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The Resource Problems and proofs in numbers and algebra, Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
Problems and proofs in numbers and algebra, Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
Resource Information
The item Problems and proofs in numbers and algebra, Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.This item is available to borrow from all library branches.
Resource Information
The item Problems and proofs in numbers and algebra, Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.
This item is available to borrow from all library branches.
 Summary
 Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets indepth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications indepth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles. The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to ĺlproveĺl or ĺlsolveĺl complex problems. This method of instruction is augmented by examining applications of number theory in systems such as RSA cryptography, Universal Product Code (UPC), and International Standard Book Number (ISBN). The numerous problems and examples included in each section reward curiosity and insightfulness over more simplistic approaches. Each problem set begins with a few easy problems, progressing to problems or proofs with multistep solutions. Exercises in the text stay close to the examples of the section, allowing students the immediate opportunity to practice developing techniques.℗l Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge
 Language
 eng
 Extent
 1 online resource (x, 223 pages)
 Contents

 Preface; Contents; Part I The Integers; 1 Number Concepts, Prime Numbers, and the Division Algorithm; 1.1 Beginning Number Concepts and Prime Numbers; 1.2 Divisibility of Some Combinations of Integers; 1.3 Long Division: The Division Algorithm; 1.4 Tests for Divisibility in Base Ten; 1.5 Binary and Other Number Systems; 1.5.1 Conversion Between Binary and Decimal; 1.5.2 Conversion from Decimal to Binary; 1.5.3 Arithmetic in Binary Systems; Addition of Binary Numbers; Multiplication of Binary Numbers; Subtraction in the Binary System; Division in the Binary System
 1.5.4 Duodecimal Number SystemConversion from Decimal to Duodecimal System; Conversion from Duodecimal to Decimal System; 2 Greatest Common Divisors, Diophantine Equations, and Combinatorics; 2.1 GCD and LCM Through the Fundamental Theorem of Arithmetic; 2.2 GCD, the Euclidean Algorithm and Its Byproducts; 2.3 Linear Equations with Integer Solutions: Diophantine Equations; 2.4 A Brief Introduction to Combinatorics; 2.5 Linear Diophantine Equations and Counting; 3 Equivalence Classes with Applications to Clock Arithmetic and Fractions; 3.1 Equivalence Relations and Equivalence Classes
 3.2 Modular (Clock) Arithmetic Through Equivalence Relations3.3 Fractions Through Equivalence Relations; 3.4 Integers Modular n and Applications; 3.4.1 RSA Cryptosystem; 3.4.2 UPC and ISBN (See Gallin and Winters Gallian1988, Rosen Rosen2007); Part II The Algebra of Polynomials and Linear Systems; 4 Polynomials and the Division Algorithm; 4.1 Addition and Multiplication of Polynomials; 4.2 Divisibility, Quotients and Remainders of Polynomials; 4.3 The Remainder Theorem; 4.4 Synthetic Division; 5 Factoring Polynomials, Their Roots, and Some Applications
 5.1 Factoring Polynomials and Their Roots5.2 Rational Roots of Polynomials; 5.2.1 Appendix to Sect. 5.2: A Brief Review of Factoring Quadratics; 5.3 Greatest Common Divisors and Least Common Multiples for Polynomials; 6 Matrices and Systems of Linear Equations ; 6.1 Matrix Operations; 6.2 Systems of Linear Equations in the Plane; 6.3 Systems of Linear Equations in Euclidean nSpace; 6.4 System of Linear Equations: Cramer's Rule; 6.5 Applications of Matrix Operations to the GCD; 6.6 Evaluations of Determinants of 33 Matrices; 6.7 Application of Determinants (Line and Area); Selected Answers
 Section 1.1Section 1.2; Section 1.3; Section 1.4; Section 1.5; Section 2.1; Section 2.2; Section 2.3; Section 2.4; Section 2.5; Section 3.1; Section 3.2; Section 3.3; Section 3.4; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 5.1; Section 5.2; Section 5.3; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; References
 Isbn
 9783319144276
 Label
 Problems and proofs in numbers and algebra
 Title
 Problems and proofs in numbers and algebra
 Statement of responsibility
 Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
 Subject

