MATHEMATICAL OBJECTS in

Size: px
Start display at page:

Download "MATHEMATICAL OBJECTS in"

Transcription

1 MATHEMATICAL OBJECTS in Computational Tools in a Unified Object-Oriented Approach Yair CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business A CHAPMAN St HALL BOOK

2 Contents Part I Numbers* 1 1 Natural Numbers The Need for Natural Numbers Mathematical Induction Unboundedness Infinity Adding Natural Numbers Recursion The Addition Function Stack of Calls Multiplying Natural Numbers One and Zero Decimal Representation Binary Representation Prime Numbers Prime Factors Mathematical Induction in Proofs The Greatest Common Divisor Least Common Multiple The Factorial Function Exercises 18 2 Integer Numbers Negative Numbers The Prisoner Problem The Integer Numbers The Number Axis Angles of Numbers Exercises 25 3 Rational Numbers Rational Numbers The Unique Form Adding Rational Numbers Multiplying Rational Numbers Periodic Decimal Representation 29

3 3.6 Diverging Series The Harmonic Series Converging Series Finite Power Series Infinite Power Series Periodic Decimal Fractions Exercises 36 4 Real Numbers The Square Root of The Least-Upper-Bound Axiom The Real Numbers Decimal Representation of v Irrational Numbers Transcendental Numbers The Natural Exponent Exercises 45 5 Complex Numbers The Imaginary Number The Number Plane Sine and Cosine Adding Complex Numbers Multiplying Complex Numbers The Sine and Cosine Theorems Exercises 54 Part II Geometrical Objects* 57 6 Euclidean Geometry Points and Lines Rays and Intervals Comparing Intervals Ratios between Intervals Angles Comparing Angles Corresponding and Alternate Angles The Reversed Corresponding-Angle Theorem Parallel Lines - The Uniqueness Theorem Triangles Similar and Identical Triangles Isosceles Triangles Pythagoras' Axiom Sum of Edges The Longer Edge 81

4 6.16 Tales' Theorem The Reversed Tales' Theorem Circles Tangents Properties of the Tangent Exercises 93 7 Analytic Geometry God and the Origin Numbers and Points Lines - Sets of Points Hierarchy of Objects Half-Planes Angles Triangles Circles Exercises 101 Part III Composite Mathematical Objects Sets Alice in Wonderland Sets and Containers Russell's Paradox The Empty Set Sets and Natural Numbers The Order of the Natural Numbers Mappings and Cardinality Ordered Sets and Sequences Ill 8.9 Infinite Sets Ill 8.10 Enumerable Sets The Set of Integer Numbers Product of Sets Equivalence of Sets The Set of Rational Numbers Arbitrarily Long Finite Sequences Function Sets Cardinality of Function Sets Nonenumerable Sets Cardinality of the Real Axis Cardinality of the Plane Cardinality of the Multidimensional Space Larger Cardinalities Sets of Zero Measure Cantor's Set 127

5 8.25 Exercises Vectors and Matrices Two-Dimensional Vectors Adding Vectors Multiplying a Vector by a Scalar Three-Dimensional Vectors Multidimensional Vectors Matrices Adding Matrices Multiplying a Matrix by a Scalar Matrix times Vector Matrix times Matrix The Transpose of a Matrix Symmetric Matrices Hermitian Matrices Inner Product Norms of Vectors Inner Product and the Hermitian Conjugate Orthogonal Matrices Eigenvectors and Eigenvalues Eigenvalues of a Hermitian Matrix Eigenvectors of a Hermitian Matrix The Sine Transform The Cosine Transform Determinant of a Square Matrix Inverse of a Square Matrix Vector Product Exercises Multilevel Objects Induction and Deduction Mathematical Induction Trees Binary Trees Arithmetic Expressions Boolean Expressions The Tower Problem The Tree of the Tower Problem Pascal's Triangle 163 lo.lothe Binomial Coefficients 165 lo.hpaths in Pascal's Triangle Paths and the Binomial Coefficients Newton's Binomial Brownian Motion 170

6 10.15Counting Integer Vectors 171 lo.lgmathematical Induction in Newton's Binomial Factorial of a Sum The Trinomial Formula Multiscale The Decimal Representation The Binary Representation The Sine Transform Exercises Graphs Oriented Graphs Nonoriented Graphs The Node-Coloring Problem The Node-Coloring Algorithm The Edge-Coloring Problem The Edge-Coloring Algorithm Graph of Edges Triangulation The Triangle-Coloring Problem 195 ll.loweighted Graphs Algebraic Formulation The Steady State Exercises Polynomials Adding Polynomials Multiplying a Polynomial by a Scalar Multiplying Polynomials Computing a Polynomial Composition of Polynomials Natural Numbers as Polynomials Computing a Monomial Derivative Indefinite Integral Integral over an Interval Sparse Polynomials Composition of Sparse Polynomials Polynomials of Two Variables Partial Derivatives The Gradient Integral over the Unit Triangle Second Partial Derivatives Degree Polynomials of Three Variables 218

