The Real Numbers and Real Analysis
(Sprache: Englisch)
This rigorous, detailed introduction to real analysis presents the fundamentals clearly and includes definitions, theorems and proofs. Mirroring the structure of standard calculus courses makes it especially accessible to university students of mathematics.
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Produktinformationen zu „The Real Numbers and Real Analysis “
This rigorous, detailed introduction to real analysis presents the fundamentals clearly and includes definitions, theorems and proofs. Mirroring the structure of standard calculus courses makes it especially accessible to university students of mathematics.
Klappentext zu „The Real Numbers and Real Analysis “
This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.
This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.
Key Features:
- Provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis
- Presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus
- Emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than making use of sequences for proofs
- Presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property
- Relates real analysis to previously learned material, including detailed discussion of such topics as the transcendental -functions, area and the number pi
- Offers three different entryways into the study of real numbers, depending on the student audience
- Contains historical context, biographical anecdotes, and reflections on the material in each chapter
The text is accessible to students with prior experience with mathematical proofs who have not previously studied real analysis.
The text includes over 350 exercises.
The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.
Key Features:
- Provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis
- Presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus
- Emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than making use of sequences for proofs
- Presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property
- Relates real analysis to previously learned material, including detailed discussion of such topics as the transcendental -functions, area and the number pi
- Offers three different entryways into the study of real numbers, depending on the student audience
- Contains historical context, biographical anecdotes, and reflections on the material in each chapter
The text is accessible to students with prior experience with mathematical proofs who have not previously studied real analysis.
The text includes over 350 exercises.
The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.
Inhaltsverzeichnis zu „The Real Numbers and Real Analysis “
- Preface- To the Student
- To the Instructor
1. Construction of the Real Numbers
2. Properties of the Real Numbers
3. Limits and Continuity
4. Differentiation
5. Integration
6. Limits to Infinity
7. Transcental Functions
8. Sequences
9. Series
10. Sequences and Series of Functions
- Bibliography
- Index
Autoren-Porträt von Ethan D. Bloch
Dr. Ethan D. Bloch of Bard College is the author of two Springer publications "A First Course in Geometric Topology and Differential Geometry," and the first and second editions of, "Proofs and Fundamentals: A First Course in Abstract Mathematics." More information about Dr. Ethan D. Bloch can be found on his person web page: http://math.bard.edu/bloch
Bibliographische Angaben
- Autor: Ethan D. Bloch
- 2011, XXVIII, 554 Seiten, Maße: 16,7 x 24,5 cm, Gebunden, Englisch
- Verlag: Springer, Berlin
- ISBN-10: 0387721762
- ISBN-13: 9780387721767
Sprache:
Englisch
Rezension zu „The Real Numbers and Real Analysis “
From the reviews:"The author's purpose is to cover with this book the necessary mathematical background for secondary school teachers. The book is also useful for an introductory one real variable analysis course. ... The book has an interesting and useful collection of exercises ... . Last but not least, the historic notes are excellent. ... I consider this book of great interest for the academic training of the future secondary school teachers, so the author's purpose is greatly fulfilled." (Juan Ferrera, The European Mathematical Society, April, 2013)
"Bloch (Bard College) has written an introductory book on analysis at the undergraduate level, with enough material for at least two semesters of studies. The author writes very carefully and includes numerous examples and historical insights. The exposition is generally excellent. The book provides all proofs with enough details for most undergraduates to follow through without undue difficulties... Overall, an excellent book. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, and faculty."
-D. M. Ha, Ryerson University, Choice, February 2012
"The most distinctive characteristic of this text on real analysis is its three-in-one feature. It was designed specifically for three distinct groups of students. ... The book was motivated by a need for a textbook for the M.A.T. students, but is intended to have enough flexibility to serve the other groups as well. ... this is a strong text, especially for students who need more guidance and support. The book gives an instructor plenty of options for planning a course." (William J. Satzer, The Mathematical Association of America, August, 2011)
Pressezitat
From the reviews:"The author's purpose is to cover with this book the necessary mathematical background for secondary school teachers. The book is also useful for an introductory one real variable analysis course. ... The book has an interesting and useful collection of exercises ... . Last but not least, the historic notes are excellent. ... I consider this book of great interest for the academic training of the future secondary school teachers, so the author's purpose is greatly fulfilled." (Juan Ferrera, The European Mathematical Society, April, 2013)
"Bloch (Bard College) has written an introductory book on analysis at the undergraduate level, with enough material for at least two semesters of studies. The author writes very carefully and includes numerous examples and historical insights. The exposition is generally excellent. The book provides all proofs with enough details for most undergraduates to follow through without undue difficulties... Overall, an excellent book. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, and faculty."
-D. M. Ha, Ryerson University, Choice, February 2012
"The most distinctive characteristic of this text on real analysis is its three-in-one feature. It was designed specifically for three distinct groups of students. ... The book was motivated by a need for a textbook for the M.A.T. students, but is intended to have enough flexibility to serve the other groups as well. ... this is a strong text, especially for students who need more guidance and support. The book gives an instructor plenty of options for planning a course." (William J. Satzer, The Mathematical Association of America, August, 2011)
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