MATHEMATICAL OBJECTS in
|
|
- David Tyler
- 5 years ago
- Views:
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
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 informationMathematics 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 information9th 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 informationContents. 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 informationAlgebra 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 information9-12 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 informationNUMERICAL 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 informationIntroductory 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 informationRon 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 informationMathematics 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 informationCryptography. 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 informationAdaptive 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 informationIntegrated 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 informationCollege 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 informationLinear 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 informationINDEX. 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 informationIndex. 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 informationAppendix 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 informationIntroductory 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 informationAND 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 informationWESTMORELAND 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 informationWAYNESBORO 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 informationCollege 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 informationCatholic 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 informationTABLE 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 informationAlgebra 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 informationFoundations 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 informationPre 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 informationAlgebra 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 informationHow 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 informationCollege 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 informationIntroduction. 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 informationMissouri 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 informationI-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 informationSPECIALIST 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 informationxvi 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 informationPre-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 informationMaths 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 informationBusiness 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 informationSubtraction 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 informationA 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 informationPrentice 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 informationCHAPMAN & 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 informationPrentice 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 informationIndex. 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 informationCheck 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 information1 ** 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 informationArizona 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 informationIntermediate 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 informationEvaluate 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 informationPreCalculus. 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 informationTrigonometry 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 informationPURE 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 informationSecondary 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 informationMATHEMATICAL 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 informationGrade 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 informationOKLAHOMA 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 informationCME 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 informationUse 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 informationTennessee 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 informationPRECALCULUS 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 information1.1.1 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 informationNotation 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 informationAlgebra 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 informationIndex ": :", 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 informationPrentice 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 informationMathematics (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 informationBinomials 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 informationALGEBRA (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 informationContents. 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 informationCopyright 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 informationDRAFT. 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 informationCUMBERLAND 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 informationHonors 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 informationPrentice 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 informationScope 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 informationSignals 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 informationRange 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 informationUnit 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 informationHow 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 informationPrentice 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 informationUnit 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 informationMathematics 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 informationHomework 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 informationIntermediate 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 informationCollege 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 informationUNDERSTANDING 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 informationAlgebra 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 informationMULTIVARIABLE 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 informationSeries 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 informationSTATE 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 informationSingle 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 informationMATHEMATICS (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 informationSAMPLE 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 informationPrecalculus. 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 informationCurriculum 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 informationPre-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 informationCHAPTER 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 informationCheck 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