7 12.20Partial Derivatives The Gradient Integral over the Unit Tetrahedron Directional Derivatives Normal Derivatives Tangential Derivatives High-Order Partial Derivatives The Hessian Degree Degrees of Freedom Basis Functions in the Unit Tetrahedron Computing the Basis Functions Composite Functions in a General Tetrahedron The Chain Rule Directional Derivative of a Composite Function The Hessian of a Composite Function Basis Functions in a General Tetrahedron Tensors Hessian-Related Basis Functions Basis Functions at Edge Midpoints Basis Functions at Side Midpoints Continuity Continuity of Gradient Integral over a General Tetrahedron Exercises 243 Part IV Introduction to С Basics of Programming The Computer and its Memory The Program or Code The Code Segments in this Book Variables and Types Defining Variables Assignment Initialization Explicit Conversion Implicit Conversion Arithmetic Operations Functions The "Main" Function Printing Output Comparison Operators Boolean Operators The "?:" Operator 266

8 13.17Conditional Instructions Scope of Variables Loops The Power Function Integer Logarithm The Factorial Function Nested Loops Reversed Number Binary Number Pointers Pointer to a Constant Variable The Referencing Operator Arrays Two-Dimensional Arrays Passing Arguments to Functions Input/Output (I/O) Input/Out with Files Exercises Recursion Recursive Functions The Power Function Integer Logarithm The Factorial Function Ordered Arrays Binary Representation Pascal's Triangle Arithmetic Expression Static Variables The Exponent Function Exercises 300 Part V Introduction to C Objects Classes Private and Public Members Interface Functions Information and Memory Constructors Initialization List Default Arguments Explicit Conversion Implicit Conversion The Default Copy Constructor 315

9 15.11 Destructor Member and Friend Functions The Current Object and its Address Returned Pointer Pointer to a Constant Object References Passing Arguments by Reference Returning by Reference Efficiency in Passing by Reference Copy Constructor Assignment Operators perators Inverse Conversion Unary Operators Update Operators Friend Update Operators Binary Operators Friend Binary Operators Member Binary Operators Ordinary Binary Operators Complex Numbers Member Operators with Complex Numbers rdinary Operators with Complex Numbers Exercises Vectors and Matrices Induction and Deduction in Object-Oriented Programming Templates The Vector Object Constructors Assignment Operators Arithmetic Operators Points in the Cartesian Plane and Space Inheritance Public Derivation Protected Members of the Base Class Constructing a Derived Object Functions of Derived Objects Destroying a Derived Object Inherited Member Functions verridden Member Functions The Matrix Object Power of a Square Matrix Exponent of a Square Matrix Exercises 356

10 17 Dynamic Vectors and Lists Dynamic Vectors Ordinary Lists Linked Lists The Copy Constructor The Destructor Recursive Member Functions Inserting New Items The Assignment Operator Dropping Items The Merging Problem The Ordering Problem Stacks Exercises 384 Part VI Implementation of Computational Objects Trees Binary Trees Recursive Definition Implementation of Binary Tree The Destructor The Tower Constructor Solving the Tower Problem General Trees Exercises Graphs The Matrix Formulation The Node-Coloring Algorithm The Edge-Coloring Algorithm Sparse Matrix Data Access Virtual Addresses Advantages and Disadvantages Nonoriented Graphs Exercises Sparse Matrices The Matrix-Element Object Member Arithmetic Operators Comparison in Terms of Column Index Ordinary Arithmetic Operators The Row Object Reading the First Element 426

11 20.7 Inserting a New Element Recursive Functions Update Operators Member Binary Operators Ordinary Binary Operators The Sparse-Matrix Object Reading a Matrix Element Some More Member Functions The Node-Coloring Code Edge Coloring in a Nonoriented Graph Edge Coloring in an Oriented Graph Exercises Meshes The Node Object Reading and Accessing Data Fields The Cell - a Highly Abstract Object The Cell Object Reading and Accessing Vertices Constructors The Assignment Operator Nodes in a Cell Edge-Sharing Cells The Mesh Object Indexing the Nodes An Example Indexing the Cells Exercises Triangulation Triangulation of a Domain Multilevel Iterative Mesh Refinement Dividing a Triangle and its Neighbor Refining the Mesh Approximating a Circle The Cell-Coloring Code The Matrix Formulation The Code-Size Rule Exercises 480 Part VII Three-Dimensional Applications 483

12 23 Mesh of Tetrahedra The Mesh Refinement Refining the Neighbor Tetrahedra The Refinement Step Exercises Polynomials The Polynomial Object Adding Polynomials Multiplication by a Scalar Multiplying Polynomials Calculating a Polynomial Composition of Polynomials Recursive Horner Code Polynomials of Two Variables Polynomials of Three Variables Indefinite Integral Integral on the Unit Interval Integral on the Unit Triangle Integral on the Unit Tetrahedron Exercises Sparse Polynomials The Monomial Object Multiplying Monomials The Sparse-Polynomial Object Multiplying a Sparse Polynomial by a Scalar Multiplying Sparse Polynomials Adding Sparse Polynomials The Modified Horner Code Polynomials of Two Variables Polynomials of Three Variables Exercises Stiffness and Mass Matrices The Neumann Matrix The Boundary Matrix The Stiffness Matrix The Mass Matrix Newton's Mass Matrix Helmholtz Mass Matrix Helmholtz Matrix Newton's Iteration Dirichlet Boundary Conditions Exercises 531

