# Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers have the following basic arithmetic properties: 1. The set of integers is closed under addition and closed under multiplication. This means that if a and b are any integers, then a + b and ab are also integers. 2. Addition and multiplication are associative. This means that for any integers a; b; and c, it is true that a + (b + c) = (a + b) + c and a (bc) = (ab) c. 3. Addition and multiplication are commutative. This means that for any integers a and b, it is true that a + b = b + a and ab = ba. 4. The set of integers contains an additive identity element. This additive identity element is the number 0. It is called an additive identity element because if a is any integer, then a + 0 = a. 5. The set of integers contains a multiplicative identity element. This multiplicative identity element is the number 1. It is called a multiplicative identity element because if a is any integer, then 1 a = a. 1

2 6. Given any integer a, there is another integer which is the additive inverse of a. This additive inverse is the number a. It is called an additive inverse because a + ( a) = The integers obey the distributive law of multiplation over addition. This means that if a; b; and c are any integers, then a (b + c) = ab + ac. All of the other familiar properties of integer arithmetic can be deduced from these six properties. For example, these properties can be used to prove that if a is any integer, then 0 a = 0. Although you may not feel the need to prove that 0a = 0 because you are already very comfortable with this fact and have been using it since elementary school, you should consider this to be an introduction to the art of theorem proving. A theorem is a statement of the form If A is true, then B must be true. In the Theorem that we are about to prove, the A is the set of properties 1 7 listed above which we are assuming to be true, and the B is the statement that 0 a = a. In general, the given information, A, is called the hypothesis or set of hypotheses, and the to be proved informaion, B, is called the conclusion. Note that, in proving that 0 a = 0, we actually use the word Proposition rather than Theorem. A Proposition is a mini theorem or a simple fact one that requires proof, but can be deduced fairly quickly and easily from the given set of hypotheses. The word Theorem is usually reserved for deeper results whose proofs require a more complex and clever assembly of the hypotheses. Make sure, in reading the proof of the following proposition, that you understand every step. You should observe that only the given information (properties 1 7) is being used to arrive at the conclusion. In the homework exercises, you will have the opportunity to develop your theorem proving skills by proving similar propositions. Proposition 1 If a is any integer, then 0 a = 0. Proof. Let a be an integer. Since 0 is the additive identity element of the integers, we know that = 0. By multiplying both sides of the above equation by the integer a, we obtain a (0 + 0) = a 0. 2

3 Due to the distributive property of multiplication over addition, we now see that a 0 + a 0 = a 0. Since the set of integers is closed under multiplication and a and 0 are both integers, we know that a 0 must also be an integer. Since a 0 is an integer and every integer has an additive inverse, we know that the integer (a 0) exists. Adding this integer to both sides of the above equation, we obtain (a 0 + a 0) + ( (a 0)) = a 0 + ( (a 0)). as By the associative property of addition, we can write the above equation a 0 + (a 0 + ( (a 0))) = a 0 + ( (a 0)) Since a 0 + ( (a 0)) = 0 (by Property 6), we now have a = 0. Since 0 is the additive identity element, we know that a = a 0. Upon comparing the previous two equations, we deduce that a 0 = 0. Finally, since multiplication is commutative, we arrive at the conclusion that 0 a = 0. 2 Divisibility and the Division Algorithm De nition 2 Let a and b be integers with b 6= 0. We say that a is divisible by b (or that a is a multiple of b or that b is a factor of a) if there exists an integer q such that a = qb. If the integer a is divisible by the integer b, then we say that b divides a and we write b j a. 3

4 Example 3 84 is divisible by 3 because 84 = Thus we can say that 3 divides 84 and we can write 3 j ; 921 is divisible by 357 because 18; 921 = is divisible by 6 because 30 = is not divisible by 3 because there is no integer q such that 50 = q 3. Whenever an integer a is not divisible by an integer b, we can still write a as a multiple of b plus a remainder. Example 4 17 is not divisible by 7 but we can write 17 = : 24 is not divisible by 5 but we can write 24 = Some basic results involving divisibility are given in the following Proposition. Lemma 5 Suppose that a; b; and c are integers. 1. If a divides b and b divides c, then a divides c. 2. If a divides both b and c, and if m and n are any integers, then a divides mb + nc. Proof. First we prove assertion 1: Suppose that a divides b and b divides c. This means that there are integers s and t such that b = sa and c = tb. It follows from this that c = t (sa) = (ts) a. Since ts is an integer, we see that a divides c. Now we prove the second assertion of the lemma: Suppose that a divides both b and c. This means that there are integera s and t such that b = sa and c = ta. Now let m and n be any given integers. Then mb + nc = m (sa)+n (ta) = (ms) a+(nt) a = (ms + nt) a. Since mb+nc = (ms + nt) a and since ms + nt is an integer, we see that a divides mb + nc. A general fact about divisibility is that if a and b are any integers with b > 0, then we can always nd integers q and r such that a = qb + r. However, given any particular a and b, there is always more than one to choose q and r to make the above equation be true. For example, for a = 17 and b = 7, we know that 17 = 27+3, but it is also true that 17 = 37+( 4). Thus we could use q = 2 and r = 3 or we could just as well use q = 3 4

