SYMBOL AND MEANING IN MATHEMATICS


 Regina Ferguson
 2 years ago
 Views:
Transcription
1 ,,. SYMBOL AND MEANING IN MATHEMATICS ALICE M. DEAN Mathematics and Computer Science Department Skidmore College May 26,1995 There is perhaps no other field of study that uses symbols as plentifully and precisely as does mathematics. figure 1, for instance, contains a concise, grammatically correct sentence (except for the lack of a period at the end) that uses mathematical symbols to state one of the most fundamental definitions of calculus. This mathematical sentence illustrates the fact that mathematicians have, over thousands of years, developed a language with which to express their ideas. This language is in some sense more complex than a natural language (indeed it builds upon natural language) since the meanings of its symbols and words are highly precise; on the other hand, its words have little of the ambiguity and nuance of natural language and so in that sense it could be considered simpler. Of course, mathematics is far more than a language: it is the goal of mathematics to uncover and describe the fundamental patterns and symmetries underlying our universe. In this article I hope to show some of the ways in which the symbols of mathematics, like the symbols of any language, help us express our thoughts as well as shape the way we think. Figure 1. A mathematical sentence Mathematical symbols and language undergo a Darwinian sort of evolution. As new ideas are discovered, mathematicians introduce new symbols and language to describe them. The fittest of these survive through the ages to be used by subsequent generations. The great mathematician Paul Erdos speaks of a supreme being who has a book. In this book are all the great ideas and proofs of mathematics, expressed in their clearest and most elegant form. The highest compliment Erdos can pay to a mathematical proof is to call it the "book proof." By analogy, the "book symbols" might be those that manage to express our mathematical thoughts in the simplest and most beautiful manner possible. The symbols and language we use to express our ideas profoundly affect how we think about those ideas. From the most fundamental object, the number, to college topics like calculus and abstract algebra, to some of the most complex concepts of mathematics that are studied by a few experts,'the notation and definitions that describe mathematical ideas dictate, to large extent, how easily they will be understood and used. page 10/5
2 , An example that most nonmathematician readers might feel comfortable with is mathematical notation for numbers. Most readers feel at home with standard notation for whole numbers, also called integers: 0,1,2,3, 4, 15,197, 753, etc. Our notation uses a positional number system (i.e., 1'97 is different from 971) based on the use of the ten digits from 0 to 9. We call t:b.1s a base 10 representation;' if we used instead, say seven digits, it would be called base 7. Computers, for example, usually use base 2, base 8, or base 16. Of course there are other ways to represent numbers: most of you probably know how to use Roman numerals for e'xample. Mathematicians also like to represent numbers pictorially, usually as points on a line, like markings on a ruler: o Figure 2. A number line These representations of the whole numbers, or integers, give us a way to express thein, but they also help to shape the way we think about them. For instance, in the number line in Figure 2, the fact that each integer is one unit away from the next one is reflected by the little line segment between them. But what about the positions not marked as integers  for example, what about,the position halfway between 1 and 2? Again we have mathematical symbols to represent this notion. We usually would describe this number either as 1~ or 1.5. The first notation uses fraction notation, which expresses the idea of dividing the numbers, or the line segments that represent them, into parts; the second uses base 10 decimal notation. The fractional notation ~ means to divide the number 1 mto 2 parts; 1~ means the sum 1+ ~. The decimal notation 1.5 is related to fractional notation, but all numbers are divided into parts that are powers of 10: 1.5 =1 + UO' = (2 x 10) +3+ Ko + ~OO' etc. These three ways of representing numbers  as fractions, as decimals, and as points on a line form a foundation on which we can now build, introducing concepts that combine and otherwise use numbers, such as arithmetic, calculus, statistics, etc. But at all stages, our choice of representation either facilitates or impedes our ability to express these concepts, and it helps or hinders us in discovering new concepts as well. ' By continuing with this example of the three number representations, we can demonstrate how the symbols we use can actually inspire us to discover new ideas (and hence new symbols as well!). Let's think about the number represented by the fraction 3{1. To represent this number in. decimal notation, we could simply do long division: = 11) = Figure 3. From fractions to decimals page 2 0/5