13 27 Splines The Indexing Scheme Basis Functions in the Mesh The Neumann Matrix The Spline Problem The Dirichlet Matrix Exercises Appendix: Solutions of Exercises Representation of Integers in any Base Prime Factors Greatest Common Divisor Recursive Implementation of C a, n Square Root of a Complex Number Operations with Vectors Operations with Matrices Determinant, Inverse, and Transpose of 2 x 2 Matrix Determinant, Inverse, and Transpose of 3 x 3 Matrix Vector Product The Matrix Exponent Function perations with Dynamic Vectors Using the Stack Object perations with Sparse Matrices Three-Dimensional Mesh ntegrals over the Tetrahedron Computing Partial Derivatives Composing Sparse Polynomials Calculations with Sparse Polynomials The Stiffness Matrix Newton's Mass Matrix Helmholtz Mass Matrix Indexing the Edges in the Mesh Indexing the Sides in the Mesh Computing Basis Functions Setting Dirichlet Conditions 588 References 593 Index 595

FINITE-DIMENSIONAL LINEAR ALGEBRA

FINITE-DIMENSIONAL LINEAR ALGEBRA DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H ROSEN FINITE-DIMENSIONAL LINEAR ALGEBRA Mark S Gockenbach Michigan Technological University Houghton, USA CRC Press Taylor & Francis Croup

More information

Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS

Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS (i) Sets and their Representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams;

More information

9th and 10th Grade Math Proficiency Objectives Strand One: Number Sense and Operations

9th and 10th Grade Math Proficiency Objectives Strand One: Number Sense and Operations Strand One: Number Sense and Operations Concept 1: Number Sense Understand and apply numbers, ways of representing numbers, the relationships among numbers, and different number systems. Justify with examples

More information

Contents. CHAPTER P Prerequisites 1. CHAPTER 1 Functions and Graphs 69. P.1 Real Numbers 1. P.2 Cartesian Coordinate System 14

Contents. CHAPTER P Prerequisites 1. CHAPTER 1 Functions and Graphs 69. P.1 Real Numbers 1. P.2 Cartesian Coordinate System 14 CHAPTER P Prerequisites 1 P.1 Real Numbers 1 Representing Real Numbers ~ Order and Interval Notation ~ Basic Properties of Algebra ~ Integer Exponents ~ Scientific Notation P.2 Cartesian Coordinate System

More information

Algebra II Learning Targets

Algebra II Learning Targets Chapter 0 Preparing for Advanced Algebra LT 0.1 Representing Functions Identify the domain and range of functions LT 0.2 FOIL Use the FOIL method to multiply binomials LT 0.3 Factoring Polynomials Use

More information

9-12 Mathematics Vertical Alignment ( )

9-12 Mathematics Vertical Alignment ( ) Algebra I Algebra II Geometry Pre- Calculus U1: translate between words and algebra -add and subtract real numbers -multiply and divide real numbers -evaluate containing exponents -evaluate containing

More information

NUMERICAL METHODS. lor CHEMICAL ENGINEERS. Using Excel', VBA, and MATLAB* VICTOR J. LAW. CRC Press. Taylor & Francis Group

NUMERICAL METHODS. lor CHEMICAL ENGINEERS. Using Excel', VBA, and MATLAB* VICTOR J. LAW. CRC Press. Taylor & Francis Group NUMERICAL METHODS lor CHEMICAL ENGINEERS Using Excel', VBA, and MATLAB* VICTOR J. LAW CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup,

More information

Introductory Chapter(s)

Introductory Chapter(s) Course 1 6 11-12 years 1 Exponents 2 Sequences Number Theory 3 Divisibility 4 Prime Factorization 5 Greatest Common Factor Fractions and Decimals 6 Comparing and Ordering Fractions 7 Comparing and Ordering

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

Mathematics for Engineers and Scientists

Mathematics for Engineers and Scientists Mathematics for Engineers and Scientists Fourth edition ALAN JEFFREY University of Newcastle-upon-Tyne B CHAPMAN & HALL University and Professional Division London New York Tokyo Melbourne Madras Contents

More information

Cryptography. Number Theory with AN INTRODUCTION TO. James S. Kraft. Lawrence C. Washington. CRC Press

Cryptography. Number Theory with AN INTRODUCTION TO. James S. Kraft. Lawrence C. Washington. CRC Press AN INTRODUCTION TO Number Theory with Cryptography James S Kraft Gilman School Baltimore, Maryland, USA Lawrence C Washington University of Maryland College Park, Maryland, USA CRC Press Taylor & Francis

More information

Adaptive Filtering. Squares. Alexander D. Poularikas. Fundamentals of. Least Mean. with MATLABR. University of Alabama, Huntsville, AL.

Adaptive Filtering. Squares. Alexander D. Poularikas. Fundamentals of. Least Mean. with MATLABR. University of Alabama, Huntsville, AL. Adaptive Filtering Fundamentals of Least Mean Squares with MATLABR Alexander D. Poularikas University of Alabama, Huntsville, AL CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is

More information

Integrated Math III. IM3.1.2 Use a graph to find the solution set of a pair of linear inequalities in two variables.

