, binomial coefficients, and polygonal areas--that allow students to focus on the proofs without the distraction of absorbing unfamiliar ideas at the same time. Once they have acquired some geometric experience with the simpler classical notion of limit, they proceed to considerations of the area and circumference of circles. The text concludes with examinations of complex numbers and their application, via De Moivre's theorem, to real numbers.

Chapter 1

Setting Out

INDUCTION

So, naturalists observe, a flea
Hath smaller fleas that on him prey;
And these have smaller still to bite ’em;
And so proceed ad infinitum.

Jonathan Swift

There are many styles of proof, and mathematical induction is one of them. We begin by saying what mathematical induction is not. In the natural sciences, inductive reasoning is based on the principle that a frequently observed phenomenon will always occur. Thus, one says that the sun will rise tomorrow morning because, from the dawn of time, the sun has risen every morning. This is not a legitimate kind of proof in mathematics, for even though a phenomenon occurs frequently, it may not occur always.

Inductive reasoning is valuable in mathematics, because seeing patterns often helps in guessing what may be true. On the other hand, inductive reasoning is not adequate for proving theorems. Before we see examples, let us make sure we agree on the meaning of some standard terms.

Definition. An integer is one of 0, 1, – 1, 2, – 2, 3, – 3, .⋯

Definition. An integer p ≥ 2 is called a prime number¹ if its only positive divisors are 1 and p. An integer m ≥ 2 which is not prime is called composite.

A positive integer m is composite if it has a factorization m = ab, where a < m and b < m are positive integers; the inequalities are present to eliminate the uninteresting factorization m = m x 1. Notice that a ≥ 2: since a is a positive integer, the only other option is a = 1, which implies b = m (contradicting b < m); similarly, b ≥ 2.

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41. That this sequence never ends is proved in Exercise 2.10.

Consider the statement:

f(n) = n² – n + 41 is a prime number for every n ≥ 1

(this is really a whole family of statements, one for each positive integer n). As we evaluate f(n) for n = 1, 2, 3, 4, ⋯ , 40, we obtain the following values:

It is tedious but not difficult (see Exercise 1.7) to prove that every one of these numbers is prime. Can we now conclude that all the numbers of the form f(n) are prime? For example, is the next number f(41) = 1681 prime? The answer is no: f(41) = 412 – 41 + 41 = 41², which obviously factors, and hence f(41) is not prime.

Here is a more spectacular example (which I first saw in an article by W. Sierpinski). A perfect square is an integer of the form a² for some positi...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• TO THE READER
• TO THE INSTRUCTOR
• Chapter 1 - Setting Out
• Chapter 2 - Things Pythagorean
• Chapter 3 - Circles and π
• Chapter 4 - Polynomials
• EPILOGUE
• BIBLIOGRAPHY
• Glossary of Logic
• Index