3 This calculation reveals several things. First of all, the decimal representation for 7{1' unlike that for ~ = 0.5, is nonterminating: to be exact, it must go on forever! (Of course, we could think of all. decimal representations as nonterminating; for instance, 0.5 = ) It's also interesting to note that this decimal representation has the repeating pattern "18." This observation  which arose from comparing the two notations  motivates the following interesting question: Does every fraction of whole numbers have a repeating decimal expansion? If so, then a number like is interesting because it does not have a repeating pattern (it has an obvious pattern, but not one that repeats). Well, it turns out that the answer to the question, ''Does every fraction of whole numbers have a repeating decimal expansion?," is yes, and it's not even that hard to prove, although I won't prove. it here. Thus our use of decimal notation has led us to the discovery that not all numbers can be represented by fractions of whole numbers. This in turn leads us to new language: a number that can be represented as a fraction of whole numbers (or equivalently, with a repeating decimal expansion) is called rational (so ~, 7{1 ' , etc., are all rational numbers); otherwise it's called irrational (so is irrational). The existence of irrational numbers was discovered by the early Greeks some time around the fifth or sixth century BC, l and the discovery was monumental be~ause it upset many previously held assumptions about numbers. For an example, consider the number 7C, which represents the ratio of the circumference of a circle to its diameter. The number 7C, which figures in many physical applications, has been considered equal to 3 (by the ancient Chinese), 25%1 (by the ancient Egyptians), and 3JIg (by the ancient Babylonians). Even today there are many people who think that 7C equals 2~ or All of these are close to correct, but none is exactly right because it turns out that 7C is irrational (this is quite hard to prove). This is the reason thaut's convenient to invent a special symbol  the Greek letter 7C  to represent this number whose decimal expansion is nonterminating and nonrepeating. 2 Actually, the number first discovered to be irrational by the early Greeks was not 7C but.j2, that number which when multiplied by itself (i.e., "squared") gives the answer 2 (thusm = 4 because 4 2 = 4 x4 = 16). The value of.j2 is somewhere between 1 and 2 because 1 2 = 1 is too small, while 2 2 =4 is too big. We can get closer and closer estimates of.j2 by squaring test values and seeing whether the result is bigger or smaller than 2. For instance, = 2.25 is too big so.j2 < 1.5. By continuing this process, we can get better and better approximations of.j2, and it turns out that.j2 is approximately How did the early Greeks prove that.j2 is 1 All historical data in this article comes from Great Moments in Mathematics Before 1650, by Howard Eves, Mathematical Association of America, Washington, DC, For those of you who "surf' the World Wide Web, here is a node where you can find some humorous and interesting information about 7C, including the first million digits of its decimal expansion: page 3 of5
4 irrational? Well first of all, they used the idea that a rational number is one that can be expressed as a fraction of two whole numbers. Second, they used a proof technique called proof by contradiction,3 in which you assume exactly the opposite of what you think is true, and then you show that if this were the case, then something you know to be false would also be true (this is the contradiction). This proof is not very hard, but it illustrates how mathematicians combine symbols and their meaning to draw conclusions, so let's take a look at it. Theorem..J2 is an irrational number. Proof by contradiction. Assume the assertion is false; in other words, assume that.j2 is a rational number. We will show that this leads to a contradiction. Our assumption that.j2 is rational means that we can find two whole nuinbers, which we'll call a and b, with.j2 = %. We may also assume that we have reduced this fraction to lowest terms; in other words, that a and b have no factors in common (so for example, we would reduce the fraction %1 to. 7j by canceling the common factor 3 from top and bottom). From our assumption that.j2 =%it follows that 2 =(%f =a%2. If we multiply both sides of the equation by b 2, we see that 2b 2 = a 2. Since 2b 2 is obviously an even number (i.e. a multiple of 2), so is a 2 But if a 2 is even, then so is a, since the square of an odd number is also odd. Now we know that a must be an even number. To say a number is even means it's a multiple of 2 (for example, the even number 34 is 2 times 17), so we can write a as 2 times something: say, as a =2c. Now let's use this way of writing a_in the equation 2b 2 =a 2 If we replace a by 2c, we get 2b 2 =(2C)2 =4c, and dividing both sides by 2 gives us the new equation b 2 =2c. From this last equation, b 2 = 2c, we see now that b 2 is an even