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

Size: px
Start display at page:

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

Transcription

1 An Introduction to Proof-based Mathematics Harvard/MIT ESP: Summer HSSP Isabel Vogt Class Objectives Field Axioms Finite Fields Field Extensions Class 5: Fields and Field Extensions 1

2 1. Axioms for a field Loosely speaking, a field F is a set of elements for which the familiar operations of arithmetic are defined and behave in the usual way. Here is a set of axioms for a field. You can use them to prove theorems that are true for any field. Short version: (a) The elements of F form a Abelian group under addition. (b) The nonzero elements of F form a Abelian group under multiplication. (c) Distributive law: a(b + c) = ab + ac. Long version: (a) Addition is commutative: a + b = b + a. (b) Addition is associative: (a + b) + c = a + (b + c). (c) Additive identity: 0 such that a F, 0 + a = a + 0 = a. (d) Additive inverse: a F, a such that a + a = a + ( a) = 0. (e) Multiplication is associative: (ab)c = a(bc). (f) Multiplication is commutative: ab = ba. (g) Multiplicative identity: 1 such that a F, 1a = a. (h) Multiplicative inverse: a F {0}, a 1 such that a 1 a = 1. (i) Distributive law: a(b + c) = ab + ac. 2

3 2. Theorems about fields As opposed to the axioms for a group, this set of axioms for a field is not and independent set, meaning some things can be proven as theorems from the other axioms: namely the commutativity of addition. Some well-known laws of arithmetic are omitted from the list of axioms because they are easily proved as theorems. The most obvious omission is a F, 0a = 0. RTP: Given a field F, a F, 0a = 0 Do the integers Z form a field? 3

4 3. Finite fields Building upon what we know of modular arithmetic, we can make from the integers a finite field. Choose a prime number p. Break up the set of integers into p subsets. Each subset is named after the remainder when any of its elements is divided by p. [0] p = {m m = np, n Z} [1] p = {m m = np + 1, n Z} [a] p = {m m = np + a, n Z} Notice that [a + kp] p = [a] p for any k. There are only p sets, but each has many alternate names. These p infinite sets are the elements of the field Z p. Define addition by [a] p + [b] p = [a + b] p. Here a and b can be any names for the subsets, because the answer is independent of the choice of name. The rule is Add a and b, then divide by p and keep the remainder. What is the simplest name for [5] 7 + [4] 7? What is the simplest name for the additive inverse of [3] 7? Define multiplication by [a] p [b] p = [ab] p. Again a and b can be any names for the subsets, because the answer is independent of the choice of name. The rule is Multiply a and b, then divide by p and keep the remainder. What is the simplest name for [5] 7 [4] 7? Notice that here [12] 7 = 12 (mod 7), but the bracket notation merely specifies [12] 7 = [5] 7 as an element of the finite field Z 7 4

5 4. Multiplicative inverses in Z p For Z p, all the properties of a field follow directly from the corresponding properties of integer arithmetic except for the existence of a multiplicative inverse. RTP: a Z p where p is prime, a 1 Z p such that aa 1 = a 1 a = 1 There is an ingenious and efficient way to find the multiplicative inverse of [a] p based on Euclid s algorithm for finding the greatest common divisor of a and p, which we may have time to examine when we turn to number theory. In Z 3, [2] 3 [2] 3 = [4] 3 = [1] 3. In Z 5, [2] 5 [3] 5 = [6] 5 = [1] 5. In Z 5, [4] 5 [4] 5 = [16] 5 = [1] 5. Let s do a few more: In Z 7, what is the multiplicative inverse of [2] 7? In Z 7, what is the multiplicative inverse of [3] 7? In Z 7, what is the multiplicative inverse of [ 1] 7? (This is not the standard name, but it is a convenient one!) In Z 7, what is the multiplicative inverse of [6] 7? 5

6 5. Rational numbers The rational numbers Q form a field. You learned how to add and multiply them years ago! The multiplicative inverse of a is b as long as a 0. b a The rational numbers are not a big enough field for doing Euclidean geometry or calculus. Here are some irrational quantities: 2 π. most values of trig functions, exponentials, or logarithms. coordinates of most intersections of two circles. In cases such as these, it is necessary to extend a field in order to solve certain equations. For example, are there any solutions to the equation x 2 = 2 over the rational numbers? Does the equation x 2 = 2 have any solutions over the set {a+b 2 a, b Q}? 6

