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THE REAL NUMBERS AND REAL ANALYSIS

BY ETHAN D. BLOCH

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THE REAL NUMBERS AND REAL ANALYSIS BY ETHAN D. BLOCH

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THE REAL NUMBERS AND REAL ANALYSIS BY ETHAN D.

BLOCH PDF

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Review

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)

From the Back Cover

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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 is accessible to students who have prior experience with mathematical proofs and who have not previously studied real analysis. The text includes over 350 exercises.

Key features of this textbook:

- 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 relying upon sequences for proofs; presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property

- includes a thorough discussion of some topics, such as decimal expansion of real numbers, transcendental functions, area and the number p, that relate to calculus but that are not always treated in detail in real analysis texts

- offers substantial historical material in each chapter

This book 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.

About the Author

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THE REAL NUMBERS AND REAL ANALYSIS BY ETHAN D.

BLOCH PDF

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THE REAL NUMBERS AND REAL ANALYSIS BY ETHAN D.

BLOCH PDF

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.

Sales Rank: #1107884 in eBooks

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Published on: 2013-04-11

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Released on: 2013-04-11

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Format: Kindle eBook

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Review

From the reviews:

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This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. The choice of material and the flexible organization, including three different entryways into the study of the real numbers, making 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 is accessible to students who have prior experience with mathematical proofs and who have not previously studied real analysis. The text includes over 350 exercises.

About the Author

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

Most helpful customer reviews

16 of 16 people found the following review helpful. Among the best introductory analysis books in English By Charles Loyola

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read it without any prior exposure to mathematics beyond knowing how to read/write proofs.

Side note: the latter skill is traditionally forced onto students by throwing them into the deep end (sometimes even in the form of a calculus course), but lately there has been a tendency to devise courses for this purpose ("introduction to mathematics" and the like, where one learns informal logic and is exposed to some essential mathematical structures). There are also a few books for this, such as Velleman's How to Prove It. Incidentally, Bloch also published a book in that vein (Proofs and Fundamentals: A First Course in Abstract Mathematics (Undergraduate Texts in Mathematics)), which I also think is among the best of its kind. Another good one with a more "hands-on actual math" approach is A Concise Introduction to Pure Mathematics, Third Edition (Chapman Hall/CRC Mathematics Series). My point is: if you are familiar with the rudiments in such books, you should be able to follow Bloch's Analysis book painlessly.

As a cursory glance at the contents will reveal, this is not a textbook in "Real Analysis" in the regular sense, i.e. advanced investigations into the reals. The material overlaps part of that in Apostol's Calculus and Zorich's Analysis, and broadly speaking, this book is in the same category as those two, but with significant differences in coverage: it is smaller, and so covers much less ground (Apostol also covers multivariate calculus and some linear algebra, Zorich also has more topics and more theorems within each topic). For example, complex numbers are not used in Bloch. On the other hand, the 1st chapter contains a rigorous foundational discussion of the reals, in which they are meticulously constructed from the naturals (starting with the Dedekind/Peano axioms).

As mentioned in the introduction, the distinguishing feature of this book in terms of material is that sequences and series are left to the last chapters. Thus, limits, differentiation and integration are treated without them. As a result, the reader is given a clearer view of how the canonical results involving continuity follow from the least upper bound axiom (or its equivalent statements). For some, this would be considered a drawback, since some of the classical theorems have rather complicated proofs when sequences are not used. For me however, the distinguishing feature of this text is its lucidity and concision (yet it is as rigorous as they come!), as well as the goal-directed organization of material. The choice to omit a lot of commonly-included incidental material here is astute. Reading books like Zorich's can be a bit overwhelming in that many side-routes are taken (sometimes for theorems that are very important in their own right), and the beginner may lose track of the larger logical structure over all these results. In Bloch, only the core material is discussed, and so every paragraph is to the point, and well motivated. Also, he avoids abstruse notation that is often found in Analysis texts.

The exercises are not difficult, and so might disappoint somebody who is used to fancy ones (in the Russian tradition), or those that involve physics and real-world applications (as in Zorich). Also, there are relatively few, but they are well chosen. They are almost always of the "have you understood the above definitions and theorems" type, and usually do not require much imagination beyond that. These are not disadvantages in my opinion - everything in this book feels *to the point*.

Moreover, there are historical sections that are very informative, and impressively long for such a textbook (though they are given at breakneck speed - the history of calculus cannot be compressed into a few pages otherwise.) To sum up, I found this book much clearer and easier to read than Apostol, Zorich, and "Baby Rudin" (the only books that I am familiar with that are more-or-less comparable). For readers struggling with such classic texts, and who are also interested in the extra material they contain, I would recommend starting with Bloch: secure the foundations and then move on.

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that reader has taken a calculus course, this is an understandable omission (the same is true for, say, "Baby Rudin"). However, for the reader who would like, from the outset, to study "calculus" with the rigor associated with "analysis" courses (i.e. the European/Russian, not the US, way), this book would not be ideal. The only such English analysis book I know of that matches Bloch's level of rigor and user-friendliness, while also having many examples and exercises involving specific functions is Mathematical Analysis I (Universitext). On the other hand, the foundational stuff (construction of the reals, etc.) is not treated there, and Bloch does it wonderfully.

0 of 0 people found the following review helpful. it's a very easy read. Cons

By Eduardo Bravo Pros:

- Book is very rigorous

- Proofs are written out fully in that there are practically no instances where the phrase 'left to the reader' entails anything more than a straightforward verification.

- Due to the two points above, it's a very easy read.

Cons:

- There are many errors (even beyond those in the books errata).

- A lot of the proofs seem ad hoc. I realize that this is practically the case in every real analysis book, but I thought that it was even more pronounced in this book than in others.

1 of 2 people found the following review helpful. Too repetitive and verbose

By Einstein

Though Mr. Bloch does a phenomenal job writing about the Israeli/Palestinian conflict, his capacity for writing about Real Analysis is much less clear. Mr. Bloch should look at the simplicity of his political analysis, and try to apply that same simplicity to this book. The book is unstructured, too complex to follow, and boringly repetitive with regards to the axioms. Simply put: it's confusing and too verbose. Furthermore, this book could easily be edited down by about 250 pages.

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THE REAL NUMBERS AND REAL ANALYSIS BY ETHAN D.

BLOCH PDF

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Review

From the reviews:

From the Back Cover

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Analysis is accessible to students who have prior experience with mathematical proofs and who have not previously studied real analysis. The text includes over 350 exercises.

About the Author

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