Divisibility of Natural Numbers
|
|
- Kerry Wilson
- 6 years ago
- Views:
Transcription
1 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 set theory to come up with rigorous definition for natural numbers. We defined four standard operations on natural numbers (+,, ). We have described in much detail various conceptual models for the natural numbers and observed how these numbers are a natural and foundational component of quantitative thinking. Now we will investigate these numbers on a deeper and more sophisticated level. The goal of this investigation is to build up the intellectual structure necessary to establish the existence and properties of more complex mathematical number systems, particularly the rational (fractions) and real numbers. Our most powerful tools in this section will turn out to be inductive and deductive reasoning. Divisibility. For any whole numbers a and b we know (by say, the repeated subtraction or division algorithm) that there are unique whole numbers q and r so that a = b q + r where 0 r < b. Here q is called the quotient and r is called the remainder. For example, consider the problem 14 3 = 4 R2. We can write 14 = When the remainder is r = 0 in such a representation we say that b divides a. That is, if there is a unique whole number q so that a = bq then b divides a. Or in other words, we say that b divides a when b divides into a with no remainder.
2 Warning: Never write a b when you mean to say a divides b. To say a divides b you must always write out the words. Example. 7 divides 35, 25 divides 100, 341 divides 0. 8 does not divide 21, 10 does not divide 51. Example. A number a is even if 2 divides a. A number a is odd if 2 does not divide a. So we know that if a is an even whole number that a = 2 n for some other whole number n. Also, suppose a is odd. Then a = 2n + r for some n and some 0 < r < 2. So the remainder must be 1. From this we may conclude that every odd number can be written as a = 2n + 1. We have already seen this idea before in the pattern recognition section. From now on, if we want to prove something is even or odd we will use these characterizations. Factors. If b divides a, then b is sometimes called factor or divisor of a and a is called a multiple of b. Example. Find the divisors of 8. Is 32 a multiple of 8? By guess and check, the divisors of 8 are 1,2,4,8. 32 is a multiple of 8 since 8 divides 32. Prime. A number is called prime if its only factors are 1 and itself.
3 Composite A number is called composite if it is not prime, that is if it has more than two distinct factors. As a technical point, we do not consider the number 1, prime or composite. Prime numbers are of special importance because they are the basic building block of all numbers. Indeed, any natural number (other than 1) can be written as the product of prime numbers. For example 180 = = = = We summarize this idea in a simple but important theorem. Theorem 1. Every natural number greater than 1 is a power of a prime or can be expressed as a product of powers of primes. This factorization is unique, apart from the ordering. That is, If we ignore the order in which we write the prime factors there is only one prime factorization of every natural number. If we know the prime factorization of a number we can make an orderly list of its divisors. (See page for two examples). To see why, suppose that a = p q 1 1 p q 2 2 p q k k. If b divides a, we know that p q 1 1 p q 2 2 p q k k = a = b q. This says that b must be some combination of powers of p 1, p 2,..., p k and that q must be all the factors which don t appear in a. To list the divisors, we make a list of all the possible ways to write products of powers of p 1, p 2,..., p k so that the product is less than or equal to a. them: Since primes are very important numbers, two questions quickly arise in our study of 1. How many primes are there?
4 2. How do we know that a number is prime? It turns out that there are infinitely many prime numbers. Proof. The first prime number is 2. The second prime number is 3. Consider the number = 7. Since by the division algorithm, there is only one way to write 7 = 2 q 1 + r 1 and 7 = 3 q 2 + r 2, we discover that the number 7 is not divisible by 2 or by 3. Since every number larger than 1 has a prime factorization, there must be a prime number other than 3 which divides into 7. In fact, this number happens to be 7. Now, we build the number = 43. Using the division algorithm, we can see that none of 2,3, or 7 divides 43. So there must be a new prime other than 7 which divides 43. This number happens to be 43. We continue and build the number = Again, this number is not divisible by 2,3,7, or 43. So there must be a new prime other than 43 which divides In this case, we could use either the prime 13 or 139. We continue and write The idea is that we can keep doing this process forever and at each step get a new prime number. So there must be an infinite number of primes. To determine if a number is prime the following theorem is very helpful. Theorem 2. If n is a composite number, then there is a prime p such that p divides n and p n. Proof. Since n is composite we can write n = a b where we assume that a is the smaller factor (that is a b). Now since a b multiplying by a we get So a 2 ab = n a n. Now a is a whole number bigger than 1, and so has a prime factorization. If p is a prime in this factorization then p a is a divisor of a and it is also a divisor of b. So there is a prime p n which divides n.
