# 2. Two binary operations (addition, denoted + and multiplication, denoted

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between R and Q. We will see that R is complete while Q is not. We will also see that Q is countable while R is not. We will explain the consequences these differences have. 2.1 The set of Real Numbers Though it is possible to construct the set of real numbers from scratch (as did Cantor and Dedekind) and derive its properties from the fundamental axioms of set theory, this process is too big a task and beyond the scope of an introductory course in real analysis. Instead, we will make certain assumptions and derive all the important properties of R from these assumptions. Most of the properties we will derive are already known to you. The goal here is to see how they can be proven using only our assumptions and what has already been proven. More precisely, we assume we are given the following: 1. The set of real numbers R. 2. Two binary operations (addition, denoted + and multiplication, denoted.) 3. An order relation <. See section We will also assume a set of properties (see section 2.1.2). From these assumptions, we will prove most of the important results about real numbers, needed for this class. Finally, we will assume the axiom of completeness (see section 2.4). When we want to talk about the real numbers together with the two binary operations, we will use the notation (R, +,.). Before we do this though, let us give an intuitive presentation of the real numbers. 31

2 32 CHAPTER 2. THE STRUCTURE OF R Intuitive Definition of the Real Numbers The concept of number is fundamental to our way of life. The concept of natural numbers, or the numbers we use to count is grasped as an early age. This set, is denoted N. Thus, N = {1, 2, 3,...} Very soon one realizes that the natural numbers are not enough. If we think of numbers as used in counting to represent things one owns, then how does one represent what one owes? The answer is by putting a negative sign. If we combine the natural numbers, their negative and zero, we get the set of integers. This set is denoted by Z. Thus Z = {0, ±1, ±2, ±3,...} As one learns about multiplication and division, one realizes that Z is not enough. For example, if we try to divide 2 by 3, the answer is not an integer. Thus, we are quickly forced to introduce new numbers, the rational numbers. The set of rational numbers is denoted by Q. Its definition is { m } Q = n : m Z, n Z and n 0 Then, we ask the question: do we have all the numbers we ll ever need with the rational numbers? The answer is no, and it has been known for a long time. Around 500 BC, Pythagora knew that not every quantity could be expressed as a rational number. Consider for example a right triangle in which the length of the sides around the right angle is 1. Then, the length x of the hypotenuse satisfies = x 2. In other words x 2 = 2. What is the number x such that x 2 = 2? We know this number exists because we can construct such a triangle. We call this number 2. Pythagora proved that 2 is not a rational number. It is not the only one. π is not rational, neither is e. If p is a prime number, one can show that p is not rational. When a number is not rational, we say that it is irrational. There are more irrational numbers than rational ones. If we represent the rational numbers as dots on a line, there will be more empty spaces than dots. There are several categories of numbers which are irrational. We will not discuss those here. Before we look at the properties of the real numbers, we prove that 2 is irrational using Pythagora proof. We begin with a lemma which proof is left as an exercise. Lemma 79 Let n be an integer. If n 2 is even then n must also be even. Proof. See exercises. We are now ready for Pythagora proof. We state the result as a theorem and give its proof. Theorem 80 There does not exist a rational number r such that r 2 = 2.

3 2.1. THE SET OF REAL NUMBERS 33 Proof. We do a proof by contradiction. Suppose there exists integers a and b 0 such that r = a b, r2 = 2 and assume a is already in lowest terms that is b a and b do not have any common factors. Then, a2 b 2 = 2 or a2 = 2b 2. Thus a 2 is even. It follows that a is even from lemma 79. Thus there exists a number k such that a = 2k. It follows that 2b 2 = 4k 2 or b 2 = 2k 2. Hence, b 2 is even, thus b is even. Which contradicts the fact that a was in lowest terms. b Let us now look at the essential properties which define the set of real numbers Algebraic Properties of R (R, +,.) is a field that is the following properties are satisfied: Closure R is closed under both operations, that is if a R and b R then a+b R and ab R. (A1) a, b R, a + b = b + a. (+ is commutative) (A2) a, b, c R, (a + b) + c = a + (b + c). (+ is associative) (A3) There exists an element denoted 0 in R with the property: a R, a+0 = 0 + a = a. 0 is called the additive identity element. (A4) For each element a in R, there exists an element denoted a, called the additive inverse of a with the property: a + ( a) = ( a) + a = 0. (M1) a, b R, a.b = b.a. (. is commutative) (M2) a, b, c R, (a.b).c = a. (b.c). (. is associative) (M3) There exists an element denoted 1 0 in R with the property: a 0 R, a.1 = 1.a = a. 1 is called the multiplicative identity element. (M4) For each element a 0 in R, there exists an element denoted 1/a, called the multiplicative inverse of a with the property: a. (1/a) = (1/a).a = 1. (D) a, b, c R, a. (b + c) = (a.b) + (a.c) and (b + c).a = (b.a) + (c.a). (multiplication is distributive over addition) Remark Properties (A2) (A4) mean that (R, +) is a group. 2. Adding property (A1) means that (R, +) is a commutative group also called an abelian group.