7 RTP: The set {a + b 2 a, b Q} forms a field. This is a field extension. We have a Q, and then we extend that set by some rational factor of an element not in Q. Similarly we could have extended the rationals by a factor of 3 to solve the equation x 2 = 3 and had another field extension. 7

8 6. Complex numbers as a field The real numbers (represented, for example, by infinite decimal numbers) also form a field. However, there are still polynomial equations that have no solutions in this field, notably i = 0 We can extend the field of real numbers by introducing a symbol i that represents a solution to this equation. Then, when we compute with the resulting complex numbers, we just replace i 2 by -1. The set notation is again a real number, plus a real number multiple of this element i: {a + bi a, b R} (a) Prove that the field of complex numbers is closed under multiplication. (b) Prove that every nonzero complex number has a multiplicative inverse. 8

9 7. Extending a finite field We have seen that we can: Extend the field of rational numbers by choosing an equation of the form x 2 k = 0, where k is a positive integer that is not a perfect square. The extended field has elements of the form a + b k, where a and b are in the unextended field of rational numbers and k is a positive number. We calculate in the extended field by using the replacement rule k 2 = k Extend the field of real numbers by choosing the equation i = 0. The extended field has elements of the form a + bi, where a and b are in the unextended field of real numbers and i is any solution to i = 0 we cannot tell one from the other. We calculate in the extended field by using the replacement rule i 2 = 1. In the finite field Z 2, the elements are [0] 2 and [1] 2 What is the additive inverse of [1] 2? We can use Z 2 to solve the following polynomials. What are the solutions? u 2 = 0 u = 0 u 2 + u = 0 Does the following polynomial have a solution in Z 2? u 2 + u + 1 = 0 To get around this, we need to extend our finite field! 9

10 We define u to be a solution of u 2 + u + 1 = 0. Given that u must have a multiplicative inverse, what other element must be in this field? We can find it by saying u 2 + u = 1, then factoring to get u(u + 1) = 1 Thus u and u + 1 are multiplicative inverses of each other. What are the additive inverses of u and u + 1? The resulting finite field, called F 4, has four elements, 0, 1, u, and u + 1. The replacement rule is u 2 = u 1 = u + 1. (a) Use the replacement rule to calculate (u + 1) 2. (b) Show that u + 1 is also a solution to the same quadratic equation. Just as we cannot distinguish i from i, we cannot distinguish u from u

11 (c) Make a table of addition facts for F 4 (d) Make a table of multiplication facts for F 4. 11

12 8. Multistage field extensions (a) The set of numbers of the form a + b 2 with rational coefficients, is a field. Show that, in this field, is a perfect square but is not. (b) Let u = What polynomial equation does u satisfy? (c) How can you use u to create an extended field? (d) Another way to make an extended field is to use two independent square roots. For example, use elements like a + b 2 + c 3 where a, b, and c are rational numbers. A representative element is x = What polynomial equation with integer coefficients does x satisfy? (e) In general, when you eliminate a square root by squaring both sides of an equation, you double the degree of the equation. If you write a number that involves n independent square roots, what degree of polynomial equation is it guaranteed to satisfy? 12

13 9. Theorems from field axioms RTP: Given a field F, a, b F, a + b = b + a using only the other axioms of a field and considering (1 + 1)(a + b) RTP: a F, a = ( 1)a. Remenber: a means the additive inverse of a, while 1 means the additive inverse of 1 13

14 RTP: a, b F, ( a)( b) = ab. RTP: If n is not prime, n = ab where neither a nor b is 1. Use this fact to show that Z n is not a field in this case. 14

15 10. Order of Finite Fields We can now see that all finite fields must be either of order p, where p is prime, or of some power of a prime p. We get the fields of order p from the sets Z p. We get the fields of order p n, n Z from field extensions to solve certain polynomial equations. For instance, we derived F 4 from solving a quadratic polynomial (degree 2) with coefficients from Z 2. (2) 2 = 4. We derived F 9 from solving a quadratic with coefficients from Z 3. (3) 3 = 9. We derive F 8 from solving a cubic (degree 3) over Z 2, and so on. 15

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

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

More information

Commutative Rings and Fields

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

More information

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

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

More information

REVIEW Chapter 1 The Real Number System

REVIEW 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 information

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step

More information

8 Primes and Modular Arithmetic

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.

