Wednesday, November 13, 2013

Sample pages from Chinese Edition
To Download sample pages!

Link to Amazon for the Chinese translation of TBB

Here is a link to the Amazon listing for this Chinese language textbook, published today.

Chinese Language Edition of "Elementary Real Analysis"

Just published today!

This is a Chinese language translation of the first eight chapters of our elementary real analysis textbook.

It can be purchased directly from CreateSpace or from by following the links below.  I will shortly put up some sample pages on our web site

This excellent and accurate translation was done by two accomplished mathematicians: Hongjian Shi and Lei Zhang.


  • Publication Date:  Nov 13 2013
  • ISBN/EAN13:  1492864188 / 9781492864188
  • Page Count:  332
  • Binding Type:  US Trade Paper
  • Trim Size:  8" x 10"
  • Language:  Chinese
  • Color:  Black and White
  • Related Categories:Mathematics / Mathematical Analysis


Purchase information on CreateSpace.

Purchase information on Amazon.

Wednesday, August 7, 2013

What We Are Doing about the High Cost of Textbooks

There is an interesting article in the current issue of the Notices of the American Mathematics Society (August 2013)  from the AIM (American Institute of Mathematics) Editorial Board.   The first paragraph:

What We Are Doing about the High Cost of Textbooks
 Let’s begin with the obvious: The price of textbooks
has risen much faster than the cost of living over
the last thirty years, but there has not been a
significant increase in their quality. We don’t
propose to analyze the economic and educational
factors that underlie this phenomenon. Instead,
we will describe our efforts to help lower the
cost of textbooks for standard undergraduate
mathematics courses in North American colleges
and universities.
Here is a link to the entire article:

In our own smaller way we have done the same thing since 2008 with our mathematics textbooks.  All of our texts are available as FREE PDF files as well as inexpensive paperbacks from either Amazon or CreateSpace.  I expect that this is the future of publishing for academic mathematicians if not for all disciplines. 

Our experience with publishing in the traditional way with a well-known publisher, who should remain nameless (Pearson!), was not at all rewarding.  The texts were outrageously expensive for the students, they offered minimal assistance in preparing the manuscript, scarcely any promotion, modest royalties, and (worst of all) they simply dropped the title from their list (without informing us) when they thought they could make more money with another one. Creating our web site and distributing our texts there has been much more rewarding.

We have no connection (as yet) with the AIM people who are a part of a
larger NSF project1 to develop open source software and curriculum materials for undergraduate mathematics education.  But we agree that this is the future.

 1 Information about Project UTMOST (Undergraduate Teach-
ing in Mathematics with Open Software and Texts) can be
found at

Monday, June 17, 2013

Updated files (June 2013) for THEORY OF THE INTEGRAL

I have updated the FREE PDF file for the text

Click here for the PDF.

Click here to visit the web page on the site.

Thanks go to one of our more alert readers (Tom Savage) for spotting that the correction factor in the integration by parts formula for the Stieltjes integral was misstated (i..e, wrong!) in Section 5.7.

This version of the file has this error corrected.  I also made a small addition to Section 1.2.  Adam Besenyei supplied me with a reference to an early paper of the American mathematician Osgood who addressed the problem discussed in that section.

Monday, June 10, 2013

Review of "The Calculus Integral"

Our textbook THE CALCULUS INTEGRAL has just been reviewed in the most recent issue of the American Math Monthly:

Table of Contents June/July 2013 Math Monthly

The review, by David Bressoud,  covers
  • Calculus by Michael Spivak; 
  • Calculus Deconstructed: A Second Course in First-Year Calculus by Zbigniew Nitecki;  
  • Approximately Calculus by Shahriar Sharhriari;  
  • A Guide to Cauchy's Calculus: A Translation and Analysis of Calcul Infinitesimal by Dennis M. Cates; 
  • The Calculus Integral by Brian S. Thomson
 I cannot provide a link to the entire review (which should be of interest to anyone following the teaching of integration theory).  But maybe fair use allows me to quote two of the paragraphs of interest to readers of our text:
...``The final book is the most challenging of all, Brian Thomson’s The Calculus Integral. The author introduces it as appropriate for an honors course in calculus, but it would require some very special students. I could see it used for a senior seminar. Every proof, every example, and every counterexample is left as an exercise. The proofs
are scaffolded and answers are provided in the second half of the book, but if approached as intended, working through this text would be daunting. Like Nitecki and Spivak, Thomson focuses on the theorems of calculus. He begins with sequences, the careful derivation of properties of continuous functions, and the definition and basic
properties of differentiation. He takes an abrupt turn when he gets to integration. Like the other four authors, Thomson rejects Riemann’s definition of the definite integral, but he takes it a step further. He returns to Newton’s definition of the integral as antiderivative, what he calls “the calculus integral.” Thomson defines the definite integral
of f as the change in a uniformly continuous function that is an antiderivative of f at all but at most finitely many points. A function is integrable if and only if such an antiderivative exists. The fundamental theorem of integral calculus is now the observation, via the mean value theorem, that for any integrable function f and any partition
of the interval over which we integrate, for each i there exists a value xi in the ith subinterval such that the definite integral is exactly the sum over i of f (xi ) times the length of the ith subinterval. From here, he explores Riemann sums as approximations to definite integrals. Thomson’s handling of integration has the advantage that it clarifies the point of the fundamental theorem of integral calculus, which is the general equivalence of two very different approaches to integration.