Integrated Math III. IM3.1.2 Use a graph to find the solution set of a pair of linear inequalities in two variables. Standard 1: Algebra and Functions Students solve inequalities, quadratic equations, and systems of equations. They graph polynomial, rational, algebraic, and piece-wise defined functions. They graph and

More information

College Algebra and Trigonometry

College Algebra and Trigonometry GLOBAL EDITION College Algebra and Trigonometry THIRD EDITION J. S. Ratti Marcus McWaters College Algebra and Trigonometry, Global Edition Table of Contents Cover Title Page Contents Preface Resources

More information

Linear Algebra and Probability

Linear Algebra and Probability Linear Algebra and Probability for Computer Science Applications Ernest Davis CRC Press Taylor!* Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor Sc Francis Croup, an informa

More information

INDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43

INDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43 INDEX Abel s identity, 131 Abel s test, 131 132 Abel s theorem, 463 464 absolute convergence, 113 114 implication of conditional convergence, 114 absolute value, 7 reverse triangle inequality, 9 triangle

More information

Index. Excerpt from "Calculus" 2013 AoPS Inc. Copyrighted Material INDEX

Index. Excerpt from Calculus 2013 AoPS Inc.  Copyrighted Material INDEX Index #, 2 \, 5, 4 [, 4 - definition, 38 ;, 2 indeterminate form, 2 22, 24 indeterminate form, 22 23, see Euler s Constant 2, 2, see infinity, 33 \, 6, 3, 2 3-6-9 triangle, 2 4-dimensional sphere, 85 45-45-9

More information

Appendix C: Event Topics per Meet

Appendix C: Event Topics per Meet Appendix C: Event Topics per Meet Meet 1 1A Pre-algebra Topics Fractions to add and express as the quotient of two relatively prime integers Complex fractions and continued fractions Decimals, repeating

More information

Introductory Mathematics

Introductory Mathematics Introductory Mathematics 1998 2003 1.01 Identify subsets of the real number system. 1.02 Estimate and compute with rational Grade 7: 1.02 numbers. 1.03 Compare, order, and convert among Grade 6: 1.03 fractions,

More information

AND NONLINEAR SCIENCE SERIES. Partial Differential. Equations with MATLAB. Matthew P. Coleman. CRC Press J Taylor & Francis Croup

AND NONLINEAR SCIENCE SERIES. Partial Differential. Equations with MATLAB. Matthew P. Coleman. CRC Press J Taylor & Francis Croup CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES An Introduction to Partial Differential Equations with MATLAB Second Edition Matthew P Coleman Fairfield University Connecticut, USA»C)

More information

WESTMORELAND COUNTY PUBLIC SCHOOLS Integrated Instructional Pacing Guide and Checklist Algebra II

WESTMORELAND COUNTY PUBLIC SCHOOLS Integrated Instructional Pacing Guide and Checklist Algebra II WESTMORELAND COUNTY PUBLIC SCHOOLS 201 2014 Integrated Instructional Pacing Guide and Checklist Algebra II (s) ESS Vocabulary AII.4 Absolute Value Equations and Inequalities Absolute Value Equations and

More information

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ALGEBRA II

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ALGEBRA II UNIT: Review of Basic Algebra Skills as Needed SR1 and any Supplemental Materials UNIT : What skills from Algebra I are used in Algebra II? Review Algebra I Skills as Needed SR1 and any additional resources

More information

College Algebra Poudre School District Pacing Overview

College Algebra Poudre School District Pacing Overview Pacing Overview Section Title Pacing Notes A.3 Polynomials A.3 Polynomials A.4 Synthetic Division Semester 1 Algebraic Skills (Appendix A) 15-16 days A.5 Rational Expressions A.6 Solving Equations A.7

More information

Catholic Central High School

Catholic Central High School Catholic Central High School Course: Basic Algebra 2 Department: Mathematics Length: One year Credit: 1 Prerequisite: Completion of Basic Algebra 1 or Algebra 1, Basic Plane Geometry or Plane Geometry,

More information

TABLE OF CONTENTS POLYNOMIAL EQUATIONS AND INEQUALITIES

TABLE OF CONTENTS POLYNOMIAL EQUATIONS AND INEQUALITIES COMPETENCY 1.0 ALGEBRA TABLE OF CONTENTS SKILL 1.1 1.1a. 1.1b. 1.1c. SKILL 1.2 1.2a. 1.2b. 1.2c. ALGEBRAIC STRUCTURES Know why the real and complex numbers are each a field, and that particular rings are

More information

Algebra and Trigonometry

Algebra and Trigonometry Algebra and Trigonometry 978-1-63545-098-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Jay Abramson, Arizona State

More information

Foundations of Mathematics 11. solve problems that involve the application of rates

Foundations of Mathematics 11. solve problems that involve the application of rates Measurement using SI and imperial units, solve problems that involve linear measurement apply proportional reasoning to problems that involve conversions between SI and imperial units of measure solve

More information

Pre AP Algebra. Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra

Pre AP Algebra. Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra Pre AP Algebra Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra 1 The content of the mathematics standards is intended to support the following five goals for students: becoming

More information

Algebra 2. Curriculum (384 topics additional topics)

Algebra 2. Curriculum (384 topics additional topics) Algebra 2 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

More information

How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials

How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials PRC Ch P_3.notebook How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials How to remove common factors from polynomials

More information

College Algebra To learn more about all our offerings Visit Knewton.com

College Algebra To learn more about all our offerings Visit Knewton.com College Algebra 978-1-63545-097-2 To learn more about all our offerings Visit Knewton.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Text Jay Abramson, Arizona State University

More information

Introduction. Finite and Spectral Element Methods Using MATLAB. Second Edition. C. Pozrikidis. University of Massachusetts Amherst, USA

Introduction. Finite and Spectral Element Methods Using MATLAB. Second Edition. C. Pozrikidis. University of Massachusetts Amherst, USA Introduction to Finite and Spectral Element Methods Using MATLAB Second Edition C. Pozrikidis University of Massachusetts Amherst, USA (g) CRC Press Taylor & Francis Group Boca Raton London New York CRC