4 34 CHAPTER 2. THE STRUCTURE OF R Important Basic Properties of Real Numbers The properties which follow do not depend on the fact that we are dealing with real numbers. They depend on the fact that R is a field. Thus, they do not really belong to a real analysis class. Students should have studied (or will study) in a modern algebra class these properties and how to prove them. They are given here because they are essential to the everyday manipulations we perform when we work with real numbers. However, we will not spend time on them. We take many of these properties for granted. Yet, without them, many of the things we do with real numbers would not be possible. First, we establish uniqueness of the identity element as well as uniqueness of the inverse under both operations. This fact is a direct consequence of the properties of each operation; it has nothing to do with the fact that we are working with real numbers. Theorem 82 The additive and multiplicative identity are unique. Proof. We only prove uniqueness of the additive identity. For this, we will prove that if x is an element of R such that x + a = a for every a R, then x = 0. x = x + 0 = x + (a + ( a)) = (x + a) + ( a) = a + ( a) = 0 Theorem 83 The additive and multiplicative inverses are unique. Proof. We only prove uniqueness of the additive inverse. For this, we will prove that if a and b are elements of R such that a + b = 0 then b = a. b = 0 + b = (( a) + a) + b = ( a) + (a + b) = ( a) + 0 = a Remark 84 Properties (A4) and (M4) guarantee the possibility of solving the equations a + x = 0 and a.x = 1. Theorem 83 guarantees the uniqueness of the solution. We now generalize this result. Many problems in mathematics involve solving one or more equations. The next two theorems play an important role in finding the solutions of equations.

5 2.1. THE SET OF REAL NUMBERS 35 Theorem Let a and b be arbitrary elements of R. then the equation a + x = b has the unique solution x = ( a) + b 2. Let a 0 and b be arbitrary elements of R. then the equation a.x = b has the unique solution x = b. (1/a) Proof. In both cases, we have to show that there is a solution. This can be done easily by verifying that the given solution is indeed a solution. Then, we have to show that it is unique. The details of the proof are left as an exercise. Theorem 86 (Cancellation Laws) Suppose that a, b and x are real numbers. Then the following is true: 1. a + x = b + x = a = b 2. If in addition x 0 then a.x = b.x = a = b Proof. We prove each part separately. (a) Choose y such that x + y = 0. Then, a + x = b + x = (a + x) + y = (b + x) + y = a + (x + y) = b + (x + y) = a + 0 = b + 0 = a = b (b) Since x 0, choose y such that xy = 1. Then a.x = b.x = (a.x).y = (b.x).y = a. (x.y) = b. (x.y) = a.1 = b.1 = a = b We now look at various miscellaneous properties of the set of real numbers. The order in which we list them is important in the sense that the proof of some of these properties depend on other properties. Theorem 87 If a and b are any elements of R then 1. a.0 = 0 2. a = ( 1).a 3. (a + b) = ( a) + ( b) 4. ( a) = a

6 36 CHAPTER 2. THE STRUCTURE OF R 5. ( 1). ( 1) = 1 Proof. 1. Since 0+0 = 0, we have (0 + 0).a = 0.a. Using the distributive law we can write it as 0.a+0.a = 0.a. This in turn can be written as 0.a+0.a = 0.a+0. Finally, using the cancellation law for addition, we obtain 0.a = Theorem 83 establishes the fact that every non-zero real number a has a unique additive inverse a. If we can show that ( 1).a has the properties of the additive inverse of a, the result will follow. 3. Left as an exercise a + ( 1).a = 1.a + ( 1).a = (1 + ( 1)).a = 0.a = 0 4. a + ( a) = 0 implies that a is the additive inverse of a. But the additive inverse of a is, by (A4) ( a). Since the additive inverse of a element is unique, it follows that ( a) = a. 5. The details are left as an exercise. Combine parts 2 and 4. Theorem If a R and a 0, then 1/a 0 and 1/ (1/a) = a 2. Suppose that a R, b R. Then, a.b = 0 a = 0 or b = Suppose that a R, b R. Then ( a). ( b) = a.b 4. If a R and a 0, then 1/ ( a) = (1/a) Proof. 1. Since a 0, 1/a exists by (M4). If we had 1/a = 0, we would have 1 = a. (1/a) = a.0 = 0 which is a contradiction. Thus, 1/a 0. Since a. (1/a) = 1, a is the multiplicative inverse of (1/a). But, by (M4), the multiplicative inverse of 1/a is 1/ (1/a). Since the multiplicative inverse of an element is unique, 1/ (1/a) = a. 2. Left as an exercise.

