Geometry-Moise-Edwin-E-Downs-Floyd-L-Chapter01.pdf

CHAPTER Common Sense and Exact Reasoning Objectives... Introduce the organizational plan of definitions, postulates, and theorems. Develop concepts of point, line, and plane. Point out the difference between...

CHAPTER Common Sense and Exact Reasoning Objectives... Introduce the organizational plan of definitions, postulates, and theorems. Develop concepts of point, line, and plane. Point out the difference between observable facts and mathematical reasoning. Realize that intuition is not always trustworthy. 1-1 AN ORGANIZED LOGICAL DEVELOPMENT OF GEOMETRY If you stop to think, you will realize that you already know many facts about geometry. For example, you know the names of many geometric figures such as triangle, square, rectangle, circle, and cube. Also, you probably know how to find the perimeters or circumferences, areas and volumes of some of them. You also know many facts about geometric figures. Many of these facts seem so obvious that it might never occur to you to put them into words. For example, two straight lines cannot cross each other in more than one point. Some other facts you know, like the formula for the circumference of a circle (C = 2irr), are not obvious at all. Nevertheless, all of these facts, whether familiar or not, obvious or not, are important to the effective use of geometry in the learning of mathematics and in the application of mathematics to the world around us. l 2 1-1 AN ORGANIZED LOGICAL DEVELOPMENT OF GEOMETRY In this book we are going to help you organize the facts of geometry in an orderly way, with a few simple statements at the beginning leading up to more complicated ones. We shall see that all the facts of geometry can be derived from a relatively few simple statements. In this organiza¬ tional structure we shall have three kinds of statements: definitions, postulates, and theorems. We shall state definitions for geometric ideas as clearly and exactly as we can, and we shall establish the facts of geometry by giving logical proofs. The statements that we prove will be called theorems. Our simplest and most fundamental statements will be given without proof. These will be called postulates. At first thought it may seem best to define every geometric term that we use, and to prove every statement that we make. But it is fairly easy to see that this cannot be done. Most of the time, when we introduce a new term, we define it, using terms that have already been defined. But the first definition that we give cannot work this way, because in this case there aren’t any terms that have already been defined. This means that we have to introduce at least one geometric term without defining it. Actually we shall use three of the simplest and most fundamental geometric ideas as undefined terms. These undefined terms are point, line, and plane. Though we will not state definitions of point, line, and plane, we can describe in common language the ideas we have in mind. If you make a dot on a piece of paper, with a pencil, you will get a reasonably good picture of a point. The sharper your pencil is, the better your picture will be. The picture will always be only approximate, because the dot will always cover some area, whereas a point covers no area at all. But if you think of smaller and smaller dots, made by sharper and sharper pencils, you will get a good idea of what we mean in geometry by the term point. When we use the term line, we shall always have in mind the idea of a straight line. A straight line extends infinitely far in both directions. Usually we shall indicate this in our illustrations by putting arrowheads at the ends of the part of the line that we draw, like this. If we do not put in the arrowheads, we mean to indicate only a definite part of a line, having end points, called a line segment. A thin, tightly stretched string is a physical example of a line segment. AN ORGANIZED LOGICAL DEVELOPMENT OF GEOMETRY 1-1 3 When we use the term plane, we have in mind a perfectly flat surface, extending infinitely far in all directions. If you will imagine a sheet of clear window glass that is so huge that you cannot see its edges, you will have a fairly good idea of a geometric plane. A plane has no thickness. Similarly, a line has no width, and a point has no area. Since points, lines, and planes are geometric ideas, and a picture or a sketch is, at best, a visual representation, the idea of a plane is especially hard to show on paper. The figure at the left below suggests the “infinite extent” of a plane, but is a rather impractical illustration for constant use. Usually we depict a plane as in the figure at the right below. Consider next the question of the theorems. Usually, when we prove a theorem, we show that it follows logically from other theorems that we have already proved. But the first theorem that we prove cannot possibly work this way, because there aren’t any theorems that we have already proved. This means that we have to accept at least one statement without proof. Actually, we accept several. These unproved statements are the postulates. Postulates, of course, are not made up at random. Postulates describe fundamental properties of space. When we start proving theorems, the only information that we shall claim to have about points, lines, and planes will be the information given in the postulates. Algebra and the language of sets will be used throughout this course. We shall think of them, however, as things that we are working with, rather than things that we are working on. Some of our postulates will involve real numbers, and we shall use algebra in proofs. In fact, geometry and algebra are very closely connected, and both are easier to learn if we establish the connection at the outset. Finally, we give a couple of warnings. First, there are limits to what logic can do for us. Logic enables us to check our guesses, but it isn’t much help in making guesses in the first 4 1-1 AN ORGANIZED LOGICAL DEVELOPMENT OF GEOMETRY place. In our study of mathematics, you are always going to need the guidance of the feelings that you have in your bones. Second, the theorems in the next couple of chapters often may seem just as obvious as the postulates. Later, we shall get to some theorems that we shall need to prove, to make sure that they are true. But to do this, we first need to build up some mathematical machinery, and we need to see how proofs work by starting with easy theorems. This will be done in Chapters 2, 3, and 4. The ideas in these early chapters will take on more meaning for you when you use them in solving problems, and when you see how they are used in proving theorems later in the book. i Problem Set 1-1 1. You are already familiar with many facts about geometry. Make use of things you have learned before to answer the following. (a) What is the name of each figure? (b) If the sides of a triangle are 4 cm, 7 cm, and 6 cm, what is the perime¬ ter of the triangle? (c) It the lengths of two sides of a rectangle are 7 and 10, what is its perimeter? What is its area? AN ORGANIZED LOGICAL DEVELOPMENT OF GEOMETRY 1-1 5 (d) The formula for the area of a circle with radius r is 7rr2. If a circle has a diameter of 6 cm, what is its area? (Use 3.14 for 7r.) (e) What is the formula for the circumference of a circle with radius r? 2. A student who wanted to know the meaning of the word “dimension” consulted a dictionary. The dictionary listed as a synonym the word “measurement,” whose definition the student in turn looked up. He made the following chart. Size Extent^ or / \ / Length—longest dimension Dimension—measurement—/ or \ Dimension Size