Introduction

Why proof?

For most people, Mathematics is about using mathematical facts to solve practical problems. Users of Mathematics are rarely concerned about why the methods work and care only that they do work. To too many people, Mathematics is a collection of arcane techniques known only to a select few with "math brains." It is troublesome when those arcane techniques that confuse people are differentiation, integration, or matrix manipulation. It is downright frightening when the confusing problems are adding fractions or computing a restaurant tip. The worst way to view Mathematics is as a long collection of hard-to-remember techniques for solving specific problems. A much better way is to think of Mathematics as an organization of basic ideas that can solve all sorts of problems as needed. When you understand what Mathematics actually means, you can use that understanding to produce your own problem-solving techniques. The key to understanding any piece of Mathematics (or anything else for that matter) is to understand why it works the way it does.

Since the ancient Greeks first studied Mathematics in a careful way, the subject has been built on deductive proof. Mathematical results are accepted as facts only after they have been logically proved from a few basic facts. Once mathematical facts are established, they can be used to solve practical and theoretical mathematical problems. Mathematicians have two reasons for proving a mathematical statement rigorously: first, to be sure that the result is true, and second, to understand when and how it works.

Following the ancient Greek process, mathematicians want a proof for everything - whether it is on the cutting edge of mathematics and science or it is an apparently obvious fact about grade school arithmetic. The idea is to understand why a mathematical result is true and to move on to what you know because it is true. Most of the Mathematics we see in school is about the "moving on" variety. Once school children understand the connection between combining small groups of objects and adding numbers, they can move on to the arithmetic algorithm of adding larger numbers. Thus,

is just the theoretical way to combining 278 objects and 394 objects and counting the combination. Once school children understand the connection between groups of groups and multiplication, they can learn the algorithm for multiplication. Then

is just the theoretical way of counting 35 rows of 257 objects.

At the very beginning, every child is given some simple justifications for the validity of these algorithms. The strong belief among math educators and education researchers is that students who understand those justifications best are the students that will learn the algorithms best. Granted in the long run, it is a child's ability with the algorithm that is considered most important. In time, greater facility with the algorithms supplants a person's need for the logic behind those algorithms. But the complete understanding of the operation behind the algorithm is always essential for its proper use in odd situations.

There is a popular notion that the logic behind the techniques of Mathematics can be ignored once the procedures of Mathematics are learned. This notion seems to work well for the basic arithmetic of whole numbers. There is a lot of evidence, however, that this is why so many people stumble over problems involving fractions. Too many people "move on" to memorizing the algorithms of fractional arithmetic before they understand the meaning of that arithmetic or why the things they are memorizing work. It is hard to memorize anything and harder still to hold that memory without knowing the context of what you are learning. "To add fractions, find a common denominator." "To divide fractions, invert and multiply." Everyone knows this, but how many can correctly add to or divide 21 by ?

As perplexing as fractions are to the general population, decimal numbers are even worse. Thanks to calculators, everyone knows where the dots tell us a better calculator would give more digits. Everyone also seems to know that where here the dots mean that the 3s go on forever, or at least they would if it were actually possible for written digits to go on forever. Most people understand decimal numbers well enough that they can move on to using them very well and very effectively without error. But even the most highly trained person can be tripped up by an unexpected decimal question that involves infinitely many decimals. In the next section, we consider some surprisingly confusing questions about simple numbers.

Before we get to these confusing examples, let us set up a plan for curing any resulting mathematical confusion. Early school mathematical training generally concentrates on the problem-solving problems using Mathematics. Some theoretical or intuitive explanations of the ideas and techniques are given, but the level of logical rigor in these justifications varies greatly depending on the topic under discussion. If we are interested in a more advanced education in Mathematics, we must revisit these past justifications of the mathematical ideas we now hold so dear. The time must come when we understand and appreciate a rigorous justification of every mathematical result we will use. This turns out to be a rather difficult step to make. We will work on it in stages.

Why analysis?

Our main objective in this study is to develop a precise description of the real numbers for use as a foundation for the ideas and methods of calculus. There are two ingredients in this development: algebra and analysis. "Algebra" generally refers to the arithmetic of the numbers: addition, subtraction, multiplication, and division. The ways in which these operations interact form the "algebraic structure" of the number systems that we will consider. "Analysis" refers to the study of the distinctions between exact numbers and their approximations. It is simply a fact that certain real numbers cannot be expressed exactly using only finitely many whole numbers. Analysis allows us to say precise things about real numbers that cannot be precisely described with a finite expression.

Problems in analysis typically occur when we use numbers to measure things. Given an isosceles right triangle, two squares drawn with sides the length of the short sides of the triangle will have a combined area equal to a square with a side whose length is the same as the hypotenuse. If we measure the sides as units, the hypotenuse will measure units. Thus, to measure the hypotenuse, there must be a number we write as , which when multiplied by itself is 2. A good calculator will approximate as 1.41421. A better calculator will approximate it as 1.41421356237, and a sensational one as

But, as the Greeks discovered, the only way to write an exact representation of the number is by saying that it is a number that when squared is 2 and then to make up a symbol for it, such as .

Since our goal is to develop a rigorous description of the real numbers, we must be able to use it to work with numbers we can describe exactly but cannot calculate exactly. We will use algebra and analysis to allow us to do arithmetic with numbers such as this. Suppose, for example, that we need a number so that . Once we are sure that it exists, we can assign it a symbol. For now, let us say . As it turns out, is like . We can approximate it as accurately as we like, but it may be that the only way to write it exactly is . We can use algebra to do some exact calculations with . For example, , but it is a matter of opinion whether is a better name for or if it is the other way around.

For a more famous example, suppose that we need a number that is the ratio between the circumference of a circle and the diameter of the circle. First, we need to know that it exists, but we can thank the ancient Greeks for that. We can assign it a symbol . We can approximate it as accurately as we like, but the only way to write it exactly is . The situation is even worse than or ; mathematicians have proved that there is no polynomial of any degree with rational coefficients so that . This means that the only possible way to write exactly is .

The way most people know is "3.14159.. where the digits continue forever without a pattern." So the question is, "Does anyone know exactly?" If there is no pattern to the digits and they go on forever, then no one can know them all. These digits may look random after a while, but because we believe is a real number, we believe that all the digits are exactly described even if they may never be all known. Most educated people have a working knowledge of the real numbers, but mostly because they have a reasonable understanding of decimal approximation. Thus, they are not bothered by questions about exact values of .

On the other hand, consider . With a calculator, almost anyone can find that , and many will guess that this is simply an approximation of the exact value. But scratch the surface of this general understanding of real numbers and you discover a problem: what have we approximated? That is, "What is the meaning of ?" Now , but is not a rational fraction. So this is of little help describing what the number means. The only reason most people have to believe that it has a meaning at all is that their calculator will calculate it.

Next consider a problem with infinite decimal arithmetic that most people avoid by using approximations. Consider the numbers: and , where the ellipsis () means that the pattern of digits repeats forever....