7 2.1. THE SET OF REAL NUMBERS Left as an exercise. 4. Since a 0, a 0, so 1/ ( a) exists. Furthermore, 1 = a. (1/a) = ( a) ( (1/a)) by part 3 of this theorem Thus, -(1/a) is the multiplicative inverse of a. But by (M4), the multiplicative inverse of a should be 1/ ( a). By uniqueness of the multiplicative inverse, it follows that 1/ ( a) = (1/a). From this point on, we will drop the use of the dot to denote multiplication. We will write: ab instead of a.b a 2 for aa, a 3 for ( a 2) a = aaa,... a n+1 for (a n ) a for n N. b a for b + ( a) or ( a) + b a 1 for 1/a and a n for 1/ (a n ) a b or a/b for a (1/b) Exercises As you do these problems, keep in mind that you already know all these results. The goal is to prove them using all the assumptions we have made as well as what you have already proven. When I ask you to prove a result stated in the notes, you can only use all the assumptions we have made as well as the results proven up to the result I am asking you to prove. If the question is a stand alone question, then you can use all the results stated in the notes as well as all the problems up to the question you are working on. 1. Finish proving theorem Finish proving theorem Finish proving theorem Finish proving theorem Finish proving theorem Prove that a + a = 0 = a = 0 for any arbitrary real number a. 7. Prove that if n is an integer and n is even, then n 2 is also even. 8. Prove lemma 79.

8 38 CHAPTER 2. THE STRUCTURE OF R 9. Prove that if n 2 is an odd integer, then n must also be odd. 10. Prove that 6 is not a rational number. 11. Prove that if x is irrational and r is rational, then x+r is irrational. Also, show that if r is a non-zero rational then xr is irrational. 12. Show by example that if x and y are irrational, then x + y and xy may be rational.

9 2.2. ORDER AXIOMS AND ORDER PROPERTIES OF R Order Axioms and Order Properties of R In this section, we introduce the order properties of R. These properties are very important and will be used heavily throughout the remainder of this course. Like in the previous section, we do not develop these properties from scratch. We assume a few of them, then we derive most of the properties from our assumptions. The properties we assume will be called axioms Order Axioms for R and Definitions On R there is a relation, denoted <, which satisfies the following axioms: Axiom 89 (O1) For all a and b in R, exactly one of the following holds: 1. a < b 2. a = b 3. b < a Axiom 90 (O2) For all a, b and c in R, if a < b and b < c then a < c. Axiom 91 (O3) For all a, b and c in R, if a < b then a + c < b + c. Axiom 92 (O4) For all a, b and c in R, if a < b and 0 < c then ac < bc. Remark 93 The first axiom, axiom 89, is known as the law of trichotomy. Remark 94 The second axiom, axiom 90, is simply the statement of the property called transitivity. Remark 95 If we represent the set of real numbers as a line stretching to infinity in both directions (the real line), then by convention if a < b, then a is positioned on the line to the left of b. Definition 96 We can now define some well known terms. Let a and b be in R. 1. If a < b, we say that a is less than b. 2. If a < b then we also write b > a and say b greater than a. 3. If a < b or a = b then we write a b and say a less than or equal to b 4. Similarly, we can define greater than or equal to. 5. If a > 0, we say that a is positive. If a < 0, we say that a is negative. 6. If a 0, we say that a is non-negative. If a 0, we say that a is non-positive.

