Long division for integers
|
|
- Laura Richards
- 6 years ago
- Views:
Transcription
1 Feasting on Leftovers January 2011 Summary notes on decimal representation of rational numbers Long division for integers Contents 1. Terminology 2. Description of division algorithm for integers (optional reading) Decimals of rational numbers eventually repeat or terminate 3. Leftovers and leftover sequences 4. Decimal sequences 5. Theorem and Proof of the Day: Rational expansion theorem Long division for integers 1 Terminology When we divide integers by each other, we may use the terminology which is shorthand for the relationship More compactly, divisor ) quotient REM remainder dividend dividend = divisor quotient + remainder. b ) q REM r a means a = bq + r and 0 r < b. Here we are working with integer quotients. For example, 7 ) 2 REM 6 20 is shorthand for the relationship 20 = Sometimes we may want to work with real quotients, in which case we use the terminology so that we can use the relationship divisor ) quotient dividend dividend = divisor quotient. For example, 7 ) is shorthand for the relationship 2 =
2 2 Description of the division algorithm for integers (optional reading) We will work with only positive integers. While the theorems still hold for negative dividends, it is enough to get the idea of the theorems and their proofs by working with positive dividends. Although we will not go through this section in class, I am including it here in case you are interested, and because the mathematics is beautiful. Theorem (Division Algorithm Theorem for Integers [Usiskin, Theorem 5.3, p. 206]). Given positive integers a, b where a b > 0, there exist unique integers q, r so that a = bq + r and 0 r < b. The number q is called the quotient, and the number r is called the remainder. Proof of the Division Algorithm Theorem for Integers. The proof comes in two parts. 1. Such q and r exist. Consider all multiples of b, and find the one that is closest to a without being larger than a. We claim that if we set this multiple to qb, and let r = a qb, then r must be less than b. To see this, by way of contradiction suppose that r b. Then (q + 1)b a, so we get the contradiction that qb wasn t the closest multiple. So r must be less than b. We have now shown that we can construct q and r so that a = qb + r and 0 r < b. r b 0 b 2b 3b 4b a 5b r... b 0 b qb a 2. Such q and r are unique. Let q 1, q 2, r 1, r 2 Z such that We show that this implies q 1 = q 2 and r 1 = r 2 : a = q 1 b + r 1, a = q 2 b + r 2, and 0 r 1, r 2 < b. a = q 1 b + r 1 a = q 2 b + r 2 = 0 = (q 1 q 2 )b + (r 1 r 2 ) This means that q 1 q 2 b = r 1 r 2. Because r 1 and r 2 are both less than b, we know that r 1 r 2 < b. So q 1 q 2 < 1. But q 1 and q 2 are integers, so the only way that this can happen is if q 1 = q 2. If q 1 = q 2, then 0 = 0 b + (r 1 r 2 ), implying r 1 = r 2. We have shown that q 1 = q 2 and r 1 = r 2 as desired. 2
3 Decimals of rationals eventually repeat or terminate Using long division, find the decimal expansions for: Questions to think about: What do you think the decimal expansions are? How would you justify that your answer is correct? Before turning the page, circle the remainders after each step. Call these the leftover terms. What patterns do you see in the leftovers? Your observations motivate the technique we use in the theorem and proof of the day: Theorem (Rational Expansion Theorem). If r is a rational number, then the decimal expansion for r must either terminate or eventually be periodic. Before we get to the proof, we need to introduce some terminology to help us communicate our observations in a more precise way. 3
4 3 Describing Leftover Terms A leftover is the number that remains after each step of a long division problem, used to begin the next step. A critical property of leftovers is the following: 0 leftover < divisor. 3.1 Example In the division problem 7 ) 2, the gave us the leftover decimal digit L D Each leftover L is less than 7. How exactly did we get D from L in each one of these instances? < < < 7 10L 7 D < 7 In each case, the decimal digit D is the largest multiple of 7 that is smaller than 10 times the leftover L for that step Leftover and Decimal Observation aka Predestination of Leftovers Observation (Leftover and Decimal Observation). In the division problem b ) a, suppose the leftover is L at a particular step. Then the decimal digit for that step is the largest integer D so that 10L bd < b. (This is the same as saying that bd is the largest multiple of b that is less than or equal to 10L.) Moreover, the leftover for the next step is given by 10L bd. These are the only decimal digits and leftovers possible. In other words, it s impossible to do a long division problem and get more than one correct answer. 3.3 Leftover Sequences A leftover sequence is the sequence of leftovers of a division problem, in order. We count the term used in the first step of a division problem as a leftover. Example. The leftover sequence for the division problem 7 ) 2 is
