Revision v2.0 , Chapter I Foundations of Geometry in the Plane The primary goal of this book is to display the richness, unity, and insight that the concept of symmetry brings to the study of geometry. To do so, we must start from a solid grounding in elementary geometry. This chapter provides that necessary background. A rigorous treatment of foundational geometry is hardly elementary: it re- quires a careful development of appropriate concepts, axioms, and theorems. We will not give all the details for such a full axiomatic treatment. We will, however, structure the discussion in this chapter to outline such a treatment. The logical clarity of this approach helps in understanding and remembering the material.1 There are various ways to build a foundation for geometry. Our approach is that of metric geometry, which means that distance between points and angular measure are central organizing concepts. These concepts, in turn, are dependent on the basic properties of R, the real number system. We thus begin our discussion with some comments about the real numbers. I.1 The Real Numbers § We denote the set of real numbers with the symbol R. Other important number systems are Z, the set of integers; N, the set of natural numbers (non-negative integers); and Q, the set of rational numbers (quotients of integers). Later we will also discuss C, the set of complex numbers. Intuitively the real numbers correspond to the points on a straight line. Given a line ℓ, fix a unit of length (e.g., an inch), designate a point on O ℓ as the origin, and specify the “positive” and “negative” sides of ℓ. Then every point p of ℓ respresents one unique real number xp, where xp is the distance of p to (positive or negative depending on the side of ) — this is O O shown in Figure 1.1. The algebraic operations of addition and multiplication of real numbers then correspond to geometric movements on the line. 1We closely follow the development of geometry found in Edwin Moise, Elementary Geometry from an Advanced Standpoint, Third Edition, Addison Wesley Publishers. This book is the primary reference for the basic geometry to be used in Continuous Symmetry. 1 2 Revision v2.0 , Chapter I, Foundations of Geometry in the Plane distance x p p O ℓ -2 -1 0 1xp 2 Figure 1.1. The real number line. Using the number line, the order operation on real numbers, x<y, read “x is less than y,” means that the point corresponding to x occurs “to the left” of the point corresponding to y. The inequality x y, read “x is less than ≤ or equal to y,” means either x<y or x = y. We will assume all the standard algebraic properties of the real numbers, i.e., all the standard properties of addition, multiplication, subtraction, division, and order. For example, if x + r = y + r, then x = y, or if x<y and r > 0, then rx < ry. Do not underestimate the importance or the depth of this assumption! We base our development of geometry on the real number system, a system whose existence is non-trivial to establish and whose prop- erties are sophisticated. A proper study of the real number system belongs to the mathematical subject known as analysis. Using the order relations, we define various types of intervals for real num- bers a < b: bounded closed interval: [a, b]= all x such that a x b , { ≤ ≤ } bounded open interval: (a, b)= all x such that a<x<b . { } We allow open endpoints with a = or b = . However, in those cases −∞ ∞ the intervals are unbounded. The real numbers include all the natural numbers 1, 2, 3, . An important property of this inclusion is called the Archimedean Ordering Principle: For any real number x there exists a natural number n greater than x. One simple but highly important consequence of this property is that for any positive real number ǫ> 0, no matter how small, there exists a natural number n such that 1/n < ǫ (Exercise 1.1). Thus the fraction 1/n can be made “arbitrarily small” by choosing n sufficiently large. Completeness of the Real Number System. An intuitive understand- ing of real numbers as developed in, say, advanced secondary school algebra, will suffice for most of our work. However, there is one far deeper fact that we will need at certain times: the real number line “has no holes.” When rig- orously formulated, this property is known as the completeness of the real number system. Some readers may wish to defer this sophisticated concept until needed later in the text. Such readers should now skip to I.2. § §I.1. The Real Numbers 3 There are various equivalent ways to describe completeness — we will give a description that is easy to picture using closed intervals. A sequence of bounded, closed intervals [a1, b1], [a2, b2], . is said to be nested if each interval contains the next one as a subset. Written in set notation, this means [a , b ] [a , b ] [an, bn] . 