10 40 CHAPTER 2. THE STRUCTURE OF R Remark 97 If we have a < b and b < c, it is convenient to combine the two relations and write a < b < c. The same applies for the other inequalities. Remark 98 A field together with an order that is a relation satisfying axioms is called an ordered field. So, (R, +,., <) is an ordered field. Remark 99 The opposite of the statement a < b is a b. opposite of the statement a > b is a b. Similarly, the Properties of the Order We now establish the properties the relations we just defined satisfy. Most of these properties are already known to the reader. We will prove a few of them. The remaining proofs will be assigned as exercises. Theorem 100 For every xin R, the following is true: 1. If x < 0 then x > If x > 0 then x < 0. Proof. We only prove the first part. The second part is left as an exercise. If x < 0 then by O3 x + ( x) < 0 + ( x). It follows that 0 < x which is the same as x > 0. Theorem 101 Let a 0 R. 1. a 2 > > 0 Proof. We prove each part separately. 1. Since a 0, either a > 0 or a < 0. If a > 0, then a 2 = aa > 0.a = 0, hence a 2 > 0. If a < 0, then a > 0. Hence, ( a) ( a) > 0. ( a) = 0. But ( a) ( a) = aa = a 2. So, a 2 > Follows from part 1 and the fact that 1 = 1 2 Theorem 102 Let a, b, c be elements of R. 1. a < b a b < 0 2. a > b and c < 0 ac < bc 3. a > 0 1/a > 0 4. a < 0 1/a < 0

11 2.2. ORDER AXIOMS AND ORDER PROPERTIES OF R 41 Proof. left as an exercise. Theorem 103 If a and b are any real numbers, then a > b = a > 1 (a + b) > 2 b Proof. left as an exercise. Remark 104 Theorem 103 implies (by setting b = 0) that given any strictly positive number a, there exists another smaller strictly positive number ( 1 2 a). Thus there is no smallest strictly positive number. Theorem 105 Let a and b be arbitrary real numbers. If a > 0 and b > 0 then ab > 0 Proof. If a > 0 and b > 0 then by O4 ab > 0.b = 0. Therefore, ab > 0. Theorem 106 If ab > 0 then we either have a > 0 and b > 0 or we have a < 0 and b < 0. Proof. If ab > 0, then ab 0, hence neither a = 0 nor b ( = 0. By the law of trichotomy, either a > 0 or a < 0. If a > 0, then b = b a 1 ) = (ba) 1 a a = (ab) 1 a > 0 (why?) Hence, b > 0. The proof is similar if we assume that a < 0. Corollary 107 If ab < 0 then we either have a > 0 and b < 0 or we have a < 0 and b > 0. Proof. Left as an exercise Exercises 1. Finish proving theorem Finish proving theorem Finish proving theorem Finish proving theorem Finish proving corollary Prove that If a b and b a then a = b where a and b are arbitrary real numbers. 7. Prove the following where a, b, c and d denote arbitrary elements of R. (a) a < 0 and b > 0 = ab < 0 (b) a < 0 and b < 0 = ab > 0 (c) 0 < a < 1 = a 2 < a (d) 1 < a = a 2 > a

12 42 CHAPTER 2. THE STRUCTURE OF R (e) a < a + 1 (f) 0 < a < b = a 2 < b 2 (g) a b and c d = a + c b + d 8. If a, b R and a 2 + b 2 = 0, show that a = b = 0.

13 2.3. METRIC PROPERTIES OF R - ABSOLUTE VALUE Metric Properties of R - Absolute Value Definition and Properties The trichotomy property assures that if a 0 then either a > 0 or a < 0. Thus, we define: Definition 108 (Absolute value) The absolute value of a real number a, denoted a is defined by: { a if a 0 a = a if a < 0 The domain of this function is R, its range is the set of non-negative real numbers, and it maps the elements a and a into the same element. Geometrically, the absolute value of a number x can be interpreted as how far from 0 x is. Similarly, x a can be interpreted as the distance between x and a and therefore, one can think of the absolute value as a tool to measure distances between numbers. In particular, for a fixed number a, the set defined by {x R such that x a < δ} denotes the set of numbers within a distance δ of a. The inequality x a < δ is equivalent to δ < x a < δ or a δ < x < a + δ that is x (a δ, a + δ). This will be used a lot throughout this class. Remark 109 With a = 0 above, we see that δ < x < δ is equivalent to x < δ. The same is true if we replace < by. Students should review how to solve equations and inequalities involving absolute values as they too will be used a lot in this class. Theorem 110 The following properties hold: 1. a = 0 a = 0 2. a = a for all a in R 3. ab = a b for all a, b in R 4. 1 a = 1 for all a 0 in R a 5. a = a for all a, b 0 in R b b 6. If c 0, then a c c a c 7. a a a 8. a 2 = a Proof. See problems