number, so b must also be even. So, based on 'our original assumption that.j2 = %' we have shown that both a and b must be even, i.e., they are both multiples of2. However, this contradicts our other assumption that a and b had no factors in common, in other words, that the fraction %had been reduced to lowest terms. Since any fraction can be reduced to lowest terms, that means our assumption that J2 = %must have been false. Thus we have proven the theorem, because we have shown that.j2 cannot be written as a fraction of whole numbers and hence is an irrational number. I The little black bar on the line after the proof is a standard symbol used to indicate that a proof has been completed. Notice how carefully the definition of rational number was used to prove 3 Also called indirect proofor reductio ad absurdum. page 4 of5
5 OIl.'",.. that..fi could not be rational. We also used several basic properties of aritlnnetic (reducing fractions, clearing parentheses) that we were able to express clearly and succinctly through the use of wellchosen symbols. In particular, the use of the letters a, b, and c to represent undetermined numbers permitted 'us to use concise expressions to describe the relationships among various quantities. From this example, we can see how the careful use of symbols with very precise meanings not only lets us express mathematical ideas, but also leads to new ideas. This cycle of new concepts begetting new words and symbols, which in tum motivate other new concepts, is one that occurs throughout mathematics. Mathematicians will often describe a definition or theorem with words that others use to describe art or literature, e.g., beautiful, elegant, clever (or sometimes clumsy, cumbersome, or ugly). Mathematics can help us understand the underlying structure and logic of the physical world around us, sometimes revealing that two seemingly different phenomena have a great deal in common. For instance, automobile suspensions and electrical circuits behave' according to the saine mathematical rules, and chemical isomers and English grammar can be represented using the same mathematical notation. Ultimately the symbols of mathematics convey the meaning of nature. Exercises 4 1. In the paragraph where we discuss the number 1C, we gave five different rational approximations of 1C.' Find their decimal expansions to determine which one is closest to the actual value of 1C, whose decimal expansion begins , Consider the following two decimal expansions: 1= and Do you think these represent the. same number? 3. We have shown in this article the existence of irrational numbers. If x is an irrational number, show that there are rational numbers arbitrarily close to x. HINT: Think about the numbers you get by chopping off the decimal expansion of x after a fmite number of digits (e.g., chopping off the end of the decimal expansion of..fi to get 1.4, or 1.414, or , etc.). OPTIONAL: Show that ify is any rational number, there are irrational numbers arbitrarily close to y. 4. When I ran my word processor's grammar checker on this article, it repeatedly prompted me to replace the word "discover" with the word "invent," and vice versa. To what extent do you think mathematics is discovered, and to what extent is it invented? 4 After all, this is a mathematics article! page 50/5
MAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationLecture 1: Elementary Number Theory
Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers
More informationLESSON 1 PRIME NUMBERS AND FACTORISATION
LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers
More informationThe numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers
Section 1.8 The numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers Each is said to be a subset of the real numbers.
More informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationStudents in their first advanced mathematics classes are often surprised
CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are
More informationChapter 1. Real Numbers Operations
www.ck1.org Chapter 1. Real Numbers Operations Review Answers 1 1. (a) 101 (b) 8 (c) 1 1 (d) 1 7 (e) xy z. (a) 10 (b) 14 (c) 5 66 (d) 1 (e) 7x 10 (f) y x (g) 5 (h) (i) 44 x. At 48 square feet per pint
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationCHAPTER 1 NUMBER SYSTEMS POINTS TO REMEMBER
CHAPTER NUMBER SYSTEMS POINTS TO REMEMBER. Definition of a rational number. A number r is called a rational number, if it can be written in the form p, where p and q are integers and q 0. q Note. We visit
More informationContinued fractions and good approximations.
Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our
More informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warmup problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More information1. The Fly In The Ointment
Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent
More information2.1.1 Examples of Sets and their Elements
Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationSTRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.
STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary
More informationSECTION 14 Absolute Value in Equations and Inequalities
14 Absolute Value in Equations and Inequalities 37 SECTION 14 Absolute Value in Equations and Inequalities Absolute Value and Distance Absolute Value in Equations and Inequalities Absolute Value and
More informationChapter 10 Expanding Our Number System
Chapter 10 Expanding Our Number System Thus far we have dealt only with positive numbers, and, of course, zero. Yet we use negative numbers to describe such different phenomena as cold temperatures and
More informationDirect Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction
Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationPYTHAGORAS. c b. c 2 = a 2 + b 2
PYTHAGORAS c b a c 2 = a 2 + b 2 This relationship had been known since very ancient times. For example a number of Pythagorean triplets are found on Babylonian clay tablets: 3, 4, 5; 5, 12, 13; and as
More informationEven Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2.
MATH 337 Proofs Dr. Neal, WKU This entire course requires you to write proper mathematical proofs. All proofs should be written elegantly in a formal mathematical style. Complete sentences of explanation
More informationAlgebra I Notes Review Real Numbers and Closure Unit 00a
Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition,
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationChapter 1.1 Rational and Irrational Numbers
Chapter 1.1 Rational and Irrational Numbers A rational number is a number that can be written as a ratio or the quotient of two integers a and b written a/b where b 0. Integers, fractions and mixed numbers,
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L34) is a summary BLM for the material
More informationSection 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.
M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationMathematica  The Principles of Math 10. The Endless World of Irrational Numbers
Mathematics, English for Sek I and Sek II Mathematica  The Principles of Math 10. The Endless World of Irrational Numbers 09:55 minutes 00:23 (caption) Our world is vast, and space is virtually unending
More informationMathematics Success Grade 8
T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers,
More informationTexas Assessment of Knowledge and Skills (TAKS) 6th Grade
Texas Assessment of Knowledge and Skills (TAKS) 6th Grade 98 99 100 Grade 6 Mathematics TAKS Objectives and TEKS Student Expectations TAKS Objective 1 The student will demonstrate an understanding of numbers,
More informationAxiom A.1. Lines, planes and space are sets of points. Space contains all points.
73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationWeek 5: Quantifiers. Number Sets. Quantifier Symbols. Quantity has a quality all its own. attributed to Carl von Clausewitz
Week 5: Quantifiers Quantity has a quality all its own. attributed to Carl von Clausewitz Number Sets Many people would say that mathematics is the science of numbers. This is a common misconception among
More informationLesson on Repeating and Terminating Decimals. Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu
Lesson on Repeating and Terminating Decimals Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu Background: This lesson would be embedded in a unit on the real number system. The set of real
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationA Basic Course in Real Analysis Prof. P. D. Srivastava Department of Mathematics Indian Institute of Technology, Kharagpur
A Basic Course in Real Analysis Prof. P. D. Srivastava Department of Mathematics Indian Institute of Technology, Kharagpur Lecture  3 Continuum and Exercises So the last lecture we have discussed the
More information2. Propositional Equivalences
2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationSession 7 Fractions and Decimals
Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More information31 is a prime number is a mathematical statement (which happens to be true).
Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationPythagorean Theorem. Overview. Grade 8 Mathematics, Quarter 3, Unit 3.1. Number of instructional days: 15 (1 day = minutes) Essential questions
Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean Theorem Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Prove the Pythagorean Theorem. Given three side lengths,
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationMath 140a  HW 1 Solutions
Math 140a  HW 1 Solutions Problem 1 (WR Ch 1 #1). If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Given that r is rational, we can write r = a b for some
More informationTennessee Mathematics Standards 20092010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes
Tennessee Mathematics Standards 20092010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical
More information2 The Euclidean algorithm
2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In
More information13 Absolute Value in Equations and Inequalities
SECTION 1 3 Absolute Value in Equations and Inequalities 103 13 Absolute Value in Equations and Inequalities Z Relating Absolute Value and Distance Z Solving Absolute Value Equations and Inequalities
More informationDISCRETE MATHEMATICS W W L CHEN
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationSet Theory Basic Concepts and Definitions
Set Theory Basic Concepts and Definitions The Importance of Set Theory One striking feature of humans is their inherent need and ability to group objects according to specific criteria. Our prehistoric
More informationRules and Tips for Writing Mathematics Discrete Math
Rules and Tips for Writing Mathematics Discrete Math Adapted from: Writing in Mathematics by Annalisa Crannell Why Should You Have To Write Papers In A Math Class? For most of your life so far, the only
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationRIT scores between 191 and 200
Measures of Academic Progress for Mathematics RIT scores between 191 and 200 Number Sense and Operations Whole Numbers Solve simple addition word problems Find and extend patterns Demonstrate the associative,
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationCalculus for Middle School Teachers. Problems and Notes for MTHT 466
Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble
More information7.1 An Axiomatic Approach to Mathematics
Chapter 7 The Peano Axioms 7.1 An Axiomatic Approach to Mathematics In our previous chapters, we were very careful when proving our various propositions and theorems to only use results we knew to be true.