There is much more to Thomson’s book. He continues on to the monotone convergence theorem, then to sets of measure zero, to absolute continuity, and on to define the Lebesgue integral of f as the change in a function F that is an antiderivative of f almost everywhere and is absolutely continuous in the Vitali sense. He concludes with
the Henstock–Kurzweil integral. His book is magnificent, but even if we stop before preparing for the monotone convergence theorem, it would be supremely demanding for an honors course in the first year.''

Students and teachers of integration theory should be familiar with David Bressoud  whose two "radical" books are must reads for anyone with a serious interest in this subject.

Sunday, February 10, 2013

Theory of the Integral

Now published (February 10, 2013)
Theory of the Integral by Brian S Thomson 
List Price: $15.88
6" x 9" (15.24 x 22.86 cm)
Black & White on Cream paper
422 pages
ISBN-13: 978-1467924399 (CreateSpace-Assigned)
ISBN-10: 1467924393
BISAC: Mathematics / Advanced
This text is intended as a treatise for a rigorous course introducing the elements of
integration theory on the real line. All of the important features of the Riemann integral,
the Lebesgue integral, and the Henstock-Kurzweil integral are covered.
The text can be considered a sequel to the four chapters of the more elementary
text The Calculus Integral  which can be downloaded from our web site. 
For advanced readers, however, the text is self-contained.

 This title can be purchased directly from the CreateSpace link above.  It is currently active on and should soon be as well as the European and UK Amazon sites.
As always our titles are available for FREE download as PDF files in addition to the offer of inexpensive paperback editions.  

Thursday, January 31, 2013

Unprepared for College?

The blogger at College@Home  sent me an interesting infographic illustrating just how unprepared the great majority of American high school students are for college level instruction.

While the readers of our blog (focusing on real analysis) are not in this
category, they may likely end up at some time in their career teaching freshman level mathematics.   The phenomenon is particularly acute for mathematics instruction:  many students arrive with their stellar math grades from high school and end up completely crushed by a simple calculus course.  There is a huge gap between the understanding expected at the high school level and the demands and ideas at the level just above.

Much can be written about this (I won't).  Here I want to mention that the same thing can happen with an elementary real analysis course.  I have never taught this subject without a few "weepers."  These are students who have always been convinced that they are "good at mathematics."  They aced the calculus courses, and yet are overwhelmed at the elementary real analysis level.  They can understand everything in class (they claim) but simply cannot write a correct proof for even the most trivial limit theorems of the course.  They have miserably failed the first midterm, something that they couldn't conceive would happen to them ever.

Why is analysis so difficult?  They aren't much amused when I tell them that this material was routinely taught in the former Soviet Union to 15 year olds.

Early study of any subject may not prepare you for the next level.  Real analysis courses are very subtle (not difficult really)  and demand greater insight and thought than the computational courses that precede them. 

Learning a new subject does not always create challenges, but you should be aware that at some time in your life you will certainly encounter a new subject that overwhelms you at first.  Perhaps the most important skill you can learn is how to dig yourself out of trouble when a subject bites you hard.  You will have to do it eventually. 

For high school students, if this happens in the first weeks of college---well enjoy the experience and learn the necessary study skills.  You will need them again.  For math students--don't get too smug.  At some point in your life you will have to work very hard to get to the next level.

We will always find ourselves at some point in our life "unprepared" for the next level of study.  Consider it a question of character as to whether you can salvage yourself.  As the College@Home blogger points out, probably 75% of college students arrive "unprepared."  If we can't improve that percentage, at least we can warn them.