14 44 CHAPTER 2. THE STRUCTURE OF R The next result is often used in Mathematics. Theorem 111 (Triangle inequality) If a and b are any real numbers, then a b a + b a + b Proof. First, we prove the right side. Since a a a and b b b it follows that ( a + b ) a + b a + b. Thus, a + b a + b by remark 109 or part 6 of theorem 110. Next, we prove the left side. a = a + b b a + b + b = a + b + b Thus, a b a + b Similarly, b = a + b a a + b + a = a + b + a Thus b a a + b Combining the two inequalities, we get a b a + b a + b a b a + b It follows by part 6 of theorem 110 that a b a + b An application of the above theorem, often used when dealing with absolute values and inequalities is given by the corollary below. Corollary 112 If a, b and c are any given real numbers then a c a b + b c Proof. If we let a c = (a b) + (b c) and use the triangle inequality, we have a c = (a b) + (b c) a b + b c

15 2.3. METRIC PROPERTIES OF R - ABSOLUTE VALUE 45 Corollary 113 If a 1, a 2,...a n are any n real numbers, then Proof. See exercises. a 1 + a a n a 1 + a a n We finish with a theorem which appears simple in statement but whose proof can sometimes give a hard time to beginners. The theorem simply states that the only non-negative real number less that every positive number is 0. Theorem 114 Let x be a real number. If x < ɛ for each ɛ > 0 then x = 0. Proof. See exercises. Remark 115 This result is often used to show two numbers are equal. If a and b are two real numbers, to show a = b, it is enough to show that a b < ɛ for each ɛ > 0. By the theorem, it will follow that a b = 0 or a = b. Geometrically, we are simply saying that if the distance between a and b is less than any positive real number then a = b Equations and Inequalities Involving Absolute Values The purpose of this class is not to learn how to solve equations and inequalities involving absolute values. However, they will be used a lot throughout this class. Students should make sure they review them. We remind students of the basic principles used when solving equations and inequalities involving absolute values. For what follows, let a and b denote any expression. The equation a = b is equivalent to the two equations a = b and a = b. However, solutions must be checked, some may not be valid. The equation a = b is equivalent to the two equations a = b and a = b. However, solutions must be checked, some may not be valid. The inequality a < b is equivalent to b < a < b The inequality a b is equivalent to b a b The inequality a > b is equivalent to a > b or a < b The inequality a b is equivalent to a b or a b Notion of a Metric Space As we noted earlier, the absolute value can be used to measure distances between numbers. x y denotes the distance between x and y on the real line. For the rest of these notes, we will use the absolute value every time we need to measure distances between two real numbers. However, if the set in which our elements come from were not R, the absolute value may not be appropriate to measure

16 46 CHAPTER 2. THE STRUCTURE OF R distances. For example, you already know that in R 2, the distance between two points of coordinates (x 1, y 1 ) and (x 2, y 2 ) is given by (x 2 x 1 ) 2 + (y 2 y 1 ) 2. In more general settings, we replace the absolute value by a function we call a distance function or a metric. We give a formal definition for a metric. Definition 116 (metric) Let X be a set. A metric or a distance on X is a function d : X X R which satisfies the properties below. In the properties below, we let x, y and z be elements of X. 1. d (x, y) 0 (non-negativity) 2. d (x, y) = 0 x = y 3. d (x, y) = d (y, x) (symmetry) 4. d (x, z) d (x, y) + d (y, z) (triangle inequality) Definition 117 (metric space) A space X together with a metric d is called a metric space. We say that (X, d) is a metric space. Example 118 It is easy to see that the absolute value function is a metric for R, hence we say that (R,. ) is a metric space Exercises 1. Prove theorem Prove the second corollary of theorem Show that a + b = a + b ab Given real numbers a and b, prove that a b a b. 5. Given real numbers a and b, prove that a b a b. This is sometimes called the reverse triangle inequality. 6. Prove theorem Solve 2x 3 = x. 8. Solve 2x 3 = 3x. 9. Solve 2x + 5 > Solve 2x + 5 < 11.

### Commutative Rings and Fields

Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two

### Structure of R. Chapter Algebraic and Order Properties of R

Chapter Structure of R We will re-assemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions

### Real Analysis - Notes and After Notes Fall 2008

Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start

### CHAPTER 3: THE INTEGERS Z

CHAPTER 3: THE INTEGERS Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction The natural numbers are designed for measuring the size of finite sets, but what if you want to compare the sizes of two sets?