5 Consequently, by the rule of indirect reasoning, if there is no prime p n which divides n then n is prime. (Why?) For examples of using this theorem, see
A 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 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 informationDivisibility = 16, = 9, = 2, = 5. (Negative!)
Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties
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 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 informationnot 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 informationGCSE AQA Mathematics. Numbers
GCSE Mathematics Numbers Md Marufur Rahman Msc Sustainable Energy Systems Beng (Hons) Mechanical Engineering Bsc (Hons) Computer science & engineering GCSE AQA Mathematics 215/16 Table of Contents Introduction:...
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 informationSection 4.3. Polynomial Division; The Remainder Theorem and the Factor Theorem
Section 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem Polynomial Long Division Let s compute 823 5 : Example of Long Division of Numbers Example of Long Division of Numbers Let
More informationDirect Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24
Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using
More informationMultiplying and dividing integers Student Activity Sheet 2; use with Exploring Patterns of multiplying and dividing integers
1. Study the multiplication problems in the tables. [EX2, page 1] Factors Product Factors Product (4)(3) 12 ( 4)(3) 12 (4)(2) 8 ( 4)(2) 8 (4)(1) 4 ( 4)(1) 4 (4)(0) 0 ( 4)(0) 0 (4)( 1) 4 ( 4)( 1) 4 (4)(
More informationMATH10040: Numbers and Functions Homework 1: Solutions
MATH10040: Numbers and Functions Homework 1: Solutions 1. Prove that a Z and if 3 divides into a then 3 divides a. Solution: The statement to be proved is equivalent to the statement: For any a N, if 3
More informationRings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
More informationOn the polynomial x(x + 1)(x + 2)(x + 3)
On the polynomial x(x + 1)(x + 2)(x + 3) Warren Sinnott, Steven J Miller, Cosmin Roman February 27th, 2004 Abstract We show that x(x + 1)(x + 2)(x + 3) is never a perfect square or cube for x a positive
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 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 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 informationChapter 3 Basic Number Theory
Chapter 3 Basic Number Theory What is Number Theory? Well... What is Number Theory? Well... Number Theory The study of the natural numbers (Z + ), especially the relationship between different sorts of
More informationREVIEW Chapter 1 The Real Number System
REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }
More informationIntroduction to Induction (LAMC, 10/14/07)
Introduction to Induction (LAMC, 10/14/07) Olga Radko October 1, 007 1 Definitions The Method of Mathematical Induction (MMI) is usually stated as one of the axioms of the natural numbers (so-called Peano
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
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 informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More information2 Elementary number theory
2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},
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 information5.1. Primes, Composites, and Tests for Divisibility
CHAPTER 5 Number Theory 5.1. Primes, Composites, and Tests for Divisibility Definition. A counting number with exactly two di erent factors is called a prime number or a prime. A counting number with more
More informationBasic Principles of Algebra
Basic Principles of Algebra Algebra is the part of mathematics dealing with discovering unknown numbers in an equation. It involves the use of different types of numbers: natural (1, 2, 100, 763 etc.),
More informationHomework 3, solutions
Homework 3, solutions Problem 1. Read the proof of Proposition 1.22 (page 32) in the book. Using simialr method prove that there are infinitely many prime numbers of the form 3n 2. Solution. Note that
More informationNumber Theory. Introduction
Number Theory Introduction Number theory is the branch of algebra which studies the properties of the integers. While we may from time to time use real or even complex numbers as tools to help us study
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 informationDefinition For a set F, a polynomial over F with variable x is of the form
*6. Polynomials Definition For a set F, a polynomial over F with variable x is of the form a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 1 x + a 0, where a n, a n 1,..., a 1, a 0 F. The a i, 0 i n are the
More informationPolynomial Division. You may also see this kind of problem written like this: Perform the division x2 +2x 3
Polynomial Division 5015 You do polynomial division the way you do long division of numbers It s difficult to describe the general procedure in words, so I ll work through some eamples stepbystep Eample
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 informationFour Basic Sets. Divisors
Four Basic Sets Z = the integers Q = the rationals R = the real numbers C = the complex numbers Divisors Definition. Suppose a 0 and b = ax, where a, b, and x are integers. Then we say a divides b (or
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 informationQ 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?