More information

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

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

More information

Cool Results on Primes

Cool 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 information

Galois fields/1. (M3) There is an element 1 (not equal to 0) such that a 1 = a for all a.

Galois fields/1. (M3) There is an element 1 (not equal to 0) such that a 1 = a for all a. Galois fields 1 Fields A field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except by zero) can be performed, and satisfy the usual rules. More

More information

The theory of numbers

The theory of numbers 1 AXIOMS FOR THE INTEGERS 1 The theory of numbers UCU Foundations of Mathematics course 2017 Author: F. Beukers 1 Axioms for the integers Roughly speaking, number theory is the mathematics of the integers.

More information

Chapter 4 Finite Fields

Chapter 4 Finite Fields Chapter 4 Finite Fields Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers what constitutes a number

More information

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z: NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,

More information

2 Elementary number theory

2 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 information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 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 information

4 Powers of an Element; Cyclic Groups

4 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 information

Modular numbers and Error Correcting Codes. Introduction. Modular Arithmetic.

Modular numbers and Error Correcting Codes. Introduction. Modular Arithmetic. Modular numbers and Error Correcting Codes Introduction Modular Arithmetic Finite fields n-space over a finite field Error correcting codes Exercises Introduction. Data transmission is not normally perfect;

More information

Number Theory. Modular Arithmetic

Number Theory. Modular Arithmetic Number Theory The branch of mathematics that is important in IT security especially in cryptography. Deals only in integer numbers and the process can be done in a very fast manner. Modular Arithmetic

More information

Number Axioms. P. Danziger. A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b. a b S.

Number Axioms. P. Danziger. A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b. a b S. Appendix A Number Axioms P. Danziger 1 Number Axioms 1.1 Groups Definition 1 A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b and c S 0. (Closure) 1. (Associativity)

More information

NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4

NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4 Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as

More information

Classify, graph, and compare real numbers. Find and estimate square roots Identify and apply properties of real numbers.

Classify, graph, and compare real numbers. Find and estimate square roots Identify and apply properties of real numbers. Real Numbers and The Number Line Properties of Real Numbers Classify, graph, and compare real numbers. Find and estimate square roots Identify and apply properties of real numbers. Square root, radicand,

More information

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member

R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers

More information

Mathematics Foundation for College. Lesson Number 1. Lesson Number 1 Page 1

Mathematics Foundation for College. Lesson Number 1. Lesson Number 1 Page 1 Mathematics Foundation for College Lesson Number 1 Lesson Number 1 Page 1 Lesson Number 1 Topics to be Covered in this Lesson Sets, number systems, axioms, arithmetic operations, prime numbers and divisibility,

More information

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand

More information

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

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

More information

Mathematics for Cryptography

Mathematics for Cryptography Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1

More information

Foundations for Functions Knowledge and Skills: Foundations for Functions Knowledge and Skills:

Foundations for Functions Knowledge and Skills: Foundations for Functions Knowledge and Skills: Texas University Interscholastic League Contest Event: Number Sense This 80-question mental math contest covers all high school mathematics curricula. All answers must be derived without using scratch

More information

A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:

A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under

More information

Course Outcome Summary

Course Outcome Summary Course Information: Algebra 2 Description: Instruction Level: 10-12 Total Credits: 2.0 Prerequisites: Textbooks: Course Topics for this course include a review of Algebra 1 topics, solving equations, solving

More information

TECHNIQUES IN FACTORISATION

TECHNIQUES IN FACTORISATION TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation

More information

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

More information

Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc)

Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc) ALGEBRA (SMR Domain ) Algebraic Structures (SMR.) Skill a. Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc) Basic Properties

More information

Chapter 3: Factors, Roots, and Powers

Chapter 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 information

Prime Factorization and GCF. In my own words

Prime Factorization and GCF. In my own words Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite

More information

Fundamentals. Introduction. 1.1 Sets, inequalities, absolute value and properties of real numbers