 Proof theory
 Mathematics
 General Algebraic Systems
 Algebra
 Mathematics  Algebra  Abstract
 Electronic books
 Mathematical Logic and Foundations
 Proof theory
 Algebra
 Physical Sciences & Mathematics
 Number Theory
 Number theory
 Algebra
 Mathematics  Logic
 Algebra
 Mathematical foundations
 Mathematics  Number Theory
 Mathematics
 Number theory
 Number theory
 Language
 eng
 Summary
 Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets indepth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications indepth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles. The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to ĺlproveĺl or ĺlsolveĺl complex problems. This method of instruction is augmented by examining applications of number theory in systems such as RSA cryptography, Universal Product Code (UPC), and International Standard Book Number (ISBN). The numerous problems and examples included in each section reward curiosity and insightfulness over more simplistic approaches. Each problem set begins with a few easy problems, progressing to problems or proofs with multistep solutions. Exercises in the text stay close to the examples of the section, allowing students the immediate opportunity to practice developing techniques.℗l Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge
 Cataloging source
 GW5XE
 http://library.link/vocab/creatorDate
 1945
 http://library.link/vocab/creatorName
 Millman, Richard S.
 Dewey number
 512.7
 Illustrations
 illustrations
 Index
 no index present
 LC call number
 QA241
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 http://library.link/vocab/relatedWorkOrContributorDate
 1941
 http://library.link/vocab/relatedWorkOrContributorName

 Shiue, Peter JauShyong
 Kahn, Eric Brendan
 http://library.link/vocab/subjectName

 Number theory
 Proof theory
 Algebra
 Algebra
 Number theory
 Proof theory
 Mathematics
 Physical Sciences & Mathematics
 Algebra
 Mathematics
 General Algebraic Systems
 Number Theory
 Mathematical Logic and Foundations
 Mathematics
 Mathematics
 Number theory
 Mathematical foundations
 Mathematics
 Algebra
 Label
 Problems and proofs in numbers and algebra, Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references
 Carrier category
 online resource
 Carrier category code
 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code
 txt
 Content type MARC source
 rdacontent
 Contents

 Preface; Contents; Part I The Integers; 1 Number Concepts, Prime Numbers, and the Division Algorithm; 1.1 Beginning Number Concepts and Prime Numbers; 1.2 Divisibility of Some Combinations of Integers; 1.3 Long Division: The Division Algorithm; 1.4 Tests for Divisibility in Base Ten; 1.5 Binary and Other Number Systems; 1.5.1 Conversion Between Binary and Decimal; 1.5.2 Conversion from Decimal to Binary; 1.5.3 Arithmetic in Binary Systems; Addition of Binary Numbers; Multiplication of Binary Numbers; Subtraction in the Binary System; Division in the Binary System
 1.5.4 Duodecimal Number SystemConversion from Decimal to Duodecimal System; Conversion from Duodecimal to Decimal System; 2 Greatest Common Divisors, Diophantine Equations, and Combinatorics; 2.1 GCD and LCM Through the Fundamental Theorem of Arithmetic; 2.2 GCD, the Euclidean Algorithm and Its Byproducts; 2.3 Linear Equations with Integer Solutions: Diophantine Equations; 2.4 A Brief Introduction to Combinatorics; 2.5 Linear Diophantine Equations and Counting; 3 Equivalence Classes with Applications to Clock Arithmetic and Fractions; 3.1 Equivalence Relations and Equivalence Classes
 3.2 Modular (Clock) Arithmetic Through Equivalence Relations3.3 Fractions Through Equivalence Relations; 3.4 Integers Modular n and Applications; 3.4.1 RSA Cryptosystem; 3.4.2 UPC and ISBN (See Gallin and Winters Gallian1988, Rosen Rosen2007); Part II The Algebra of Polynomials and Linear Systems; 4 Polynomials and the Division Algorithm; 4.1 Addition and Multiplication of Polynomials; 4.2 Divisibility, Quotients and Remainders of Polynomials; 4.3 The Remainder Theorem; 4.4 Synthetic Division; 5 Factoring Polynomials, Their Roots, and Some Applications
 5.1 Factoring Polynomials and Their Roots5.2 Rational Roots of Polynomials; 5.2.1 Appendix to Sect. 5.2: A Brief Review of Factoring Quadratics; 5.3 Greatest Common Divisors and Least Common Multiples for Polynomials; 6 Matrices and Systems of Linear Equations ; 6.1 Matrix Operations; 6.2 Systems of Linear Equations in the Plane; 6.3 Systems of Linear Equations in Euclidean nSpace; 6.4 System of Linear Equations: Cramer's Rule; 6.5 Applications of Matrix Operations to the GCD; 6.6 Evaluations of Determinants of 33 Matrices; 6.7 Application of Determinants (Line and Area); Selected Answers
 Section 1.1Section 1.2; Section 1.3; Section 1.4; Section 1.5; Section 2.1; Section 2.2; Section 2.3; Section 2.4; Section 2.5; Section 3.1; Section 3.2; Section 3.3; Section 3.4; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 5.1; Section 5.2; Section 5.3; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; References
 Dimensions
 unknown
 Extent
 1 online resource (x, 223 pages)
 File format
 unknown
 Form of item
 online
 Isbn
 9783319144276
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code
 c
 Note
 SpringerLink
 Other control number
 10.1007/9783319144276
 Other physical details
 illustrations
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number