### Proofs. Chapter 2 P P Q Q

Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,

### Contribution of Problems

Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

### The Real Number System

MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely

### Direct Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24

Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using

### CONSTRUCTION OF THE REAL NUMBERS.

CONSTRUCTION OF THE REAL NUMBERS. IAN KIMING 1. Motivation. It will not come as a big surprise to anyone when I say that we need the real numbers in mathematics. More to the point, we need to be able to

### An Introduction to Proof-based Mathematics Harvard/MIT ESP: Summer HSSP Isabel Vogt

An Introduction to Proof-based Mathematics Harvard/MIT ESP: Summer HSSP Isabel Vogt Class Objectives Binary Operations Groups Axioms Closure Associativity Identity Element Unique Inverse Abelian Groups

### What is proof? Lesson 1

What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might

### Chapter 5: The Integers

c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition

### Supplementary Material for MTH 299 Online Edition

Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think

### 1 Take-home exam and final exam study guide

Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number

### The Integers. Math 3040: Spring Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers Multiplying integers 12

Math 3040: Spring 2011 The Integers Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers 11 4. Multiplying integers 12 Before we begin the mathematics of this section, it is worth

### Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with

### * 8 Groups, with Appendix containing Rings and Fields.

* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that

### SEVENTH EDITION and EXPANDED SEVENTH EDITION

SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 5-1 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to

### means is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.

1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for

### We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

### Handout #6 INTRODUCTION TO ALGEBRAIC STRUCTURES: Prof. Moseley AN ALGEBRAIC FIELD

Handout #6 INTRODUCTION TO ALGEBRAIC STRUCTURES: Prof. Moseley Chap. 2 AN ALGEBRAIC FIELD To introduce the notion of an abstract algebraic structure we consider (algebraic) fields. (These should not to

### Section 0.6: Factoring from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative

Section 0.6: Factoring from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 201,

### Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with

### Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number

### Rings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.

Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over

### Analysis I. Classroom Notes. H.-D. Alber

Analysis I Classroom Notes H-D Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21

### GROUPS. Chapter-1 EXAMPLES 1.1. INTRODUCTION 1.2. BINARY OPERATION

Chapter-1 GROUPS 1.1. INTRODUCTION The theory of groups arose from the theory of equations, during the nineteenth century. Originally, groups consisted only of transformations. The group of transformations

### EECS 1028 M: Discrete Mathematics for Engineers

EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 32 Proofs Proofs

### Outline. We will now investigate the structure of this important set.

The Reals Outline As we have seen, the set of real numbers, R, has cardinality c. This doesn't tell us very much about the reals, since there are many sets with this cardinality and cardinality doesn't

### Section 19 Integral domains

Section 19 Integral domains Instructor: Yifan Yang Spring 2007 Observation and motivation There are rings in which ab = 0 implies a = 0 or b = 0 For examples, Z, Q, R, C, and Z[x] are all such rings There

### Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

### MATH CSE20 Homework 5 Due Monday November 4

MATH CSE20 Homework 5 Due Monday November 4 Assigned reading: NT Section 1 (1) Prove the statement if true, otherwise find a counterexample. (a) For all natural numbers x and y, x + y is odd if one of

### Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch

Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary

### Order of Operations. Real numbers

Order of Operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply and divide from left to right. 4. Add

### NUMBERS It s the numbers that count

NUMBERS It s the numbers that count Starting from the intuitively obvious this note discusses some of the perhaps not so intuitively obvious aspects of numbers. Henry 11/1/2011 NUMBERS COUNT! Introduction

### The natural numbers. The natural numbers come with an addition +, a multiplication and an order < p < q, q < p, p = q.

The natural numbers N = {0, 1,, 3, } The natural numbers come with an addition +, a multiplication and an order < p, q N, p + q N. p, q N, p q N. p, q N, exactly one of the following holds: p < q, q

### CSE 1400 Applied Discrete Mathematics Proofs

CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4

### Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models

Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................