Fundamentals. Introduction. 1.1 Sets, inequalities, absolute value and properties of real numbers Introduction This first chapter reviews some of the presumed knowledge for the course that is, mathematical knowledge that you must be familiar with before delving fully into the Mathematics Higher Level

More information

Solutions to Practice Final

Solutions to Practice Final s to Practice Final 1. (a) What is φ(0 100 ) where φ is Euler s φ-function? (b) Find an integer x such that 140x 1 (mod 01). Hint: gcd(140, 01) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100 ) = (

More information

Discrete Structures Lecture Solving Congruences. mathematician of the eighteenth century). Also, the equation gggggg(aa, bb) =

Discrete Structures Lecture Solving Congruences. mathematician of the eighteenth century). Also, the equation gggggg(aa, bb) = First Introduction Our goal is to solve equations having the form aaaa bb (mmmmmm mm). However, first we must discuss the last part of the previous section titled gcds as Linear Combinations THEOREM 6

More information

Table of Contents. 2013, Pearson Education, Inc.

Table of Contents. 2013, Pearson Education, Inc. Table of Contents Chapter 1 What is Number Theory? 1 Chapter Pythagorean Triples 5 Chapter 3 Pythagorean Triples and the Unit Circle 11 Chapter 4 Sums of Higher Powers and Fermat s Last Theorem 16 Chapter

More information

Introduction to Information Security

Introduction to Information Security Introduction to Information Security Lecture 5: Number Theory 007. 6. Prof. Byoungcheon Lee sultan (at) joongbu. ac. kr Information and Communications University Contents 1. Number Theory Divisibility

More information

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 11 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 real

More information

Order of Operations. Real numbers

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

More information

Evaluate algebraic expressions for given values of the variables.

Evaluate algebraic expressions for given values of the variables. Algebra I Unit Lesson Title Lesson Objectives 1 FOUNDATIONS OF ALGEBRA Variables and Expressions Exponents and Order of Operations Identify a variable expression and its components: variable, coefficient,

More information

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

2. Two binary operations (addition, denoted + and multiplication, denoted Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between

More information

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS

ZEROS 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 information

Rational Numbers. a) 5 is a rational number TRUE FALSE. is a rational number TRUE FALSE

Rational 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 information

SEVENTH EDITION and EXPANDED SEVENTH EDITION

SEVENTH EDITION and EXPANDED SEVENTH EDITION SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 10-1 Chapter 10 Mathematical Systems 10.1 Groups Definitions A mathematical system consists of a set of elements and at least one binary operation. A

More information

x y x y ax bx c x Algebra I Course Standards Gap 1 Gap 2 Comments a. Set up and solve problems following the correct order of operations (including proportions, percent, and absolute value) with rational

More information

function independent dependent domain range graph of the function The Vertical Line Test

function independent dependent domain range graph of the function The Vertical Line Test Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding

More information

Math1a Set 1 Solutions

Math1a 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 information

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics

More information

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals z Transform Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals (ii) Understanding the characteristics and properties

More information

5.1 Monomials. Algebra 2

5.1 Monomials. Algebra 2 . Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific

More information

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

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

More information

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

Numbers. 2.1 Integers. P(n) = n(n 4 5n 2 + 4) = n(n 2 1)(n 2 4) = (n 2)(n 1)n(n + 1)(n + 2); 120 = 2 Numbers 2.1 Integers You remember the definition of a prime number. On p. 7, we defined a prime number and formulated the Fundamental Theorem of Arithmetic. Numerous beautiful results can be presented

More information

Wednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory).

Wednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory). Wednesday, February 21 Today we will begin Course Notes Chapter 5 (Number Theory). 1 Return to Chapter 5 In discussing Methods of Proof (Chapter 3, Section 2) we introduced the divisibility relation from

More information

Day 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 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 information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,

More information

Executive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics:

Executive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics: Executive Assessment Math Review Although the following provides a review of some of the mathematical concepts of arithmetic and algebra, it is not intended to be a textbook. You should use this chapter

More information

22. The Quadratic Sieve and Elliptic Curves. 22.a The Quadratic Sieve

22. The Quadratic Sieve and Elliptic Curves. 22.a The Quadratic Sieve 22. The Quadratic Sieve and Elliptic Curves 22.a The Quadratic Sieve Sieve methods for finding primes or for finding factors of numbers are methods by which you take a set P of prime numbers one by one,