More information

Missouri Educator Gateway Assessments DRAFT

Missouri Educator Gateway Assessments DRAFT Missouri Educator Gateway Assessments FIELD 023: MATHEMATICS January 2014 DRAFT Content Domain Range of Competencies Approximate Percentage of Test Score I. Numbers and Quantity 0001 0002 14% II. Patterns,

More information

I-2 Index. Coterminal Angles, 738 Counting numbers, 8 Cramer, Gabriel, 309 Cramer s rule, 306 Cube root, 427, 434 Cylinder, right circular, 117

I-2 Index. Coterminal Angles, 738 Counting numbers, 8 Cramer, Gabriel, 309 Cramer s rule, 306 Cube root, 427, 434 Cylinder, right circular, 117 Index Absolute value, 18 equations, 154, 162 inequalities, 159, 162 Absolute error, 158 Addition, 4 associative property, 19 commutative property, 18 of complex numbers, 481 of fractions, 21 of functions,

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS SPECIALIST MATHEMATICS (Year 11 and 12) UNIT A A1: Combinatorics Permutations: problems involving permutations use the multiplication principle and factorial notation permutations and restrictions with

More information

xvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7

xvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7 About the Author v Preface to the Instructor xvi WileyPLUS xxii Acknowledgments xxiii Preface to the Student xxvi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real

More information

Pre-Calculus and Trigonometry Capacity Matrix

Pre-Calculus and Trigonometry Capacity Matrix Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational expressions Solve polynomial equations and equations involving rational expressions Review Chapter 1 and their

More information

Maths for Map Makers

Maths for Map Makers SUB Gottingen 7 210 050 861 99 A 2003 Maths for Map Makers by Arthur Allan Whittles Publishing Contents /v Chapter 1 Numbers and Calculation 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

More information

Business Calculus

Business Calculus Business Calculus 978-1-63545-025-5 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Senior Contributing Authors: Gilbert

More information

Subtraction Property of Equality terms. x-axis x-coordinate x-intercept y-axis y-coordinate y-intercept

Subtraction Property of Equality terms. x-axis x-coordinate x-intercept y-axis y-coordinate y-intercept Algebra I Chapter 1 Chapter 2 additive identity range absolute value algebraic expression reciprocal additive inverses Associative Property Reflexive Property of Equality back-to-back stem and leaf bar

More information

A video College Algebra course & 6 Enrichment videos

A video College Algebra course & 6 Enrichment videos A video College Algebra course & 6 Enrichment videos Recorded at the University of Missouri Kansas City in 1998. All times are approximate. About 43 hours total. Available on YouTube at http://www.youtube.com/user/umkc

More information

Prentice Hall PreCalculus, 3rd Edition 2007, (Blitzer)

Prentice Hall PreCalculus, 3rd Edition 2007, (Blitzer) Prentice Hall PreCalculus, 3rd Edition 2007, (Blitzer) C O R R E L A T E D T O Number Properties and Operations High school students should enter high school with a strong background in rational numbers

More information

CHAPMAN & HALL/CRC CRYPTOGRAPHY AND NETWORK SECURITY ALGORITHMIC CR YPTAN ALY51S. Ant nine J aux

CHAPMAN & HALL/CRC CRYPTOGRAPHY AND NETWORK SECURITY ALGORITHMIC CR YPTAN ALY51S. Ant nine J aux CHAPMAN & HALL/CRC CRYPTOGRAPHY AND NETWORK SECURITY ALGORITHMIC CR YPTAN ALY51S Ant nine J aux (g) CRC Press Taylor 8* Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor &

More information

Prentice Hall Intermediate Algebra, 5th Edition 2009 (Martin-Gay)

Prentice Hall Intermediate Algebra, 5th Edition 2009 (Martin-Gay) Prentice Hall Intermediate Algebra, 5th Edition 2009 (Martin-Gay) C O R R E L A T E D T O Number Properties and Operations High school students should enter high school with a strong background in rational

More information

Index. Excerpt from "Art of Problem Solving Volume 1: the Basics" 2014 AoPS Inc. / 267. Copyrighted Material

Index. Excerpt from Art of Problem Solving Volume 1: the Basics 2014 AoPS Inc.  / 267. Copyrighted Material Index Ø, 247 n k, 229 AA similarity, 102 AAS congruence, 100 abscissa, 143 absolute value, 191 abstract algebra, 66, 210 altitude, 95 angle bisector, 94 Angle Bisector Theorem, 103 angle chasing, 133 angle

More information

Check boxes of Edited Copy of Sp Topics (was 261-pilot)

Check boxes of Edited Copy of Sp Topics (was 261-pilot) Check boxes of Edited Copy of 10023 Sp 11 253 Topics (was 261-pilot) Intermediate Algebra (2011), 3rd Ed. [open all close all] R-Review of Basic Algebraic Concepts Section R.2 Ordering integers Plotting

More information

1 ** The performance objectives highlighted in italics have been identified as core to an Algebra II course.