### Numbers. 2.1 Integers. P(n) = n(n 4 5n 2 + 4) = n(n 2 1)(n 2 4) = (n 2)(n 1)n(n + 1)(n + 2); 120 =

2 Numbers 2.1 Integers You remember the definition of a prime number. On p. 7, we defined a prime number and formulated the Fundamental Theorem of Arithmetic. Numerous beautiful results can be presented

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### Contribution of Problems

Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

### UMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM

UMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM YUFEI ZHAO In this talk, I want give some examples to show you some linear algebra tricks for the Putnam. Many of you probably did math contests in

### The Completion of a Metric Space

The Completion of a Metric Space Let (X, d) be a metric space. The goal of these notes is to construct a complete metric space which contains X as a subspace and which is the smallest space with respect

### Some Background Material

Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

### 3.2 Constructible Numbers

102 CHAPTER 3. SYMMETRIES 3.2 Constructible Numbers Armed with a straightedge, a compass and two points 0 and 1 marked on an otherwise blank number-plane, the game is to see which complex numbers you can

### a. Define a function called an inner product on pairs of points x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in R n by

Real Analysis Homework 1 Solutions 1. Show that R n with the usual euclidean distance is a metric space. Items a-c will guide you through the proof. a. Define a function called an inner product on pairs

### Example. Addition, subtraction, and multiplication in Z are binary operations; division in Z is not ( Z).

CHAPTER 2 Groups Definition (Binary Operation). Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G. Note. This condition of assigning

### Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009

Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009 Our main goal is here is to do counting using functions. For that, we

### Lecture 8: A Crash Course in Linear Algebra

Math/CS 120: Intro. to Math Professor: Padraic Bartlett Lecture 8: A Crash Course in Linear Algebra Week 9 UCSB 2014 Qué sed de saber cuánto! Pablo Neruda, Oda a los Números 1 Linear Algebra In the past

### MA554 Assessment 1 Cosets and Lagrange s theorem

MA554 Assessment 1 Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem; they go over some material from the lectures again, and they have some new material it is all examinable,

### ACCUPLACER MATH 0310

The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

### CS 6820 Fall 2014 Lectures, October 3-20, 2014

Analysis of Algorithms Linear Programming Notes CS 6820 Fall 2014 Lectures, October 3-20, 2014 1 Linear programming The linear programming (LP) problem is the following optimization problem. We are given

Section 9.4 Radical Expressions 95 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties.

### Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

### Complex Numbers For High School Students

Complex Numbers For High School Students For the Love of Mathematics and Computing Saturday, October 14, 2017 Presented by: Rich Dlin Presented by: Rich Dlin Complex Numbers For High School Students 1

### Stephen F Austin. Exponents and Logarithms. chapter 3

chapter 3 Starry Night was painted by Vincent Van Gogh in 1889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Exponents and Logarithms This chapter focuses on understanding

### Math Review. for the Quantitative Reasoning measure of the GRE General Test

Math Review for the Quantitative Reasoning measure of the GRE General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important for solving

### Points of Finite Order

Points of Finite Order Alex Tao 23 June 2008 1 Points of Order Two and Three If G is a group with respect to multiplication and g is an element of G then the order of g is the minimum positive integer

### Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

### Inverses and Elementary Matrices

Inverses and Elementary Matrices 1-12-2013 Matrix inversion gives a method for solving some systems of equations Suppose a 11 x 1 +a 12 x 2 + +a 1n x n = b 1 a 21 x 1 +a 22 x 2 + +a 2n x n = b 2 a n1 x

### Boolean Algebras. Chapter 2

Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union

### The Ring Z of Integers

Chapter 2 The Ring Z of Integers The next step in constructing the rational numbers from N is the construction of Z, that is, of the (ring of) integers. 2.1 Equivalence Classes and Definition of Integers

### MATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.

MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. College Algebra for STEM

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics College Algebra for STEM Marcel B. Finan c All Rights Reserved 2015 Edition To my children Amin & Nadia Preface From

### Prove proposition 68. It states: Let R be a ring. We have the following

Theorem HW7.1. properties: Prove proposition 68. It states: Let R be a ring. We have the following 1. The ring R only has one additive identity. That is, if 0 R with 0 +b = b+0 = b for every b R, then

### Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

### Mathematical Induction

Mathematical Induction Let s motivate our discussion by considering an example first. What happens when we add the first n positive odd integers? The table below shows what results for the first few values

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Lesson 6: Algebra. Chapter 2, Video 1: "Variables"

Lesson 6: Algebra Chapter 2, Video 1: "Variables" Algebra 1, variables. In math, when the value of a number isn't known, a letter is used to represent the unknown number. This letter is called a variable.

### 1. multiplication is commutative and associative;

Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.