More information

Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings)

Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings) COMPETENCY 1.0 ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings)

More information

West Windsor-Plainsboro Regional School District Math A&E Grade 7

West Windsor-Plainsboro Regional School District Math A&E Grade 7 West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale

More information

3.4. ZEROS OF POLYNOMIAL FUNCTIONS

3.4. ZEROS OF POLYNOMIAL FUNCTIONS 3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find

More information

CSE 1400 Applied Discrete Mathematics Proofs

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

More information

Tenth Bit Bank Mathematics Real Numbers

Tenth Bit Bank Mathematics Real Numbers Tenth Bit Bank Mathematics Real Numbers 1. The rational number among the following is... i) 4.28 ii) 4.282828... iii) 4.288888... A) i) & ii) B) ii) & iii) C) i) & iii) D) All the above 2. A rational number

More information

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

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

More information

First Quarter Second Quarter Third Quarter Fourth Quarter Unit 1: Expressions and Operations 2.5 weeks/6 blocks

First Quarter Second Quarter Third Quarter Fourth Quarter Unit 1: Expressions and Operations 2.5 weeks/6 blocks Algebra 1/Algebra 1 Honors Pacing Guide Focus: Third Quarter First Quarter Second Quarter Third Quarter Fourth Quarter Unit 1: Expressions and Operations 2.5 weeks/6 blocks Unit 2: Equations 2.5 weeks/6

More information

Mathematical Reasoning & Proofs

Mathematical Reasoning & Proofs Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0

More information

Chapter 14: Divisibility and factorization

Chapter 14: Divisibility and factorization Chapter 14: Divisibility and factorization Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter

More information

Intermediate Algebra

Intermediate Algebra Intermediate Algebra 978-1-63545-084-2 To learn more about all our offerings Visit Knewton.com Source Author(s) (Text or Video) Title(s) Link (where applicable) Openstax Lyn Marecek, MaryAnne Anthony-Smith

More information

A number that can be written as, where p and q are integers and q Number.

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 information

Elementary Properties of the Integers

Elementary Properties of the Integers Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of

More information

CHAPTER 2 Review of Algebra

CHAPTER 2 Review of Algebra CHAPTER 2 Review of Algebra 2.1 Real Numbers The text assumes that you know these sets of numbers. You are asked to characterize each set in Problem 1 of Chapter 1 review. Name Symbol Set Examples Counting

More information

Associative property

Associative property Addition Associative property Closure property Commutative property Composite number Natural numbers (counting numbers) Distributive property for multiplication over addition Divisibility Divisor Factor

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS

More information

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions

MATH 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 information

Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.

Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers. Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one

More information

Algebra 2 Notes AII.7 Polynomials Part 2

Algebra 2 Notes AII.7 Polynomials Part 2 Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Name: Date: Block: Zeros of a Polynomial Function So far: o If we are given a zero (or factor or solution) of a polynomial function, we can use division

More information

a + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.

a + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c. Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called

More information

Properties of the Integers

Properties of the Integers Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called

More information

1.1.1 Algebraic Operations

1.1.1 Algebraic Operations 1.1.1 Algebraic Operations We need to learn how our basic algebraic operations interact. When confronted with many operations, we follow the order of operations: Parentheses Exponentials Multiplication

More information

PETERS TOWNSHIP HIGH SCHOOL

PETERS TOWNSHIP HIGH SCHOOL PETERS TOWNSHIP HIGH SCHOOL COURSE SYLLABUS: ALG EBRA 2 HONORS Course Overview and Essential Skills This course is an in-depth study of the language, concepts, and techniques of Algebra that will prepare

More information

Working with Square Roots. Return to Table of Contents

Working with Square Roots. Return to Table of Contents Working with Square Roots Return to Table of Contents 36 Square Roots Recall... * Teacher Notes 37 Square Roots All of these numbers can be written with a square. Since the square is the inverse of the

More information

Rings and modular arithmetic

Rings and modular arithmetic Chapter 8 Rings and modular arithmetic So far, we have been working with just one operation at a time. But standard number systems, such as Z, have two operations + and which interact. It is useful to