5 where the above block repeats. (I circled the first term to keep track of where the division begins.) Example. Find the leftover sequence for the division problem 13 ) Example. Find the leftover sequence for the division problem 4 ) Example. Find the leftover sequence for the division problem 40 ) Puzzler... Can we ever have the leftover sequence ? (Answer is upside down for your mathematical protection!) No We can t have both 3 and 1 follow 2, by predestination of leftovers and remainders. There is only one leftover that can follow 2. Another puzzler... Can we ever have the decimal sequence... 2, 3, 2, 1, 2, 3, 2, 1...? Yes! This would represent a decimal such as Unlike leftovers, it is not easy to predict what the next decimal digit will be. This is why we need to work with leftover sequences to prove things about decimals: it is easier to prove things if we can always predict what leftover will come next. Theorem (Leftover Sequence Theorem). Leftover sequences must eventually repeat. Proof. Let b ) a be a division problem. Suppose by contradiction that its leftover sequence never repeats. By the Leftover and Remainder Observation, never repeating means there must be an infinitely many different numbers in the leftover sequences. But leftovers must always lie between 0 and b (possibly including 0), and they are always whole numbers. This is a contradiction, as there cannot be an infinite number of whole numbers between 0 and b. So the remainder sequence must eventually repeat. Corollary (Leftover Block Length). Suppose a and b are positive integers. The repeating block of a leftover sequence for the division problem b ) a has length at most b 1. Proof. Either the sequence has a 0 or not. Case 1: the leftover sequence contains a 0. If this is the case, then the sequence looks like This is because the only leftover that can follow 0 is 0. So the repeating block is 0. This is a block of length 1. Case 2: the leftover sequence does not contain a 0. Then the only possibly numbers the leftover sequence can contain are 1, 2,..., b 1. So the length of the repeated block is at most b 1. 5
6 Feasting on Leftovers: Summary notes on decimal representation of rational numbers Math486-W11 Y. Lai 4 Decimal Sequences A decimal sequence is the decimal digits in the quotient of a division problem. Important fact: We can always read off the decimal sequence from the leftover sequence! Example. Write down the leftover sequence and decimal sequence for 7 ) 2. Can you get the decimal sequence from the leftover sequence without using the standard representation of the long division algorithm? leftover sequence , 8, 5, 7, 1, 4,... decimal sequence Example. Find the decimal sequence for 5 ) 4. Relate it to the leftover sequence. The decimal sequence is 8, 0, 0, 0, 0,... and the leftover sequence is 4 0. We get 8 from 4 because 8 5 is the largest multiple of 5 less than or equal to 40. We get 0 because 0 5 is the largest multiple of 5 less than or equal to 0. Cool Example. (a) Find the decimal expansions 7 ) 3 and 7 ) 5. If you use the previous examples wisely, you will not have to do any long division! (Again, explanation is upside down for your mathematical protection.) We can read it off the leftover sequence for 7 ) 2. (!) This means that all the decimal expansions for n/7 are just rotations of each other! This is a beautiful consequence of the Leftover and Decimal Observation. Finally, we are ready for the theorem and proof of the day. 6
7 5 Theorem and Proof of the Day Theorem (Rational Expansion Theorem). If r = a/b is a rational number, where a, b Z, then a decimal expansion for r eventually terminates or repeats. Proof. The division problem b ) a gives us a decimal expansion for r. By the Leftover Sequence Theorem, the leftover sequence for b ) a must eventually repeat. Suppose the leftover sequence contains a 0. Then the leftover sequence eventually looks like 0, so the decimal sequence eventually looks like 0, 0, 0,..., meaning the decimal expansion terminates. Suppose the leftover sequence doesn t contain a 0. Suppose the repeating block is... L 1 L 2... L N.... Then the decimal sequence must end with the repeating block... D 1 D 2... D N... where D k is the largest multiple of b smaller than 10L k, by the Leftover and Decimal Observation. Therefore the decimal sequence eventually repeats. We have now shown that given a rational number, its decimal expansion must either terminate or eventually repeat. 7
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
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers
More informationA number that can be written as, where p and q are integers and q Number.
RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.
More informationEDULABZ INTERNATIONAL NUMBER SYSTEM
NUMBER SYSTEM 1. Find the product of the place value of 8 and the face value of 7 in the number 7801. Ans. Place value of 8 in 7801 = 800, Face value of 7 in 7801 = 7 Required product = 800 7 = 00. How
More informationQuestion 1: Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0?
Class IX - NCERT Maths Exercise (.) Question : Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0? q Solution : Consider the definition of a rational number.
More informationDivision Algorithm B1 Introduction to the Division Algorithm (Procedure) quotient remainder
A Survey of Divisibility Page 1 SECTION B Division Algorithm By the end of this section you will be able to apply the division algorithm or procedure Our aim in this section is to show that for any given
More informationClass IX Chapter 1 Number Sustems Maths
Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented
More informationCHAPTER 1 REAL NUMBERS KEY POINTS
CHAPTER 1 REAL NUMBERS 1. Euclid s division lemma : KEY POINTS For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b. 2. Euclid s division
More informationCHAPTER 1 NUMBER SYSTEMS. 1.1 Introduction
N UMBER S YSTEMS NUMBER SYSTEMS CHAPTER. Introduction In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it (see Fig..). Fig.. : The number
More informationCool Results on Primes
Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why
More informationWORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:
WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their
More informationIntermediate Math Circles February 26, 2014 Diophantine Equations I
Intermediate Math Circles February 26, 2014 Diophantine Equations I 1. An introduction to Diophantine equations A Diophantine equation is a polynomial equation that is intended to be solved over the integers.
More informationIntermediate Math Circles February 14, 2018 Contest Prep: Number Theory
Intermediate Math Circles February 14, 2018 Contest Prep: Number Theory Part 1: Prime Factorization A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. An integer
More informationPrepared by Sa diyya Hendrickson. Package Summary
Introduction Prepared by Sa diyya Hendrickson Name: Date: Package Summary Defining Decimal Numbers Things to Remember Adding and Subtracting Decimals Multiplying Decimals Expressing Fractions as Decimals
More informationNumber Systems. Exercise 1.1. Question 1. Is zero a rational number? Can you write it in the form p q,
s Exercise. Question. Is zero a rational number? Can you write it in the form p q, where p and q are integers and q 0? Solution Yes, write 0 (where 0 and are integers and q which is not equal to zero).
More informationMath /Foundations of Algebra/Fall 2017 Foundations of the Foundations: Proofs
Math 4030-001/Foundations of Algebra/Fall 017 Foundations of the Foundations: Proofs A proof is a demonstration of the truth of a mathematical statement. We already know what a mathematical statement is.
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationMore Polynomial Equations Section 6.4
MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division
More informationChapter 3: Factors, Roots, and Powers
Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly
More informationDivisibility of Natural Numbers
10-19-2009 Divisibility of Natural Numbers We now return to our discussion of the natural numbers. We have built up much of the mathematical foundation for the natural numbers (N = 1, 2, 3,...). We used
More informationUnit Essential Questions. What are the different representations of exponents? Where do exponents fit into the real number system?
Unit Essential Questions What are the different representations of exponents? Where do exponents fit into the real number system? How can exponents be used to depict real-world situations? REAL NUMBERS
More informationA group of figures, representing a number, is called a numeral. Numbers are divided into the following types.
1. Number System Quantitative Aptitude deals mainly with the different topics in Arithmetic, which is the science which deals with the relations of numbers to one another. It includes all the methods that
More informationMathematical Induction
Mathematical Induction Representation of integers Mathematical Induction Reading (Epp s textbook) 5.1 5.3 1 Representations of Integers Let b be a positive integer greater than 1. Then if n is a positive
More informationThe 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
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationMath From Scratch Lesson 29: Decimal Representation
Math From Scratch Lesson 29: Decimal Representation W. Blaine Dowler January, 203 Contents Introducing Decimals 2 Finite Decimals 3 2. 0................................... 3 2.2 2....................................