1 ** The performance objectives highlighted in italics have been identified as core to an Algebra II course. Strand One: Number Sense and Operations Every student should understand and use all concepts and skills from the pervious grade levels. The standards are designed so that new learning builds on preceding

More information

Arizona Mathematics Standards Articulated by Grade Level (2008) for College Work Readiness (Grades 11 and 12)

Arizona Mathematics Standards Articulated by Grade Level (2008) for College Work Readiness (Grades 11 and 12) Strand 1: Number and Operations Concept 1: Number Sense Understand and apply numbers, ways of representing numbers, and the relationships among numbers and different number systems. College Work Readiness

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

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

PreCalculus. Curriculum (447 topics additional topics)

PreCalculus. Curriculum (447 topics additional topics) PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

More information

Trigonometry Self-study: Reading: Red Bostock and Chandler p , p , p

Trigonometry Self-study: Reading: Red Bostock and Chandler p , p , p Trigonometry Self-study: Reading: Red Bostock Chler p137-151, p157-234, p244-254 Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant,

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics

More information

Secondary Honors Algebra II Objectives

Secondary Honors Algebra II Objectives Secondary Honors Algebra II Objectives Chapter 1 Equations and Inequalities Students will learn to evaluate and simplify numerical and algebraic expressions, to solve linear and absolute value equations

More information

MATHEMATICAL ANALYSIS CURRICULUM GUIDE

MATHEMATICAL ANALYSIS CURRICULUM GUIDE MATHEMATICAL ANALYSIS CURRICULUM GUIDE Loudoun County Public Schools 2010-2011 Complete scope, sequence, pacing and resources are available on the CD and will be available on the LCPS Intranet. INTRODUCTION

More information

Grade Math (HL) Curriculum

Grade Math (HL) Curriculum Grade 11-12 Math (HL) Curriculum Unit of Study (Core Topic 1 of 7): Algebra Sequences and Series Exponents and Logarithms Counting Principles Binomial Theorem Mathematical Induction Complex Numbers Uses

More information

OKLAHOMA SUBJECT AREA TESTS (OSAT )

OKLAHOMA SUBJECT AREA TESTS (OSAT ) CERTIFICATION EXAMINATIONS FOR OKLAHOMA EDUCATORS (CEOE ) OKLAHOMA SUBJECT AREA TESTS (OSAT ) October 2005 Subarea Range of Competencies I. Mathematical Processes and Number Sense 01 04 II. Relations,

More information

CME Project, Geometry 2009 Correlated to: Kentucky Core Content for Mathematics Assessment 4.1 (High School, Grade 11)

CME Project, Geometry 2009 Correlated to: Kentucky Core Content for Mathematics Assessment 4.1 (High School, Grade 11) Number Properties and Operations High school students should enter high school with a strong background in rational numbers and numerical operations and expand this to real numbers. This becomes the foundation

More information

Use estimation strategies reasonably and fluently while integrating content from each of the other strands. PO 1. Recognize the limitations of

Use estimation strategies reasonably and fluently while integrating content from each of the other strands. PO 1. Recognize the limitations of for Strand 1: Number and Operations Concept 1: Number Sense Understand and apply numbers, ways of representing numbers, and the relationships among numbers and different number systems. PO 1. Solve problems

More information

Tennessee s State Mathematics Standards Precalculus

Tennessee s State Mathematics Standards Precalculus Tennessee s State Mathematics Standards Precalculus Domain Cluster Standard Number Expressions (N-NE) Represent, interpret, compare, and simplify number expressions 1. Use the laws of exponents and logarithms

More information

PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO. Prepared by Kristina L. Gazdik. March 2005

PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO. Prepared by Kristina L. Gazdik. March 2005 PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO Prepared by Kristina L. Gazdik March 2005 1 TABLE OF CONTENTS Course Description.3 Scope and Sequence 4 Content Outlines UNIT I: FUNCTIONS AND THEIR GRAPHS

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

Notation Index. gcd(a, b) (greatest common divisor) NT-16

Notation Index. gcd(a, b) (greatest common divisor) NT-16 Notation Index (for all) B A (all functions) B A = B A (all functions) SF-18 (n) k (falling factorial) SF-9 a R b (binary relation) C(n,k) = n! k! (n k)! (binomial coefficient) SF-9 n! (n factorial) SF-9

More information

Algebra 2 Secondary Mathematics Instructional Guide

Algebra 2 Secondary Mathematics Instructional Guide Algebra 2 Secondary Mathematics Instructional Guide 2009-2010 ALGEBRA 2AB (Grade 9, 10 or 11) Prerequisite: Algebra 1AB or Geometry AB 310303 Algebra 2A 310304 Algebra 2B COURSE DESCRIPTION Los Angeles

More information

Index ": :", 50 "? :", 14 <, 14 =, 8, 14 >, 14, 55., 14., 49 ;, 6 %, 9, 37, 176 &, 58, 150 :, 52, 79 *, 8, 22, 150 "++", 10

Index : :, 50 ? :, 14 <, 14 =, 8, 14 >, 14, 55., 14., 49 ;, 6 %, 9, 37, 176 &, 58, 150 :, 52, 79 *, 8, 22, 150 ++, 10 Index , 14, 55., 14., 49 ;, 6 %, 9, 37, 176 &, 58, 150 :, 52, 79 *, 8, 22, 150 "++", 10 "+=", 10 in complex, 68 in dynamic vector, 672 in matrix, 670 in point, 63 in sparse matrix, 692

More information

Prentice Hall Geometry (c) 2007 correlated to American Diploma Project, High School Math Benchmarks

Prentice Hall Geometry (c) 2007 correlated to American Diploma Project, High School Math Benchmarks I1.1. Add, subtract, multiply and divide integers, fractions and decimals. I1.2. Calculate and apply ratios, proportions, rates and percentages to solve problems. I1.3. Use the correct order of operations

More information

Mathematics (MAT) MAT 051 Pre-Algebra. 4 Hours. Prerequisites: None. 4 hours weekly (4-0)