### 3 The language of proof

3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;

### 1-2 Study Guide and Intervention

1- Study Guide and Intervention Real Numbers All real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### Section 31. The Separation Axioms

31. The Separation Axioms 1 Section 31. The Separation Axioms Note. Recall that a topological space X is Hausdorff if for any x,y X with x y, there are disjoint open sets U and V with x U and y V. In this

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION. The Peano axioms

PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION The Peano axioms The following are the axioms for the natural numbers N. You might think of N as the set of integers {0, 1, 2,...}, but it turns

### CIS Spring 2018 (instructor Val Tannen)

CIS 160 - Spring 2018 (instructor Val Tannen) Lecture 09 Tuesday, February 13 PROOFS and COUNTING Figure 1: Too Many Pigeons Theorem 9.1 (The Pigeonhole Principle (PHP)) If k + 1 or more pigeons fly to

### MATH 201 Solutions: TEST 3-A (in class)

MATH 201 Solutions: TEST 3-A (in class) (revised) God created infinity, and man, unable to understand infinity, had to invent finite sets. - Gian Carlo Rota Part I [5 pts each] 1. Let X be a set. Define

### CONSTRUCTION OF sequence of rational approximations to sets of rational approximating sequences, all with the same tail behaviour Deﬁnition 1.

CONSTRUCTION OF R 1. MOTIVATION We are used to thinking of real numbers as successive approximations. For example, we write π = 3.14159... to mean that π is a real number which, accurate to 5 decimal places,

### SYMBOL NAME DESCRIPTION EXAMPLES. called positive integers) negatives, and 0. represented as a b, where

EXERCISE A-1 Things to remember: 1. THE SET OF REAL NUMBERS SYMBOL NAME DESCRIPTION EXAMPLES N Natural numbers Counting numbers (also 1, 2, 3,... called positive integers) Z Integers Natural numbers, their

### Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

### Basic Proof Examples

Basic Proof Examples Lisa Oberbroeckling Loyola University Maryland Fall 2015 Note. In this document, we use the symbol as the negation symbol. Thus p means not p. There are four basic proof techniques

### Cosets and Lagrange s theorem

Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem some of which may already have been lecturer. There are some questions for you included in the text. You should write the

### Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

### AQA Level 2 Further mathematics Further algebra. Section 4: Proof and sequences

AQA Level 2 Further mathematics Further algebra Section 4: Proof and sequences Notes and Examples These notes contain subsections on Algebraic proof Sequences The limit of a sequence Algebraic proof Proof

### The Completion of a Metric Space

The Completion of a Metric Space Brent Nelson Let (E, d) be a metric space, which we will reference throughout. The purpose of these notes is to guide you through the construction of the completion of

### CMPSCI 601: Tarski s Truth Definition Lecture 15. where

@ CMPSCI 601: Tarski s Truth Definition Lecture 15! "\$#&%(') *+,-!".#/%0'!12 43 5 6 7 8:9 4; 9 9 < = 9 = or 5 6?>A@B!9 2 D for all C @B 9 CFE where ) CGE @B-HI LJKK MKK )HG if H ; C if H @ 1 > > > Fitch

### Solving Quadratic & Higher Degree Equations

Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

### PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.

PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice

### Instructor Notes for Chapters 3 & 4

Algebra for Calculus Fall 0 Section 3. Complex Numbers Goal for students: Instructor Notes for Chapters 3 & 4 perform computations involving complex numbers You might want to review the quadratic formula

### Lecture 1: Axioms and Models

Lecture 1: Axioms and Models 1.1 Geometry Although the study of geometry dates back at least to the early Babylonian and Egyptian societies, our modern systematic approach to the subject originates in

### 2 so Q[ 2] is closed under both additive and multiplicative inverses. a 2 2b 2 + b

. FINITE-DIMENSIONAL VECTOR SPACES.. Fields By now you ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices,

### Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

### The Chinese Remainder Theorem

Chapter 4 The Chinese Remainder Theorem The Monkey-Sailor-Coconut Problem Three sailors pick up a number of coconuts, place them in a pile and retire for the night. During the night, the first sailor wanting

### Chapter 4. Basic Set Theory. 4.1 The Language of Set Theory

Chapter 4 Basic Set Theory There are two good reasons for studying set theory. First, it s a indispensable tool for both logic and mathematics, and even for other fields including computer science, linguistics,

### not to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results

REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division

### TAYLOR POLYNOMIALS DARYL DEFORD

TAYLOR POLYNOMIALS DARYL DEFORD 1. Introduction We have seen in class that Taylor polynomials provide us with a valuable tool for approximating many different types of functions. However, in order to really

### Complex Numbers. Rich Schwartz. September 25, 2014

Complex Numbers Rich Schwartz September 25, 2014 1 From Natural Numbers to Reals You can think of each successive number system as arising so as to fill some deficits associated with the previous one.