Preface 

vii  


1  (62) 

Classroom Connections: Representing Patterns 


1  (2) 

Reflections on Classroom Connections: Representing Patterns 


3  (11) 


14  (2) 

Classroom Connections: A Quadratic Sequence 


16  (1) 

Reflections on Classroom Connections: A Quadratic Sequence 


16  (5) 

Finite Arithmetic Sequences 


21  (2) 


23  (6) 


29  (5) 

Classroom Connection: Counting Tools 


34  (13) 


47  (4) 


51  (12) 

Arithmetic and Algebra of the Integers 


63  (32) 

A Few Mathematical Questions Concerning the Periodical Cicadas 


64  (1) 

Classroom Connections: Multiples and Divisors 


64  (1) 

Reflections on Classroom Connections: Multiples and Divisors 


65  (5) 


70  (3) 

Least Common Multiple and Greatest Common Divisor 


73  (2) 

The Fundamental Theorem of Arithmetic 


75  (7) 

Revisiting the LCM and GCD 


82  (7) 

Relations and Results Concerning LCM and GCD 


89  (6) 

The Division Algorithm and the Euclidean Algorithm 


95  (50) 

Measuring Integer Lengths and the Division Algorithm 


95  (7) 


102  (4) 

Applications of the Representation GCD (a, b) = ax + by 


106  (4) 


110  (15) 


125  (20) 

Arithmetic and Algebra of the Integers Modulon 


145  (70) 

Classroom Connections: Divisibility Tests 


146  (1) 

Reflections on Classroom Connections: Justifying the Divisibility Tests 


146  (2) 


148  (3) 


151  (13) 

Comparing Arithmetic Properties of Z and Zn 


164  (5) 

Multiplicative Inverses in Zn 


169  (5) 

Elementary Applications of Modular Arithmetic 


174  (14) 

Fermat's Little Theorem and Wilson's Theorem 


188  (5) 

Linear Equations Defined over Zn 


193  (7) 

Extended Studies: The Chinese Remainder Theorem 


200  (3) 

Extended Studies: Quadratic Equations Defined over Zn 


203  (12) 

Algebraic Modeling in Geometry: The Pythagorean Theorem and More 


215  (38) 

The Significance of Daryl's Measurements and Related Geometry 


216  (1) 

Classroom Connections: The Pythagorean Theorem 


217  (1) 

Reflections on Classroom Connections: The Pythagorean Theorem and Its Converse 


217  (9) 

Computing Distance in TwoDimensional and ThreeDimensional Euclidean Space: The Distance Formula 


226  (1) 

An Extension of the Pythagorean Theorem: The Law of Cosines 


227  (2) 

Integer Distances in the Plane 


229  (1) 

Pythagorean Triples: Positive Integer Solutions to x2 + y2 = z2 


230  (7) 

Extended Studies: Further Investigations into Integer Distance Point SetsA Theorem of Erdos 


237  (3) 

Extended Studies: Additional Questions Concerning Pythagorean Triples 


240  (10) 


250  (3) 

Arithmetic and Algebra of Matrices 


253  (58) 

Classroom Connections: Systems of Linear Equations 


254  (1) 

Reflections on Classroom Connections: Systems of Linear Equations 


255  (6) 

Rational and Irrational Numbers 


261  (7) 

Systems of Linear Equations 


268  (14) 

Polynomial Curve Fitting: An Application of Systems of Linear Equations 


282  (4) 

Matrix Arithmetic and Matrix Algebra 


286  (12) 

Multiplicative Inverses: Solving the Matrix Equation AX = B 


298  (8) 


306  (5) 
Glossary 

311  (10) 
References 

321  (2) 
Answers to (Most) OddNumbered Exercises 

323  (22) 
Photo Credits 

345  (2) 
Index 

347  