Mathematics (MAT) MAT 051 Pre-Algebra. 4 Hours. Prerequisites: None. 4 hours weekly (4-0) Mathematics (MAT) MAT 051 Pre-Algebra 4 Hours Prerequisites: None 4 hours weekly (4-0) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student

More information

Binomials defined, 13 division by, FOIL method and, 22 multiplying binomial by trinomial,

Binomials defined, 13 division by, FOIL method and, 22 multiplying binomial by trinomial, 5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-1 Index Absolute value defined, 46 functions and, 126 59 Absolute value equations, solutions, 46 49 Absolute value inequalities, solutions, 263 267 Acceleration,

More information

ALGEBRA (SMR Domain 1) Algebraic Structures (SMR 1.1)...1

ALGEBRA (SMR Domain 1) Algebraic Structures (SMR 1.1)...1 TABLE OF CONTENTS Page Numbers ALGEBRA (SMR Domain 1)...1 0001 Algebraic Structures (SMR 1.1)...1 Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a

More information

Contents. List of Applications. Basic Concepts 1. iii

Contents. List of Applications. Basic Concepts 1. iii 46891_01_FM_pi_pxviii.QXD 10/1/09 3:25 PM Page iii 1 List of Applications ix Preface xiii Basic Concepts 1 Unit 1A REVIEW OF OPERATIONS WITH WHOLE NUMBERS 2 1.1 Review of Basic Operations 2 1.2 Order of

More information

Copyright 2016 Pearson Education, Inc. or its affiliates. All rights reserved. NES, the NES logo, Pearson, the Pearson logo, and National Evaluation

Copyright 2016 Pearson Education, Inc. or its affiliates. All rights reserved. NES, the NES logo, Pearson, the Pearson logo, and National Evaluation Mathematics (304) Copyright 2016 Pearson Education, Inc. or its affiliates. All rights reserved. NES, the NES logo, Pearson, the Pearson logo, and National Evaluation Series are trademarks, in the U.S.

More information

DRAFT. Pre-calculus Curriculum Map Quarter 1 Chapters P 2. Extraneous Critical numbers Test intervals

DRAFT. Pre-calculus Curriculum Map Quarter 1 Chapters P 2. Extraneous Critical numbers Test intervals Quarter 1 Chapters P 2 Plot points in the coordinate plane and use distance and midpoint formulas. Sketch graphs of equations. Find and use slope of a line to write and graph linear equations. Solve equations:

More information

CUMBERLAND COUNTY SCHOOL DISTRICT BENCHMARK ASSESSMENT CURRICULUM PACING GUIDE

CUMBERLAND COUNTY SCHOOL DISTRICT BENCHMARK ASSESSMENT CURRICULUM PACING GUIDE CUMBERLAND COUNTY SCHOOL DISTRICT BENCHMARK ASSESSMENT CURRICULUM PACING GUIDE School: CCHS Subject: Algebra II Grade: 10 th Grade Benchmark Assessment 1 Instructional Timeline: 1 st Nine Weeks Topic(s):

More information

Honors Integrated Algebra/Geometry 3 Critical Content Mastery Objectives Students will:

Honors Integrated Algebra/Geometry 3 Critical Content Mastery Objectives Students will: Content Standard 1: Numbers, Number Sense, and Computation Place Value Fractions Comparing and Ordering Counting Facts Estimating and Estimation Strategies Determine an approximate value of radical and

More information

Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards

Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards Core: Operations on Numbers and Expressions Priority: 15% Successful students will be able to perform operations with rational, real, and complex numbers, using both numeric and algebraic expressions,

More information

Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A)

Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Updated 06/05/16 http://www.haesemathematics.com.au/ Note: Exercises in red text indicate material in the 10A textbook

More information

Signals and Systems Laboratory with MATLAB

Signals and Systems Laboratory with MATLAB Signals and Systems Laboratory with MATLAB Alex Palamides Anastasia Veloni @ CRC Press Taylor &. Francis Group Boca Raton London NewYork CRC Press is an imprint of the Taylor & Francis Group, an informa

More information

Range of Competencies

Range of Competencies MATHEMATICS l. Content Domain Range of Competencies Mathematical Processes and Number Sense 0001 0003 19% ll. Patterns, Algebra, and Functions 0004 0007 24% lll. Measurement and Geometry 0008 0010 19%

More information

Unit Activity Correlations to Common Core State Standards. Precalculus. Table of Contents

Unit Activity Correlations to Common Core State Standards. Precalculus. Table of Contents Unit Activity Correlations to Common Core State Standards Precalculus Table of Contents Number and Quantity 1 Algebra 3 Functions 3 Geometry 5 Statistics and Probability 6 Number and Quantity The Complex

More information

How do we analyze, evaluate, solve, and graph quadratic functions?

How do we analyze, evaluate, solve, and graph quadratic functions? Topic: 4. Quadratic Functions and Factoring Days: 18 Key Learning: Students will be able to analyze, evaluate, solve and graph quadratic functions. Unit Essential Question(s): How do we analyze, evaluate,

More information

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Algebra II (Grades 9-12)

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Algebra II (Grades 9-12) California Mathematics Content Standards for Algebra II (Grades 9-12) This discipline complements and expands the mathematical content and concepts of algebra I and geometry. Students who master algebra

More information

Unit 1. Revisiting Parent Functions and Graphing

Unit 1. Revisiting Parent Functions and Graphing Unit 1 Revisiting Parent Functions and Graphing Precalculus Analysis Pacing Guide First Nine Weeks Understand how the algebraic properties of an equation transform the geometric properties of its graph.