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More information5. Geometric Series and Three Applications
5. Geometric Series and Three Applications 5.. Geometric Series. One unsettling thing about working with infinite sums is that it sometimes happens that you know that the sum is finite, but you don t know
More information4.2 Euclid s Classification of Pythagorean Triples
178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple
More informationRepton Manor Primary School. Maths Targets
Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work
More informationChapter 1 Introductory Information and Review
SECTION 1.1 Numbers Chapter 1 Introductory Information and Review Section 1.1: Numbers Types of Numbers Order on a Number Line Types of Numbers Natural Numbers: MATH 1300 Fundamentals of Mathematics 1
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationIntroduction to mathematical arguments
Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math
More informationCourse notes on Number Theory
Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that
More information10.3 GEOMETRIC AND HARMONIC SERIES
10.3 Geometric and Harmonic Series Contemporary Calculus 1 10.3 GEOMETRIC AND HARMONIC SERIES This section uses ideas from Section 10.2 about series and their convergence to investigate some special types
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
More informationSequences and Series
Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will
More informationDECIMALS AND PERCENTAGES
The Improving Mathematics Education in Schools (TIMES) Project DECIMALS AND PERCENTAGES NUMBER AND ALGEBRA Module 8 A guide for teachers  Years 5 8 June 20 5YEARS 8 Decimals and Percentages (Number and
More informationThe Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic 1 Introduction: Why this theorem? Why this proof? One of the purposes of this course 1 is to train you in the methods mathematicians use to prove mathematical statements,
More informationIrrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More information2. INEQUALITIES AND ABSOLUTE VALUES
2. INEQUALITIES AND ABSOLUTE VALUES 2.1. The Ordering of the Real Numbers In addition to the arithmetic structure of the real numbers there is the order structure. The real numbers can be represented by
More informationHow many numbers there are?
How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural
More information1 Die hard, once and for all
ENGG 2440A: Discrete Mathematics for Engineers Lecture 4 The Chinese University of Hong Kong, Fall 2014 6 and 7 October 2014 Number theory is the branch of mathematics that studies properties of the integers.
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationStructure and randomness in the prime numbers. Science colloquium January 17, A small selection of results in number theory. Terence Tao (UCLA)
Structure and randomness in the prime numbers A small selection of results in number theory Science colloquium January 17, 2007 Terence Tao (UCLA) 1 Prime numbers A prime number is a natural number larger
More informationCOMPASS Numerical Skills/PreAlgebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13
COMPASS Numerical Skills/PreAlgebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More informationThe Division Algorithm E. L. Lady (July 11, 2000)
The Division Algorithm E. L. Lady (July 11, 2000) Theorem [Division Algorithm]. Given any strictly positive integer d and any integer a, there exist unique integers q and r such that a = qd + r, and 0
More informationProof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Math 232  Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the
More informationCourse: Math 7. engage in problem solving, communicating, reasoning, connecting, and representing
Course: Math 7 Decimals and Integers 11 Estimation Strategies. Estimate by rounding, frontend estimation, and compatible numbers. Prentice Hall Textbook  Course 2 7.M.0 ~ Measurement Strand ~ Students
More informationReal Numbers. Learning Outcomes. chapter. In this chapter you will learn:
chapter Learning Outcomes In this chapter you will learn: ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ How to define the real numbers About factors, multiples and prime factors How to write a whole number as a product of prime
More informationPreliminary Mathematics
Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationIntegers and Rational Numbers
Chapter 6 Integers and Rational Numbers In this chapter, we will see constructions of the integers and the rational numbers; and we will see that our number system still has gaps (equations we can t solve)
More information1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real
More information