How We Got from There to Here: A Story of Real Analysis

Reviewed by Roberto De Leo, Associate Professor, Howard University on 2/24/21

Comprehensiveness rating: 5 see less

This book covers all standard content of an introductory Real Analysis course.

Content Accuracy rating: 5

The book's content is perfectly accurate, I could not detect any error.

Relevance/Longevity rating: 5

This book presents Real Analysis in a way that seems to the reviewer absolutely new and quite interesting and exciting. Hopefully it will also encourage the release of more books with a similar approach.

Clarity rating: 5

The book's prose is easy to follow and it avoids systematically, on purpose, unnecessarily technical terms in order to increase readability. In fact, the whole point of this particular book is to make as easy and effortless as possible to undergraduate students the task of understanding the most central concepts of real analysis, that unfortunately look, at first sight, unnatural and hard to grasp.

Consistency rating: 5

The book presents the topics of Real Analysis in the order in which they were historically developed. The consistency of the book comes automatically out of this.

Modularity rating: 5

The text is finely divided in many sections and therefore highly modular.

Organization/Structure/Flow rating: 5

Once again, topics in this text are sorted in historical order. On one side, this is not the most logical order. On the other side, learning how and why new concepts had to be introduced during the development of the theory does help the reader to understand the reason for the introduction of concepts and definitions otherwise quite hard to grasp.

Interface rating: 5

This book is based on the pretext package, a very solid package to produce books in the most widespread online (e.g. HTML) and offline (e.g. PDF) formats. In particular, its HTML interface is excellent.

Grammatical Errors rating: 5

I could not find any grammar problem (not, though, that English is not my native language).

Cultural Relevance rating: 5

This category does not really apply to math books in general, and to this in particular.

Comments

This undergraduate level text is, in my experience, a precious and unique Real Analysis text because tries to solve, by a historical approach, one of the main problem of teaching this topic, namely the fact that the most important concepts -- in particular the 'epsilon-delta' definition of limits -- are quite unintuitive and mastering them is a challenge for most students (it certainly was for me when I first met it!). Moreover the book does an excellent job at pushing strongly the reader from the very beginning into being an active reader, for instance by insisting that he/she should always verify and complete the details of calculations in the book while reading the book -- always keep pencil and scrap paper while reading a math book! I agree with the authors that this is the best -- and perhaps only -- way to really learn from a math book. The whole book is architectured so to push the reader to "enter" the book and "fill the dots" by him/herself while learning the material. Of course hints are provided for the most challenging points.
Perhaps one can say that the keyword of this book is 'motivations'. Unfortunately often in math publications the reader is left clueless of which motivations lead to definitions that seem to appear 'out of the blue', causing sometimes a complete loss of interest in the subject and anyways making unnecessarily hard learning the material. This book shows that a historical approach provides an automatic solution to this problem. It would be very interesting to have a similar text about Complex Analysis.

Reviewed by Patrick Shipman, Associate Professor, Colorado State University on 1/7/16

Comprehensiveness rating: 5 see less

Besides standard material for an analysis book, this text runs on themes motivated by discussions of the history of analysis. Along with that come some topics that would not normally appear in an analysis text, but that broaden the appeal of the subject. These include Fourier series and their role in the development of analysis, Snell's law (how Leibnitz derived it, and later in the text how Bernoulli used it to solve the Brachistochrone problem), some number theory (Fermat's little theorem), and how power series can be used to solve differential equations.

Content Accuracy rating: 5

The text is accurate. Many of the proofs appear only as sketches that the reader will hopefully fill in accurately. The authors encourage readers to talk to others, hopefully making it likely that mistakes in readers' proofs will be caught.

Relevance/Longevity rating: 5

This is classical material, arranged to a large extent according to historical discovery.

Clarity rating: 4

At a first reading, I enjoy the informal style and reminders that the reader needs to work out the details of the calculations. I imagine that this might get old for some of those learning for the first time and reading it multiple times. The figures could be better made and (this is the reason for a ranking of 4 rather than 5) axes are not always labeled.

Consistency rating: 5

The text is consistent in both terminology and style.