2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More information1.1 Inductive Reasoning filled in.notebook August 20, 2015
1.1 Inductive Reasoning 1 Vocabulary Natural or Counting Numbers Ellipsis Scientific Method Hypothesis or Conjecture Counterexample 2 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole
More information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
More informationLong division for integers
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
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Factorise each polynomial: a) x 2 6x + 5 b) x 2 16 c) 9x 2 25 2) Simplify the following algebraic fractions fully: a) x 2
More informationFibonacci mod k. In this section, we examine the question of which terms of the Fibonacci sequence have a given divisor k.
Fibonacci mod k I start by giving out a table of the first 0 Fibonacci numbers actually the first, because we begin with u 0 =0 (and I have a reason for that which will soon become apparent). Okay, which
More informationLiquid-in-glass thermometer
Liquid-in-glass thermometer Objectives The objective of this experiment is to introduce some basic concepts in measurement, and to develop good measurement habits. In the first section, we will develop
More informationMATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions
MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 8
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 8 Polynomials Polynomials constitute a rich class of functions which are both easy to describe and widely applicable in
More informationNumber Theory. Number Theory. 6.1 Number Theory
6.1 Number Theory Number Theory The numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory. 2 2010 Pearson Education,
More information3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section
More informationa = qb + r where 0 r < b. Proof. We first prove this result under the additional assumption that b > 0 is a natural number. Let
2. Induction and the division algorithm The main method to prove results about the natural numbers is to use induction. We recall some of the details and at the same time present the material in a different
More information35 Chapter CHAPTER 4: Mathematical Proof
35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21
More informationDiscrete Mathematics and Probability Theory Fall 2014 Anant Sahai Note 7
EECS 70 Discrete Mathematics and Probability Theory Fall 2014 Anant Sahai Note 7 Polynomials Polynomials constitute a rich class of functions which are both easy to describe and widely applicable in topics
More informationEXAMPLES OF MORDELL S EQUATION
EXAMPLES OF MORDELL S EQUATION KEITH CONRAD 1. Introduction The equation y 2 = x 3 +k, for k Z, is called Mordell s equation 1 on account of Mordell s long interest in it throughout his life. A natural
More information1 Modular Arithmetic Grade Level/Prerequisites: Time: Materials: Preparation: Objectives: Navajo Nation Math Circle Connection
1 Modular Arithmetic Students will explore a type of arithmetic that results from arranging numbers in circles instead of on a number line. Students make and prove conjectures about patterns relating to
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More information6: Polynomials and Polynomial Functions
6: Polynomials and Polynomial Functions 6-1: Polynomial Functions Okay you know what a variable is A term is a product of constants and powers of variables (for example: x ; 5xy ) For now, let's restrict
More informationMath-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials
Math- Lesson - Finding Zeroes of NOT nice rd Degree Polynomials f ( ) 4 5 8 Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term.
More informationAN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS
AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS The integers are the set 1. Groups, Rings, and Fields: Basic Examples Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, and we can add, subtract, and multiply
More informationPreCalculus: Semester 1 Final Exam Review
Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain
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 informationUNIT 5 VOCABULARY: POLYNOMIALS
3º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Monomials A monomial is an algebraic expression consisting of only one term. A monomial can be any of the following: A constant: 2 4-5 A variable:
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 informationThe numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory.
6.1 Number Theory Number Theory The numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory. 2010 Pearson Education,
More information#26: Number Theory, Part I: Divisibility
#26: Number Theory, Part I: Divisibility and Primality April 25, 2009 This week, we will spend some time studying the basics of number theory, which is essentially the study of the natural numbers (0,
More informationPolynomial and Synthetic Division
Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1
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 informationCS 360, Winter Morphology of Proof: An introduction to rigorous proof techniques
CS 30, Winter 2011 Morphology of Proof: An introduction to rigorous proof techniques 1 Methodology of Proof An example Deep down, all theorems are of the form If A then B, though they may be expressed
More informationAlgebra III and Trigonometry Summer Assignment
Algebra III and Trigonometry Summer Assignment Welcome to Algebra III and Trigonometry! This summer assignment is a review of the skills you learned in Algebra II. Please bring this assignment with you
More information. As the binomial coefficients are integers we have that. 2 n(n 1).