5 and r = 4. The following very important theorem, called The Division Algorithm, states that we can always choose the numbers q and r in such a way that 0 r < b and that there is only one way in which to do this. The tool needed to prove the Division Algorithm is the General Well Ordering Principle (Chapter 1, Section 1). Theorem 6 (The Division Algorithm) Let a and b be integers with b > 0. Then there are integers q and r such that a = qb + r and 0 r < b. Furthermore, there is only one choice of q and r that will make a = qb + r and 0 r < b both be true. Proof. Let the integers a and b with b > 0 be given and de ne the set S as S = fa qb j q 2 Z and a qb 0g. We want to apply the General Well Ordering Principle to the set S to conclude that S has a smallest member. In order to do this, we must show that S 6= ; and that there is some integer n 0 such that every member of S is greater than or equal to n 0. It is easy to see that we can take n 0 = 0, because part of the de nition of the set S is that every one of its members must be greater than or equal to 0. But we still need to make sure that S 6= ;. To see why this is so, we consider two possibilities: One possibility is that a 0. In this case, the number a 0 b is in S because 0 2 Z and a 0 b = a 0. The other possibility is that a < 0. In this case, the number a a b is in S because a 2 Z and a a b = a (1 b), and since a < 0 and b 1, it must be the case that a (1 b) 0. This reasoning shows that, in any case, S 6= ;. By the General Well Ordering Principle, S must have a smallest member. We will call this smallest member r. Since r 2 S, there must be some integer q such that a qb = r and it must be true that r 0. We have almost nished proving that rst assertion of the Division Algorithm Theorem, because we proved that there exist integers q and r such that a = qb + r and r 0. We still have to prove, though, that r < b. To prove this, we observe that r b = a qb b = a (q + 1) b. From the above equation, we can deduce that the number r b must be negative. Why? If it were the case that r b 0, then since q + 1 is an integer and r b = a (q + 1) b, it would be true that r b 2 S. However, 5

6 clearly r b < r (because b > 0), and so it is not possible that r b 2 S because r is the smallest member of S. We conclude that r b must be a negative number and hence that r < b. This completes the proof of the rst assertion of The Division Algorithm. We have proved that if a and b are any integers with b > 0, then there must exist integers q and r such that a = qb + r and 0 r < b. We are still left to prove the second assertion of the theorem, which is that the number q and r can be chosen in only one way in order to satisfy both of these criteria. Suppose that there are two di erent choices of q and r that both satisfy the conditions a = qb + r and 0 r < b. Call these di erent choices (q 1 ; r 1 ) and (q 2 ; r 2 ). Thus a = q 1 b + r 1, a = q 2 b + r 2, 0 r 1 < b, and 0 r 2 < b. For de niteness, let us suppose that r 1 r 2. Then 0 r 1 r 2 < b. Also which implies that q 1 b + r 1 = q 2 b + r 2 (q 1 q 2 ) b = r 2 r 1. Now observe that r 2 r 1 0 and also, since r 2 < b and r 1 0, then r 2 r 1 < b. Therefore 0 r 2 r 1 < b. From the above equation, we conclude that 0 (q 1 q 2 ) b < b. Since b > 0, we may divide all parts of the above inequality by b (without reversing the order of the inequality) to obtain 0 q 1 q 2 < 1. Since q 1 q 2 is an integer, it must be true that q 1 q 2 = 0. Thus q 1 = q 2. It then follows from the equation (q 1 q 2 ) b = r 2 r 1 that r 1 = r 2. This proves that there is only one choice of q and r for which both conditions a = qb + r and 0 r < b are satis ed. Our proof of the Division Algorithm Theorem is now complete. We will refer to the numbers q and r obtained in the division algorithm as the quotient and the remainder obtained in the division of a by b. Example 7 Apply the division algorithm to a = 356 and b = 56. (By apply the division algorithm, we mean to nd q and r such that 356 = q 56 + r where 0 r < 56.) 6