Modularity rating: 5

It is possible to assign small reading sections. The text is designed to initiate ideas early (such as in the opening discussion of the meaning of 'number") and come back to those ideas later.

Organization/Structure/Flow rating: 5

The text opens with three 'lessons' that i) get the reader thinking about numbers and definitions, ii) encourage the reader to work out the details of computations and algebraic manipulations given in the text, and iii) give some philosophical insights into solving problems. After that, rather than just giving the reader results of analysis, Part I raises questions on numbers (building on the opening lesson) and series. Only after another interlude on Fourier series does the text focus more on finding answers to the problems. The construction of real numbers is left to the end, at which point the student has more appreciation for thinking about what numbers are. An aspect of the book that I particularly appreciate is how much thought the authors have given to initiating ideas that they come back to later--such as the theme of understading what a number is, which opens the first lesson of the text and also rounds out the text in the concluding material.

Interface rating: 5

Besides a normal index. there are lists of Theorems, Lemmas, Corollaries, Definitions and Problems at the end of the book, all with links to the pages in the book. The index might lack complete comprehesiveness (I didn't find 'Snell's Law' there), but that does not really matter since one can search in a .pdf file without an index.

Grammatical Errors rating: 5

The grammar is great.

Cultural Relevance rating: 5

It is rare for a math text to include cultural/historic background as a truly important part of the presentation of the material.

Comments

The text, being very enjoyable to work through and leaving it the reader to complete proofs, would seem to work very well with (but not only with) a flipped classroom style of teaching. Something that is usually underdeveloped in mathematics education is letting students discover interesting problems; this text makes some progress towards doing that by telling the tale of how interesting problems in analysis were come upon.

Reviewed by Clark Gaylord, Adjunct, Virginia Tech on 6/10/15

Comprehensiveness rating: 5 see less

From Number to Cantor's Theorem, this book brings you on a journey of the development of mathematical analysis. Several important stops along the way include Taylor Series, the Bolazano-Weierstrass Theorem, and Cauchy Sequences, I cannot think of any notable omissions along the road.

Content Accuracy rating: 4

Spot on, though the text is built on problems that leave a lot of work for the reader.

Relevance/Longevity rating: 5

The approach the authors take is essentially timeless, in that it brings us to modern analysis. I imagine fifty years from now we could still look at this book as a very good exposition of how we got to where we are in twentieth century mathematics, and that will still be quite relevant for our moderately advanced undergraduates and casual mathematically curious students.

Clarity rating: 5

There are times when the prosaic nature of the narrative is a little strained, but the intention is to make the text more accessible. It is not a serious detraction, nor does it significantly get in the way of the text's movement. In the final analysis, it is a pleasure to read and the text moves logically.

Consistency rating: 5

Very strong. The authors have a genuine interest in the story they weave and keep the tale together throughout the text.

Modularity rating: 5

The text builds naturally through the history of mathematical analysis, so modularity is not itself a strong objective. With that said, you can reasonable go to any section and learn from it, but this text intentionally and appropriately is intended to be taken more as a sequential whole than in unrelated parts.

Organization/Structure/Flow rating: 5

Time is a natural progression -- the time of mathematics development and how results build from one generation of mathematicians to the next -- and this text flows naturally through this development.

Interface rating: 5

The text is well put together and easy to use. The links in the PDF are useful.

Grammatical Errors rating: 5

I observed no significant grammatical issues

Cultural Relevance rating: 5

This is exactly the approach to deliver cultural literacy to the budding mathematically inquisitive student. This text contains much of the lore of mathematical analysis that is appropriate for our undergraduate students.

Comments

This is a delightful text, and exactly how we should develop undergraduate analysis to our mathematical studies students. There are several classic results, but more importantly it shows how analysis develops, and hence why modern analysis has the character it does. This is how we should prepare our students for advanced undergraduate or graduate analysis, giving them a perspective of mathematics. For example, instead of taking an informal aside to explain the countability of the rationals, this is done naturally through the text (Problem 201).

The text leaves many details, including most proofs, as exercises. If you adopt this text you will probably want to take many of these as illustrative examples in your lecture development. Hence, there would be a significant amount of preparatory work to deliver this material, but it gives a very good framework to base your class on.