Math 580 Homework. 1. Divisibility. Definition 1. Let a, b be integers with a 0. Then b divides b iff there is an integer k such that b = ka. In the case we write a b. In this case we also say a is a factor
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.4 Strong Mathematical Induction and the Well-Ordering Principle for the Integers Copyright
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationEXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
More informationWarm-Up. Use long division to divide 5 into
Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use
More informationFACTORIZATION AND THE PRIMES
I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3,... of ordinary
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 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 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 informationAQA Level 2 Further mathematics Further algebra. Section 4: Proof and sequences
AQA Level 2 Further mathematics Further algebra Section 4: Proof and sequences Notes and Examples These notes contain subsections on Algebraic proof Sequences The limit of a sequence Algebraic proof Proof
More informationINFINITE SUMS. In this chapter, let s take that power to infinity! And it will be equally natural and straightforward.
EXPLODING DOTS CHAPTER 7 INFINITE SUMS In the previous chapter we played with the machine and saw the power of that machine to make advanced school algebra so natural and straightforward. In this chapter,
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
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 informationAcademic-Clinic.com BASIC ARITHMETIC AND ALGEBRA POINTERS. Whole (natural) numbers. Arithmetical operations
BASIC ARITHMETIC AND ALGEBRA POINTERS Whole (natural) numbers Natural numbers numbers, which appear as a result of calculus of single subjects: peoples, animals, birds, trees, different wares and so on.
More informationChapter 6. Systems of Equations and Inequalities
Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system
More informationCh. 12 Higher Degree Equations Rational Root
Ch. 12 Higher Degree Equations Rational Root Sec 1. Synthetic Substitution ~ Division of Polynomials This first section was covered in the chapter on polynomial operations. I m reprinting it here because
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 informationMath1a Set 1 Solutions
Math1a Set 1 Solutions October 15, 2018 Problem 1. (a) For all x, y, z Z we have (i) x x since x x = 0 is a multiple of 7. (ii) If x y then there is a k Z such that x y = 7k. So, y x = (x y) = 7k is also
More informationINTRODUCTION TO FRACTIONS
Tallahassee Community College 16 INTRODUCTION TO FRACTIONS Figure A Figure A (Use for 1-5) 4. a. The numerator of 8 is. 1. How many parts are there in this circle?. How many parts of the circle are shaded?.
More informationPolynomial and Synthetic Division
Chapter Polynomial and Rational Functions y. f. f Common function: y Horizontal shift of three units to the left, vertical shrink Transformation: Vertical each y-value is multiplied stretch each y-value
More informationMathematics 228(Q1), Assignment 2 Solutions
Mathematics 228(Q1), Assignment 2 Solutions Exercise 1.(10 marks) A natural number n > 1 is said to be square free if d N with d 2 n implies d = 1. Show that n is square free if and only if n = p 1 p k
More informationIntegers and Division
Integers and Division Notations Z: set of integers N : set of natural numbers R: set of real numbers Z + : set of positive integers Some elements of number theory are needed in: Data structures, Random
More informationLesson 3: Using Linear Combinations to Solve a System of Equations
Lesson 3: Using Linear Combinations to Solve a System of Equations Steps for Using Linear Combinations to Solve a System of Equations 1. 2. 3. 4. 5. Example 1 Solve the following system using the linear
More informationb) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the
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 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 informationHere is another characterization of prime numbers.
Here is another characterization of prime numbers. Theorem p is prime it has no divisors d that satisfy < d p. Proof [ ] If p is prime then it has no divisors d that satisfy < d < p, so clearly no divisor
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationHow might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5
8.4 1 8.4 Partial Fractions Consider the following integral. 13 2x (1) x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that 13 2x (2) x 2 x 2 = 3 x 2 5 x + 1 We could then
More informationAs the title suggests, we tackle three famous theorems in this chapter. 4.1 The Fundamental Theorem of Arithmetic
Chapter 4 Three Famous Theorems As the title suggests, we tackle three famous theorems in this chapter. 4.1 The Fundamental Theorem of Arithmetic The goal of this section is to prove The Fundamental Theorem
More informationLP03 Chapter 5. A prime number is a natural number greater that 1 that has only itself and 1 as factors. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
LP03 Chapter 5 Prime Numbers A prime number is a natural number greater that 1 that has only itself and 1 as factors. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, Question 1 Find the prime factorization of 120.
More informationQ 1 Find the square root of 729. 6. Squares and Square Roots Q 2 Fill in the blank using the given pattern. 7 2 = 49 67 2 = 4489 667 2 = 444889 6667 2 = Q 3 Without adding find the sum of 1 + 3 + 5 + 7
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 information