More informationNumber Theory and Graph Theory
1 Number Theory and Graph Theory Chapter 1 Introduction and Divisibility By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 1 DIVISION
More informationMA094 Part 2 - Beginning Algebra Summary
MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationREAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.
REAL NUMBERS Introduction Euclid s Division Algorithm Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Fundamental
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More informationFundamentals of Mathematics I
Fundamentals of Mathematics I Kent State Department of Mathematical Sciences Fall 2008 Available at: http://www.math.kent.edu/ebooks/10031/book.pdf August 4, 2008 Contents 1 Arithmetic 2 1.1 Real Numbers......................................................
More informationAll About Numbers Definitions and Properties
All About Numbers Definitions and Properties Number is a numeral or group of numerals. In other words it is a word or symbol, or a combination of words or symbols, used in counting several things. Types
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
More informationReal Number. Euclid s division algorithm is based on the above lemma.
1 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius Basic Concepts 1. Euclid s division lemma Given two positive integers a and b, there exist unique integers q and r
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationCHAPTER 1 NUMBER SYSTEMS. 1.1 Introduction
N UMBER S YSTEMS NUMBER SYSTEMS CHAPTER. Introduction In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it (see Fig..). Fig.. : The number
More informationThe Euclidean Algorithm and Multiplicative Inverses
1 The Euclidean Algorithm and Multiplicative Inverses Lecture notes for Access 2009 The Euclidean Algorithm is a set of instructions for finding the greatest common divisor of any two positive integers.
More information1 The Real Number System
1 The Real Number System The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth century, as a kind of an envelope
More informationDownloaded from
Topic : Real Numbers Class : X Concepts 1. Euclid's Division Lemma 2. Euclid's Division Algorithm 3. Prime Factorization 4. Fundamental Theorem of Arithmetic 5. Decimal expansion of rational numbers A
More informationSECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION
2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides
More informationIntermediate Math Circles February 29, 2012 Linear Diophantine Equations I
Intermediate Math Circles February 29, 2012 Linear Diophantine Equations I Diophantine equations are equations intended to be solved in the integers. We re going to focus on Linear Diophantine Equations.
More informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More informationStandard forms for writing numbers
Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,
More informationPGSS Discrete Math Solutions to Problem Set #4. Note: signifies the end of a problem, and signifies the end of a proof.
PGSS Discrete Math Solutions to Problem Set #4 Note: signifies the end of a problem, and signifies the end of a proof. 1. Prove that for any k N, there are k consecutive composite numbers. (Hint: (k +
More informationDay 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5
Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There
More informationNumbers. 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
More information1 Continued Fractions
Continued Fractions To start off the course, we consider a generalization of the Euclidean Algorithm which has ancient historical roots and yet still has relevance and applications today.. Continued Fraction
More informationCHAPTER 8: EXPLORING R
CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed
More informationThe Pigeonhole Principle
The Pigeonhole Principle 2 2.1 The Pigeonhole Principle The pigeonhole principle is one of the most used tools in combinatorics, and one of the simplest ones. It is applied frequently in graph theory,
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 27
CS 70 Discrete Mathematics for CS Spring 007 Luca Trevisan Lecture 7 Infinity and Countability Consider a function f that maps elements of a set A (called the domain of f ) to elements of set B (called
More informationDividing Polynomials: Remainder and Factor Theorems
Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.
More information2.2 Some Consequences of the Completeness Axiom
60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that
More informationCSE 20: Discrete Mathematics
Spring 2018 Summary So far: Today: Logic and proofs Divisibility, modular arithmetics Number Systems More logic definitions and proofs Reading: All of Chap. 1 + Chap 4.1, 4.2. Divisibility P = 5 divides
More informationand LCM (a, b, c) LCM ( a, b) LCM ( b, c) LCM ( a, c)
CHAPTER 1 Points to Remember : REAL NUMBERS 1. Euclid s division lemma : Given positive integers a and b, there exists whole numbers q and r satisfying a = bq + r, 0 r < b.. Euclid s division algorithm
More informationMath-2A Lesson 2-1. Number Systems
Math-A Lesson -1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?
More informationKNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS
KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS C O M P E T E N C Y 1 KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS SKILL 1.1 Compare the relative value of real numbers (e.g., integers, fractions,
More informationMath-2 Section 1-1. Number Systems
Math- Section 1-1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?
More informationItem 8. Constructing the Square Area of Two Proving No Irrationals. 6 Total Pages
Item 8 Constructing the Square Area of Two Proving No Irrationals 6 Total Pages 1 2 We want to start with Pi. How Geometry Proves No Irrations They call Pi the ratio of the circumference of a circle to
More informationDecimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.
Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before
More informationSEVENTH 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
More informationMA131 - Analysis 1. Workbook 6 Completeness II
MA3 - Analysis Workbook 6 Completeness II Autumn 2004 Contents 3.7 An Interesting Sequence....................... 3.8 Consequences of Completeness - General Bounded Sequences.. 3.9 Cauchy Sequences..........................
More informationChapter 1 The Real Numbers
Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus
More informationClass 7 Integers. Answer the questions. Choose correct answer(s) from the given choices. Fill in the blanks
ID : in-7-integers [1] Class 7 Integers For more such worksheets visit www.edugain.com Answer the questions (1) An integer is divided by 4 and gives a remainder of 3. The resulting quotient is divided
More informationNatural Numbers Positive Integers. Rational Numbers
Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -
More informationFundamentals of Mathematics (MATH 1510)
Fundamentals of Mathematics (MATH 1510) Instructor: Lili Shen Email: shenlili@yorku.ca Department of Mathematics and Statistics York University September 11, 2015 About the course Name: Fundamentals of
More informationCHAPTER 3. Number Theory
CHAPTER 3 Number Theory 1. Factors or not According to Carl Friedrich Gauss (1777-1855) mathematics is the queen of sciences and number theory is the queen of mathematics, where queen stands for elevated
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.
More informationIntroduction Integers. Discrete Mathematics Andrei Bulatov
Introduction Integers Discrete Mathematics Andrei Bulatov Discrete Mathematics - Integers 9- Integers God made the integers; all else is the work of man Leopold Kroenecker Discrete Mathematics - Integers
More informationWith Question/Answer Animations. Chapter 4
With Question/Answer Animations Chapter 4 Chapter Motivation Number theory is the part of mathematics devoted to the study of the integers and their properties. Key ideas in number theory include divisibility
More informationGrade 7/8 Math Circles. Continued Fractions
Faculty of Mathematics Waterloo, Ontario N2L 3G Centre for Education in Mathematics and Computing A Fraction of our History Grade 7/8 Math Circles October th /2 th Continued Fractions Love it or hate it,
More informationProof worksheet solutions
Proof worksheet solutions These are brief, sketched solutions. Comments in blue can be ignored, but they provide further explanation and outline common misconceptions Question 1 (a) x 2 + 4x +12 = (x +
More information6.4 Division of Polynomials. (Long Division and Synthetic Division)
6.4 Division of Polynomials (Long Division and Synthetic Division) When we combine fractions that have a common denominator, we just add or subtract the numerators and then keep the common denominator
More informationClass IX Chapter 5 Introduction to Euclid's Geometry Maths
Class IX Chapter 5 Introduction to Euclid's Geometry Maths Exercise 5.1 Question 1: Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can
More informationMATHEMATICS X l Let x = p q be a rational number, such l If p, q, r are any three positive integers, then, l that the prime factorisation of q is of t
CHAPTER 1 Real Numbers [N.C.E.R.T. Chapter 1] POINTS FOR QUICK REVISION l Euclid s Division Lemma: Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
More information1. Revision Description Reflect and Review Teasers Answers Recall of Rational Numbers:
1. Revision Description Reflect Review Teasers Answers Recall of Rational Numbers: A rational number is of the form, where p q are integers q 0. Addition or subtraction of rational numbers is possible
More informationp-adic Analysis Compared to Real Lecture 1
p-adic Analysis Compared to Real Lecture 1 Felix Hensel, Waltraud Lederle, Simone Montemezzani October 12, 2011 1 Normed Fields & non-archimedean Norms Definition 1.1. A metric on a non-empty set X is