7 Solution 8 We might guess that 56 goes into 356 about 6 times. Let s check = 20. Our guess was right because we obtained a remainder of 20 which satis es 0 20 < 56. Example 9 Apply the division algorithm to a = 7 and b = 3. Solution 10 Since these are small numbers, no experimentation is required. We see that 7 = We know that this is the correct division algorithm represenation of 7 3 because the remainder is 1 and 0 1 < 3. Example 11 Apply the division algorithm to a = 3 and b = 7. Solution 12 7 goes into 3 zero times, so it looks like the quotient should be 0 and the remainder should be 3. In fact, we see that 3 = Example 13 Apply the division algorithm to a = 356 and b = 56. Solution 14 In a previous example, we found that It follows from this that 356 = = ( 20). However, remember that we want to nd an expression a = qb + r in which 0 r < b. Thus we don t want to have a negative remainder. What if we change the 6 to a 7? = 36 gives us 356 = and this is what we were looking for because 0 36 < Homework In Chapter 1, Section 2, pages 13 14, do all of the exercises (numbers 1 11). 7

### Elementary 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,

### 2. Integers and Algorithms Euclidean Algorithm. Euclidean Algorithm. Suppose a and b are integers

2. INTEGERS AND ALGORITHMS 155 2. Integers and Algorithms 2.1. Euclidean Algorithm. Euclidean Algorithm. Suppose a and b are integers with a b > 0. (1) Apply the division algorithm: a = bq + r, 0 r < b.

### Section 4.2: The Division Algorithm and Greatest Common Divisors

Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### 26 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

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### It 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

### ELEMENTARY NUMBER THEORY AND METHODS OF PROOF

CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Copyright Cengage Learning. All rights reserved.

### CS103X: Discrete Structures Homework Assignment 2: Solutions

CS103X: Discrete Structures Homework Assignment 2: Solutions Due February 1, 2008 Exercise 1 (10 Points). Prove or give a counterexample for the following: Use the Fundamental Theorem of Arithmetic to

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### ELEMENTARY NUMBER THEORY AND METHODS OF PROOF

CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### The Inverse of a Square Matrix

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### GREATEST COMMON DIVISOR

DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### 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.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### Chapter 6. Number Theory. 6.1 The Division Algorithm

Chapter 6 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,

### Solutions to Assignment 4

Solutions to Assignment 4 Math 412, Winter 2003 3.1.18 Define a new addition and multiplication on Z y a a + 1 and a a + a, where the operations on the right-hand side off the equal signs are ordinary

### Chapter 4: Binary Operations and Relations

c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

### 3. Applications of Number Theory

3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

### 1 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.

### A fairly quick tempo of solutions discussions can be kept during the arithmetic problems.

Distributivity and related number tricks Notes: No calculators are to be used Each group of exercises is preceded by a short discussion of the concepts involved and one or two examples to be worked out

### 1.3 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.

### 12 Greatest Common Divisors. The Euclidean Algorithm

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### Topics in Number Theory

Chapter 8 Topics in Number Theory 8.1 The Greatest Common Divisor Preview Activity 1 (The Greatest Common Divisor) 1. Explain what it means to say that a nonzero integer m divides an integer n. Recall

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### Reading 7 : Program Correctness

CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 7 : Program Correctness 7.1 Program Correctness Showing that a program is correct means that

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS

MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS ALLAN YASHINSKI Abstract. We explore the notion of well-definedness when defining functions whose domain is

### Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

### Mathematics is the queen of sciences and number theory is the queen of mathematics.

Number Theory Mathematics is the queen of sciences and number theory is the queen of mathematics. But why is it computer science? It turns out to be critical for cryptography! Carl Friedrich Gauss Division

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### 36 Fields of Rational Functions

36 Fields of Rational Functions The reader might have missed the familiar quotient rule (\f(f,g)) = f g fg in Lemma 35.5. It was missing because f is not a polynomial. g We now introduce these quotients

### Math 320 Course Notes. Chapter 7: Countable and Uncountable Sets

Math 320 Course Notes Magnhild Lien Chapter 7: Countable and Uncountable Sets Hilbert s Motel: Imagine a motel with in nitely many rooms numbered 1; 2; 3; 4 ; n; : One evening an "in nite" bus full with

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### Course 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

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Mathematical Induction

Mathematical S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! Mathematical S 0 S 1 S 2 S 3 S4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! S 4 Mathematical S 0 S 1 S 2 S 3 S5 S 6 S 7 S 8 S 9 S 10

### 4.3 Limit of a Sequence: Theorems

4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

### THE p-adic EXPANSION OF RATIONAL NUMBERS =

THE p-adic EXPANSION OF RATIONAL NUMBERS KEITH CONRAD 1. Introduction In the positive real numbers, the decimal expansion of every positive rational number is eventually periodic 1 (e.g., 1/55.381.381...)