More information

Mathematics High School Advanced Mathematics Plus Course

Mathematics High School Advanced Mathematics Plus Course Mathematics High School Advanced Mathematics Plus Course, a one credit course, specifies the mathematics that students should study in order to be college and career ready. The Advanced Mathematics Plus

More information

Homework Points Notes Points

Homework Points Notes Points A2 NBC (13) 13.1 p 876 #1-8 13.2 p 882 #1-7 13.3 p 888 #1-11 13.4 p 897 #1-11 13.5 p 904 #2-20 even 13.1 Trig Identities 13.2 Verifying Trig Identities 13.3 Sum and Difference... 13.4 Double and Half...

More information

Intermediate Level Learning Targets

Intermediate Level Learning Targets Learning Target #1: Develop proficiency in analyzing, graphing and solving linear equations and inequalities. F1.1,,, B1. C1. 1.1 Students will be able to identify different types of relations and functions.

More information

College Algebra with Corequisite Support: Targeted Review

College Algebra with Corequisite Support: Targeted Review College Algebra with Corequisite Support: Targeted Review 978-1-63545-056-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable)

More information

UNDERSTANDING ENGINEERING MATHEMATICS

UNDERSTANDING ENGINEERING MATHEMATICS UNDERSTANDING ENGINEERING MATHEMATICS JOHN BIRD WORKED SOLUTIONS TO EXERCISES 1 INTRODUCTION In Understanding Engineering Mathematic there are over 750 further problems arranged regularly throughout the

More information

Algebra Topic Alignment

Algebra Topic Alignment Preliminary Topics Absolute Value 9N2 Compare, order and determine equivalent forms for rational and irrational numbers. Factoring Numbers 9N4 Demonstrate fluency in computations using real numbers. Fractions

More information

MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS

MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS T H I R D E D I T I O N MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS STANLEY I. GROSSMAN University of Montana and University College London SAUNDERS COLLEGE PUBLISHING HARCOURT BRACE

More information

Series Editor KENNETH H. ROSEN INDUCTION THEORY AND APPLICATIONS. of Manitoba. University. Winnipeg, Canada. CRC Press. Taylor StFrancis Group

Series Editor KENNETH H. ROSEN INDUCTION THEORY AND APPLICATIONS. of Manitoba. University. Winnipeg, Canada. CRC Press. Taylor StFrancis Group DISCIIETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN HANDBOOK OF MATHEMATICAL INDUCTION THEORY AND APPLICATIONS David S. Gunderson University of Manitoba Winnipeg, Canada CRC Press

More information

STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF DRAFT SYLLABUS.

STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF DRAFT SYLLABUS. STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF 2017 - DRAFT SYLLABUS Subject :Mathematics Class : XI TOPIC CONTENT Unit 1 : Real Numbers - Revision : Rational, Irrational Numbers, Basic Algebra

More information

Single Variable Calculus, Early Transcendentals

Single Variable Calculus, Early Transcendentals Single Variable Calculus, Early Transcendentals 978-1-63545-100-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax

More information

MATHEMATICS (MIDDLE GRADES AND EARLY SECONDARY)

MATHEMATICS (MIDDLE GRADES AND EARLY SECONDARY) MATHEMATICS (MIDDLE GRADES AND EARLY SECONDARY) l. Content Domain Mathematical Processes and Number Sense Range of Competencies Approximate Percentage of Test Score 0001 0003 24% ll. Patterns, Algebra,

More information

SAMPLE COURSE OUTLINE MATHEMATICS SPECIALIST ATAR YEAR 11

SAMPLE COURSE OUTLINE MATHEMATICS SPECIALIST ATAR YEAR 11 SAMPLE COURSE OUTLINE MATHEMATICS SPECIALIST ATAR YEAR 11 Copyright School Curriculum and Standards Authority, 2017 This document apart from any third party copyright material contained in it may be freely

More information

Precalculus. Represent complex numbers and their operations on the complex plane.

Precalculus. Represent complex numbers and their operations on the complex plane. Precalculus Precalculus is designed for students who have successfully completed the Algebra II with Trigonometry course. This course is considered to be a prerequisite for success in calculus and college

More information

Curriculum Guide Algebra 2 Advanced

Curriculum Guide Algebra 2 Advanced Unit 1: Equations and Inequalities Biblical Worldview Essential Questions: Is your life balanced as a believer? Are you a real Christian? 13 Lessons A2#1, A2#2 1. Use a number line to graph and order real

More information

Pre-Calculus and Trigonometry Capacity Matrix

Pre-Calculus and Trigonometry Capacity Matrix Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions

More information

CHAPTER 1 Prerequisites for Calculus 2. CHAPTER 2 Limits and Continuity 58

CHAPTER 1 Prerequisites for Calculus 2. CHAPTER 2 Limits and Continuity 58 CHAPTER 1 Prerequisites for Calculus 2 1.1 Lines 3 Increments Slope of a Line Parallel and Perpendicular Lines Equations of Lines Applications 1.2 Functions and Graphs 12 Functions Domains and Ranges Viewing

More information

Check boxes of Edited Copy of Sp Topics (was 217-pilot)

Check boxes of Edited Copy of Sp Topics (was 217-pilot) Check boxes of Edited Copy of 10024 Sp 11 213 Topics (was 217-pilot) College Algebra, 9th Ed. [open all close all] R-Basic Algebra Operations Section R.1 Integers and rational numbers Rational and irrational

More information