More informationK K.OA.2 1.OA.2 2.OA.1 3.OA.3 4.OA.3 5.NF.2 6.NS.1 7.NS.3 8.EE.8c
K.OA.2 1.OA.2 2.OA.1 3.OA.3 4.OA.3 5.NF.2 6.NS.1 7.NS.3 8.EE.8c Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to Solve word problems that
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 informationRational Numbers. a) 5 is a rational number TRUE FALSE. is a rational number TRUE FALSE
Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 1 Chapter 1A - - Real Numbers Types of Real Numbers Name(s) for the set 1, 2,, 4, Natural Numbers Positive Integers Symbol(s) for the set, -,
More informationRising 7th Grade Math. Pre-Algebra Summer Review Packet
Rising 7th Grade Math Pre-Algebra Summer Review Packet Operations with Integers Adding Integers Negative + Negative: Add the absolute values of the two numbers and make the answer negative. ex: -5 + (-9)
More information2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero
Unit 2 Examples(K) 1. Find all the real zeros of the function. Answer: Simply substitute the values given in all the functions and see which option when substituted, all the values go to zero. That is
More informationMathematics Pacing. Instruction: 9/7/17 10/31/17 Assessment: 11/1/17 11/8/17. # STUDENT LEARNING OBJECTIVES NJSLS Resources
# STUDENT LEARNING OBJECTIVES NJSLS Resources 1 Describe real-world situations in which (positive and negative) rational numbers are combined, emphasizing rational numbers that combine to make 0. Represent
More information= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
More informationCONTENTS NUMBER SYSTEMS. Number Systems
NUMBER SYSTEMS CONTENTS Introduction Classification of Numbers Natural Numbers Whole Numbers Integers Rational Numbers Decimal expansion of rational numbers Terminating decimal Terminating and recurring
More informationChapter 1: Fundamentals of Algebra Lecture notes Math 1010
Section 1.1: The Real Number System Definition of set and subset A set is a collection of objects and its objects are called members. If all the members of a set A are also members of a set B, then A is
More information3 The fundamentals: Algorithms, the integers, and matrices
3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers
More informationAnswers (1) A) 36 = - - = Now, we can divide the numbers as shown below. For example : 4 = 2, 2 4 = -2, -2-4 = -2, 2-4 = 2.
Answers (1) A) 36 We can divide the two numbers by using the following steps : 1. Firstly, we will divide the mathematical signs of the numbers. We place a negative sign before the negative numbers and
More informationWhat 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
More informationBell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.
Bell Quiz 2-3 2 pts Determine the end behavior of the graph using limit notation. 5 2 1. g( ) = 8 + 13 7 3 pts Find a function with the given zeros. 4. -1, 2 5 pts possible Ch 2A Big Ideas 1 Questions
More information8 th Grade Intensive Math
8 th Grade Intensive Math Ready Florida MAFS Student Edition August-September 2014 Lesson 1 Part 1: Introduction Properties of Integer Exponents Develop Skills and Strategies MAFS 8.EE.1.1 In the past,
More informationMath 109 HW 9 Solutions
Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we
More informationThe of the fraction ¾ is 3. a. numerator b. b. area c. c. addend d. d. denominator
The of the fraction ¾ is 3. a. numerator b. b. area c. c. addend d. d. denominator To find the area of a circle you need to square the radius and multiply by. a. diameter b. pi c. radius d. circumference
More informationFinding Prime Factors
Section 3.2 PRE-ACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to
More informationMA 301 Test 4, Spring 2007
MA 0 Test 4, Spring 007 hours, calculator allowed, no notes. Provide paper for the students to do work on. Students should not write answers on test sheet. TA Grades, 5, 6, 7 All answers must be justified.
More informationFifth Grade Mathematics Mathematics Course Outline
Crossings Christian School Academic Guide Middle School Division Grades 5-8 Fifth Grade Mathematics Place Value, Adding, Subtracting, Multiplying, and Dividing s will read and write whole numbers and decimals.
More informationMATH Dr. Halimah Alshehri Dr. Halimah Alshehri
MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary
More informationMATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology.
MATH 501 Discrete Mathematics Lecture 6: Number theory Prof. Dr. Slim Abdennadher, slim.abdennadher@guc.edu.eg German University Cairo, Department of Media Engineering and Technology 1 Number theory Number
More informationOne-to-one functions and onto functions
MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are
More information