### Notes from February 11

Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

### CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010

CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 1. Greatest common divisor Suppose a, b are two integers. If another integer d satisfies d a, d b, we call d a common divisor of a, b. Notice that as

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Algebra for Digital Communication

EPFL - Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain

### INTRODUCTION 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

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### 26 Ideals and Quotient Rings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Discrete Mathematics, Chapter 4: Number Theory and Cryptography

Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Elementary Number Theory

Elementary Number Theory Ahto Buldas December 3, 2016 Ahto Buldas Elementary Number Theory December 3, 2016 1 / 1 Division For any m > 0, we define Z m = {0, 1,... m 1} For any n, m Z (m > 0), there are

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### 5. Integers, Whole Numbers And Rational Numbers

5. Integers, Whole Numbers And Rational Numbers The set of natural numbers,, is the building block for most of the real number system. But is inadequate for measuring and describing physical quantities,

### Theorem 2. If x Q and y R \ Q, then. (a) x + y R \ Q, and. (b) xy Q.

Math 305 Fall 011 The Density of Q in R The following two theorems tell us what happens when we add and multiply by rational numbers. For the first one, we see that if we add or multiply two rational numbers

### Basic Properties of Rings

Basic Properties of Rings A ring is an algebraic structure with an addition operation and a multiplication operation. These operations are required to satisfy many (but not all!) familiar properties. Some

### B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range {0,1,...,N

### a = bq + r where 0 r < b.

Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

### 8 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.

### Fold again, the same way. We now have four equal regions: the shaded part remains the same, of the paper.

PRIMARY CONTENT MODULE V NUMBER SENSE: Rational Numbers T-12 Equivalent Fractions When do two fractions represent the same amount? Or, when do they represent the same point or number on a number line?

### Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions

Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a-27b. Section 8.3 on page 275 problems 1b, 8, 10a-10b, 14. Section 8.4 on page 279 problems

### MATH Fundamental Mathematics II.

MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

### The Real Numbers. Here we show one way to explicitly construct the real numbers R. First we need a definition.

The Real Numbers Here we show one way to explicitly construct the real numbers R. First we need a definition. Definitions/Notation: A sequence of rational numbers is a funtion f : N Q. Rather than write

### Session 11 Division, Properties of Equality and Equation Solving

Session 11 Division, Properties of Equality and Equation Solving How does the operation of division used in the two following problems relate to sets? An employer distributes \$18,3 equally in bonuses to

### 2.4 Multiplication of Integers. Recall that multiplication is defined as repeated addition from elementary school. For example, 5 6 = 6 5 = 30, since:

2.4 Multiplication of Integers Recall that multiplication is defined as repeated addition from elementary school. For example, 5 6 = 6 5 = 30, since: 5 6=6+6+6+6+6=30 6 5=5+5+5+5+5+5=30 To develop a rule

### Chapter 3: Elementary Number Theory and Methods of Proof. January 31, 2010

Chapter 3: Elementary Number Theory and Methods of Proof January 31, 2010 3.4 - Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Quotient-Remainder Theorem Given

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Matrices: 2.3 The Inverse of Matrices

September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### GCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!

GCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!!! Challenge Problem 2 (Mastermind) due Fri. 9/26 Find a fourth guess whose scoring will allow you to determine the secret code (repetitions are

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS. 1. Ideals in Commutative Rings In this section all groups and rings will be commutative.

ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS PETE L. CLARK 1. Ideals in Commutative Rings In this section all groups and rings will be commutative. 1.1. Basic definitions and examples. Let R be a (commutative!)

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

### Section 3.3 Equivalence Relations

1 Section 3.3 Purpose of Section To introduce the concept of an equivalence relation and show how it subdivides or partitions a set into distinct categories. Introduction Classifying objects and placing

### A rational number is a number that can be written as where a and b are integers and b 0.

S E L S O N Rational Numbers Goal: Perform operations on rational numbers. Vocabulary Rational number: Additive inverse: A rational number is a number that can be a written as where a and b are integers