Introduction to Mathematical Thinking Algebra and Number Systems

by Gilbert, Will J.; Vanstone, Scott A.

• ISBN13: 9780131848689
• ISBN10: 0131848682
• Edition: 1st
• Format: Paperback
• Copyright: 2004-07-22
• Publisher: Pearson

Summary

Besides giving readers the techniques for solving polynomial equations and congruences,An Introduction to Mathematical Thinkingprovides preparation for understanding more advanced topics in Linear and Modern Algebra, as well as Calculus. This book introduces proofs and mathematical thinking while teaching basic algebraic skills involving number systems, including the integers and complex numbers. Ample questions at the end of each chapter provide opportunities for learning and practice; theExercisesare routine applications of the material in the chapter, while theProblemsrequire more ingenuity, ranging from easy to nearly impossible.Topics covered in this comprehensive introduction range from logic and proofs, integers and diophantine equations, congruences, induction and binomial theorem, rational and real numbers, and functions and bijections to cryptography, complex numbers, and polynomial equations.With its comprehensive appendices, this book is an excellent desk reference for mathematicians and those involved in computer science.

Table of Contents

Preface	vii
Logic and Proofs	1	(24)
The Language of Mathematics	1	(1)
Logic	2	(5)
Sets	7	(2)
Quantifiers	9	(4)
Proofs	13	(5)
Counterexamples	18	(7)
Exercise Set 1	20	(2)
Problem Set 1	22	(3)
Integers and Diophantine Equations	25	(32)
The Division Algorithm	25	(3)
The Euclidean Algorithm	28	(6)
Linear Diophantine Equations	34	(7)
Integers in Different Bases	41	(4)
Prime Numbers	45	(12)
Exercise Set 2	50	(2)
Problem Set 2	52	(5)
Congruences	57	(32)
Congruence	57	(2)
Tests for Divisibility	59	(2)
Equivalence Relations	61	(3)
Modular Arithmetic	64	(4)
Linear Congruences	68	(5)
The Chinese Remainder Theorem	73	(5)
Euler-Fermat Theorem	78	(11)
Exercise Set 3	82	(2)
Problem Set 3	84	(5)
Induction and the Binomial Theorem	89	(22)
Mathematical Induction	89	(7)
Recursion	96	(2)
The Binomial Theorem	98	(13)
Exercise Set 4	104	(2)
Problem Set 4	106	(5)
Rational and Real Numbers	111	(14)
Rational Numbers	111	(2)
Real Numbers	113	(1)
Rational Exponents	114	(3)
Decimal Expansions	117	(8)
Exercise Set 5	121	(1)
Problem Set 5	122	(3)
Functions and Bijections	125	(38)
Functions	125	(2)
The Graph of a Function	127	(3)
Composition of Functions	130	(3)
Inverse Functions	133	(2)
The Inversion Theorem	135	(3)
Cardinality	138	(4)
Inverse Trigonometric Functions	142	(3)
Exponential and Logarithmic Functions	145	(6)
Permutations	151	(12)
Exercise Set 6	153	(4)
Problem Set 6	157	(6)
An Introduction to Cryptography	163	(22)
Cryptography	163	(2)
Private-Key Cryptography	165	(2)
Public-Key Cryptography	167	(1)
The RSA Scheme	168	(17)
Exercise Set 7	180	(2)
Problem Set 7	182	(3)
Complex Numbers	185	(42)
Quadratic Equations	185	(2)
Complex Numbers	187	(4)
The Complex Plane	191	(4)
Properties of Complex Numbers	195	(4)
Polar Representation	199	(4)
De Moivre's Theorem	203	(3)
Roots of Complex Numbers	206	(7)
The Fundamental Theorem of Algebra	213	(14)
Exercise Set 8	218	(3)
Problem Set 8	221	(6)
Polynomial Equations	227	(44)
Polynomials and Factoring	227	(8)
Complex Roots of a Polynomial	235	(4)
Rational Roots of a Polynomial	239	(3)
Approximating Real Roots	242	(5)
Polynomial Inequalities	247	(2)
Cubic Equations	249	(4)
Multiple Roots	253	(2)
Partial Fractions	255	(4)
Equations over a Finite Field	259	(12)
Exercise Set 9	262	(3)
Problem Set 9	265	(6)
Appendix	271	(8)
Trigonometry	271	(3)
Inequalities	274	(5)
Further Reading	279	(2)
Answers to the Odd-Numbered Exercises and Problems	281	(12)
List of Symbols	293	(2)
Index	295