A Bridge to Advanced Mathematics: From Natural to Complex Numbers

544 Pages·2022·9.057 MB·English

by  Sebastian M. Cioabă, Werner Linde

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This series was founded by the highly respected
mathematician and educator, Paul J. Sally, Jr.
2-color cover: Pantone 432 C (Gray) and Pantone 300 C (Blue) 542 pages on 50 lb • Spine: 1 1/16 inch • Softcover • Trim Size 7 X 10

A Bridge to
Advanced
Mathematics
From Natural to
Complex Numbers

T h e UNDERGRADUATE TEXTS • 58
SERIES
Pure and Applied
Sally
A Bridge to
Advanced
Mathematics
From Natural to
Complex Numbers
Sebastian M. Cioaba˘
Werner Linde
EDITORIAL COMMITTEE
Giuliana Davidoff Tara S. Holm
Steven J. Miller Maria Cristina Pereyra
Gerald B. Folland (Chair)
2020 Mathematics Subject Classification. Primary 00-01, 00A05, 00A06, 05-01.
For additional informationand updates on this book, visit
www.ams.org/bookpages/amstext-58
Library of Congress Cataloging-in-Publication Data
Names: Cioaba˘,SebastianM.,author. |Linde,Werner,1947-author.
Title: A bridge to advanced mathematics : from natural to complex numbers / Sebastian M.
Cioaba˘,WernerLinde.
Description: Providence, Rhode Island : American Mathematical Society, [2023] | Series: Pure
and applied undergraduate texts, 1943-9334; volume 58 | Includes bibliographical references
andindex.
Identifiers: LCCN2022034218|ISBN9781470471484(paperback)|9781470472139(ebook)
Subjects: LCSH:Numeration–Textbooks. |Numbers,Natural–Textbooks. |Numbers,Complex–
Textbooks. | AMS: General – Instructional exposition (textbooks, tutorial papers, etc.). |
General–Generalandmiscellaneousspecifictopics–Generalmathematics. |General–General
and miscellaneous specific topics – Mathematics for nonmathematicians (engineering, social
sciences,etc.). |Combinatorics–Instructionalexposition(textbooks,tutorialpapers,etc.).
Classification: LCCQA141.C482023|DDC513.2–dc23/eng20221014
LCrecordavailableathttps://lccn.loc.gov/2022034218
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Contents
Preface ix
1. TheContentoftheBook ix
2. HowtoUseThisBook? x
Chapter1. NaturalNumbersℕ 1
1.1. BasicProperties 1
1.2. ThePrincipleofInduction 10
1.3. ArithmeticandGeometricProgressions 29
1.4. TheLeastElementPrinciple 34
1.5. Thereare10KindsofPeopleintheWorld 45
1.6. Divisibility 51
1.7. CountingandBinomialFormula 65
1.8. MoreExercises 78
Chapter2. IntegerNumbersℤ 81
2.1. BasicProperties 81
2.2. IntegerDivision 84
2.3. EuclideanAlgorithmRevisited 88
2.4. CongruencesandModularArithmetic 104
2.5. ModularEquations 124
2.6. TheChineseRemainderTheorem 130
2.7. FermatandEulerTheorems 135
2.8. MoreExercises 156
v
vi Contents
Chapter3. RationalNumbersℚ 159
3.1. BasicProperties 159
3.2. NotEverythingIsRational 166
3.3. FractionsandDecimalRepresentations 173
3.4. FiniteContinuedFractions 193
3.5. FareySequencesandPick’sFormula 205
3.6. FordCirclesandStern–BrocotTrees 220
3.7. EgyptianFractions 228
3.8. MoreExercises 233
Chapter4. RealNumbersℝ 237
4.1. BasicProperties 237
4.2. TheRealNumbersFormaField 244
4.3. OrderandAbsoluteValue 247
4.4. Completeness 254
4.5. SupremumandInfimumofaSet 257
4.6. RootsandPowers 264
4.7. ExpansionofRealNumbers 272
4.8. MoreExercises 288
Chapter5. SequencesofRealNumbers 291
5.1. BasicProperties 291
5.2. ConvergentandDivergentSequences 297
5.3. TheMonotoneConvergenceTheoremandItsApplications 310
5.4. Subsequences 319
5.5. CauchySequences 325
5.6. InfiniteSeries 329
5.7. InfiniteContinuedFractions 352
5.8. MoreExercises 362
Chapter6. ComplexNumbersℂ 367
6.1. BasicProperties 367
6.2. TheConjugateandtheAbsoluteValue 375
6.3. PolarRepresentationofComplexNumbers 379
6.4. RootsofComplexNumbers 385
6.5. GeometricApplications 397
6.6. SequencesofComplexNumbers 400
6.7. InfiniteSeriesofComplexNumbers 405
6.8. MoreExercises 416
Contents vii
Epilogue 419
Appendix. Sets,Functions,andRelations 421
A.1. Logic 421
A.2. Sets 430
A.3. Functions 438
A.4. CardinalityofSets 455
A.5. Relations 472
A.6. Proofs 485
A.7. Peano’sAxiomsandtheConstructionofIntegers 495
A.8. MoreExercises 505
Bibliography 515
Index 517