 (OCoLC)903532077
 (OCoLC)ocn903532077
 Label
 Problems and proofs in numbers and algebra, Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references
 Carrier category
 online resource
 Carrier category code
 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code
 txt
 Content type MARC source
 rdacontent
 Contents

 Preface; Contents; Part I The Integers; 1 Number Concepts, Prime Numbers, and the Division Algorithm; 1.1 Beginning Number Concepts and Prime Numbers; 1.2 Divisibility of Some Combinations of Integers; 1.3 Long Division: The Division Algorithm; 1.4 Tests for Divisibility in Base Ten; 1.5 Binary and Other Number Systems; 1.5.1 Conversion Between Binary and Decimal; 1.5.2 Conversion from Decimal to Binary; 1.5.3 Arithmetic in Binary Systems; Addition of Binary Numbers; Multiplication of Binary Numbers; Subtraction in the Binary System; Division in the Binary System
 1.5.4 Duodecimal Number SystemConversion from Decimal to Duodecimal System; Conversion from Duodecimal to Decimal System; 2 Greatest Common Divisors, Diophantine Equations, and Combinatorics; 2.1 GCD and LCM Through the Fundamental Theorem of Arithmetic; 2.2 GCD, the Euclidean Algorithm and Its Byproducts; 2.3 Linear Equations with Integer Solutions: Diophantine Equations; 2.4 A Brief Introduction to Combinatorics; 2.5 Linear Diophantine Equations and Counting; 3 Equivalence Classes with Applications to Clock Arithmetic and Fractions; 3.1 Equivalence Relations and Equivalence Classes
 3.2 Modular (Clock) Arithmetic Through Equivalence Relations3.3 Fractions Through Equivalence Relations; 3.4 Integers Modular n and Applications; 3.4.1 RSA Cryptosystem; 3.4.2 UPC and ISBN (See Gallin and Winters Gallian1988, Rosen Rosen2007); Part II The Algebra of Polynomials and Linear Systems; 4 Polynomials and the Division Algorithm; 4.1 Addition and Multiplication of Polynomials; 4.2 Divisibility, Quotients and Remainders of Polynomials; 4.3 The Remainder Theorem; 4.4 Synthetic Division; 5 Factoring Polynomials, Their Roots, and Some Applications
 5.1 Factoring Polynomials and Their Roots5.2 Rational Roots of Polynomials; 5.2.1 Appendix to Sect. 5.2: A Brief Review of Factoring Quadratics; 5.3 Greatest Common Divisors and Least Common Multiples for Polynomials; 6 Matrices and Systems of Linear Equations ; 6.1 Matrix Operations; 6.2 Systems of Linear Equations in the Plane; 6.3 Systems of Linear Equations in Euclidean nSpace; 6.4 System of Linear Equations: Cramer's Rule; 6.5 Applications of Matrix Operations to the GCD; 6.6 Evaluations of Determinants of 33 Matrices; 6.7 Application of Determinants (Line and Area); Selected Answers
 Section 1.1Section 1.2; Section 1.3; Section 1.4; Section 1.5; Section 2.1; Section 2.2; Section 2.3; Section 2.4; Section 2.5; Section 3.1; Section 3.2; Section 3.3; Section 3.4; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 5.1; Section 5.2; Section 5.3; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; References
 Dimensions
 unknown
 Extent
 1 online resource (x, 223 pages)
 File format
 unknown
 Form of item
 online
 Isbn
 9783319144276
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code
 c
 Note
 SpringerLink
 Other control number
 10.1007/9783319144276
 Other physical details
 illustrations
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number

 (OCoLC)903532077
 (OCoLC)ocn903532077
Subject
 Algebra
 Algebra
 Algebra
 Algebra
 Electronic books
 General Algebraic Systems
 Mathematical Logic and Foundations
 Mathematical foundations
 Mathematics
 Mathematics
 Mathematics  Algebra  Abstract
 Mathematics  Logic
 Mathematics  Number Theory
 Number Theory
 Number theory
 Number theory
 Number theory
 Physical Sciences & Mathematics
 Proof theory
 Proof theory
Genre
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