More information

32 Divisibility Theory in Integral Domains

32 Divisibility Theory in Integral Domains 3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible

More information

Radiological Control Technician Training Fundamental Academic Training Study Guide Phase I

Radiological Control Technician Training Fundamental Academic Training Study Guide Phase I Module 1.01 Basic Mathematics and Algebra Part 4 of 9 Radiological Control Technician Training Fundamental Academic Training Phase I Coordinated and Conducted for the Office of Health, Safety and Security

More information

Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm

Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. This is presented in more depth than what we

More information

Pacing Guide. Algebra II. Robert E. Lee High School Staunton City Schools Staunton, Virginia

Pacing Guide. Algebra II. Robert E. Lee High School Staunton City Schools Staunton, Virginia Pacing Guide Algebra II Robert E. Lee High School Staunton City Schools Staunton, Virginia 2010-2011 Algebra 2 - Pacing Guide 2010 2011 (SOL# s for the 2011 2012 school year are indicated in parentheses

More information

Intermediate Algebra with Applications

Intermediate Algebra with Applications Lakeshore Technical College 10-804-118 Intermediate Algebra with Applications Course Outcome Summary Course Information Alternate Title Description Total Credits 4 Total Hours 72 Pre/Corequisites Prerequisite

More information

ECEN 5022 Cryptography

ECEN 5022 Cryptography Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,

More information

Course Number 420 Title Algebra I Honors Grade 9 # of Days 60

Course Number 420 Title Algebra I Honors Grade 9 # of Days 60 Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number

More information

Mathematics. Algebra II Curriculum Guide. Curriculum Guide Revised 2016

Mathematics. Algebra II Curriculum Guide. Curriculum Guide Revised 2016 Mathematics Algebra II Curriculum Guide Curriculum Guide Revised 016 Intentionally Left Blank Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and

More information

Fundamentals of Pure Mathematics - Problem Sheet

Fundamentals of Pure Mathematics - Problem Sheet Fundamentals of Pure Mathematics - Problem Sheet ( ) = Straightforward but illustrates a basic idea (*) = Harder Note: R, Z denote the real numbers, integers, etc. assumed to be real numbers. In questions

More information

Algebra One Dictionary

Algebra One Dictionary Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.

More information

Ron Paul Curriculum Mathematics 8 Lesson List

Ron Paul Curriculum Mathematics 8 Lesson List Ron Paul Curriculum Mathematics 8 Lesson List 1 Introduction 2 Algebraic Addition 3 Algebraic Subtraction 4 Algebraic Multiplication 5 Week 1 Review 6 Algebraic Division 7 Powers and Exponents 8 Order

More information

MA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra

MA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra 0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus

More information

WORKSHEET 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: 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 information

Equations and Inequalities

Equations and Inequalities Algebra I SOL Expanded Test Blueprint Summary Table Blue Hyperlinks link to Understanding the Standards and Essential Knowledge, Skills, and Processes Reporting Category Algebra I Standards of Learning

More information

Axioms for the Real Number System

Axioms for the Real Number System Axioms for the Real Number System Math 361 Fall 2003 Page 1 of 9 The Real Number System The real number system consists of four parts: 1. A set (R). We will call the elements of this set real numbers,

More information

Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph below. Please do this worksheet in your notebook, not on this paper.

Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph below. Please do this worksheet in your notebook, not on this paper. Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph below Please do this worksheet in your notebook, not on this paper. A) For the solid line; calculate the average speed from: 1) 1:00 pm to

More information

18 Divisibility. and 0 r < d. Lemma Let n,d Z with d 0. If n = qd+r = q d+r with 0 r,r < d, then q = q and r = r.

18 Divisibility. and 0 r < d. Lemma Let n,d Z with d 0. If n = qd+r = q d+r with 0 r,r < d, then q = q and r = r. 118 18. DIVISIBILITY 18 Divisibility Chapter V Theory of the Integers One of the oldest surviving mathematical texts is Euclid s Elements, a collection of 13 books. This book, dating back to several hundred

More information

Why write proofs? Why not just test and repeat enough examples to confirm a theory?

Why write proofs? Why not just test and repeat enough examples to confirm a theory? P R E F A C E T O T H E S T U D E N T Welcome to the study of mathematical reasoning. The authors know that many students approach this material with some apprehension and uncertainty. Some students feel

More information