1 1 ⊇ 2 2 ⊇···⊇ ⊇··· The real number system R is said to be complete because every such nested sequence of bounded closed intervals has at least one real number x that belongs to all the intervals: Theorem 1.2. The Completeness of R. For any nested sequence of bounded closed intervals in R there will al- ways be at least one real number x that belongs to all the intervals (i.e., x is in the intersection of all the intervals). This theorem is illustrated in Figure 1.3. [a1, b1] [a2, b2] ··· [an , bn ] ··· ··· ··· ℓ a1 a2 an x b b2 b1 ············n Figure 1.3. All the nested intervals [an, bn], n = 1, 2, 3,... , contain x. If the nested intervals are not closed, then there might or might not be a point common to all the intervals. This is examined in Exercise 1.3. The intuitive meaning of completeness is that the real number line “has no holes.” If you have not studied a rigorous formulation of the real number system (such as in an introductory analysis course), it’s probably hard to appreciate the importance of completeness. In fact, it is of critical impor- tance in much of mathematics — it guarantees that real numbers exist when and where we need them. An example of a number system which is not com- plete is Q, the set of rational numbers (fractions). It would, for example, be very difficult to develop calculus with just rational numbers — there are too many “holes” because the rationals are not complete. These ideas are more fully explored in the exercises. More on Completeness: Bolzano’s Theorem. There are several prop- erties of the real number system that are implied by completeness. One such property we will use later in the book is every bounded sequence in R has a convergent subsequence. This is known as Bolzano’s Theorem. We now explain this result. 4 Revision v2.0 , Chapter I, Foundations of Geometry in the Plane ∞ A sequence xn of real numbers is termed bounded if all the elements { }n=1 of the sequence are contained in a bounded interval, i.e., there exists a ∞ bounded interval [a, b] such that a xn b for all n. A sequence xn ≤ ≤ { }n=1 is said to converge to x if the terms in the sequence become arbitrarily close to x as the index n becomes large. A sequence can be bounded but not converge: a simple example is the sequence 1, 1, 1, 1, 1, 1,... { − − − } However, this sequence does have convergent subsequences; one example is 1, 1, 1,... In fact, the completeness of the real number system im- { } plies that every bounded sequence has a convergent subsequence. This is Bolzano’s Theorem. Theorem 1.4. Bolzano’s Theorem. Every bounded sequence in R has a convergent subsequence. The proof of Bolzano’s Theorem, showing it to be a consequence of the completeness of the real number system as formulated in Theorem 1.2, is given in 14. § Exercises I.1 “Why,” said the Dodo, “the best way to explain it is to do it.” Lewis Carroll Exercise 1.1. (a) Show that for any positive real number ǫ> 0, no matter how small, there exists a natural number n such that 1/n<ǫ. Hint: Use the Archimedean Ordering Principle with x = 1/ǫ. (b) For any real number x′ show there exists an integer n′ less than x′. Hint: Use the Archimedean Ordering Principle with x = x′. − Exercise 1.2. (a) Given any real number x, show there exists a smallest integer n0 such that x<n0. Hint: Use Archimedean order to show there exists an integer N such that x < N. Then use Exercise 1.1b to show that the set of integers k x < k N is non-empty and { | ≤ } finite. Any finite set of real numbers must have a smallest member. (b) Suppose x1 and x2 are two real numbers such that x1 <x2. Prove there exists a rational number r = n/m such that x1 <r<x2. Outline: First show there exists a positive integer m large enough so that 1/m < x x . Then show there exists a smallest integer 2 − 1 n such that x1 < n/m. To verify that r = n/m is what you need, you must only verify r<x2. (c) Suppose x1 and x2 are two real numbers such that whenever r1 and r2 are rational numbers satisfying r1 < x1 < r2, then r1 < x2 < r2 §I.1.