Books that make you think

I already had a post where I mentioned Reimann Hypothesis after reading The Music of The Primes by Marcus du Sautoy. As far as I recall, I liked the book a lot. It was written for a wide audience and was an easy read. Later, I accidentally found another book on the subject that was intended for more mathematically inclined readers, namely, Prime Obsession by John Derbyshire. Having been fascinated by the subject of prime numbers, the prime number theorem it was a short way to other similar books, such as Prime Number and the Riemann Hypothesis by Barry Mazur and William Stein. Then smoothly transitioning to A Study of Bernhard Riemann’s 1895 Paper by Terrence P. Murphy. Just to conclude with H.M. Edwards Riemann Zeta Function. By the way, the order in which I mentioned the books more or less conveys the mastery of mathematics required to be able to understand what’s going on in them. Which means that two last books require substantial background in calculus and complex analysis. But it’s doable if you have time and prime obsession.

Easy to not-so-easy books

I’d like to provide more details about the books above which I personally read end-to-end and also about ones that I bought, but haven’t finished yet, or only skimmed through.

Actually, I’d rather start with a short description of what the Reimann Hypothesis is by citing the Millennium Problems web site that describes a number of 21st century math problems that can bring you 1,000,000 USD for solving any of them.

So the Riemann Hypothesis is

Source: Millennium Problems

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern.  However, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function
    ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + …
called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation
    ζ(s) = 0
lie on a certain vertical straight line.
This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

Having said that now let’s look at the books.

The Music of The Primes

The book was written by Marcus du Sautoy in 2003. As I mentioned, the book does not require a degree in mathematics to be able to understands what it’s talking about. The material in it is interesting and engaging. In addition to covering, The Prime Number Theorem and Reimann Hypothesis it also covers other topics related to prime numbers usage, like cryptography. It can be a good starting point into a long journey with prime numbers.

Prime Numbers and the Riemann Hypothesis

The book was written by Barry Mazur and William Stein in 2016. It has four parts, where first part intended for a wide audience, and each consecutive part presuppose gradually increasing knowledge of math to be able to grasp the content. What’s interesting about this book that it sheds light on some interesting connections between Riemann Hypothesis and Fourie Transform, which electrical engineers can relate to. Also the book is quite short.

Prime Obsession

The book is written by John Derbyshire in 2003 (same year when Marcus du Sautoy wrote his book). This book has two parts: The Prime Number Theorem and The Riemann Hypothesis, but it goes into nitty gritty details of both of them and don’t allow a reader relax too much. Following the content of the book could require some math background and at times some calculations to be sure that one gets proper understanding of what’s going on. Personally, out of all the books I mention in this post I find this one the most engaging.

A Study of Bernhard Riemann’s 1859 Paper

The book is written by Terrence P. Murthy in 2020. It is one of the two most technical books on the subject that requires substantial background in mathematics. The book provides Riemann’s 1859 paper in full in English and then systematically goes and provide proofs for all relevant parts of Riemann’s paper in subsequent chapters (except for the Riemann Hypothesis itself :). I think Terrence Murphy summarizes who this book is intended for in his own words the best:

Who Is This Book For?
If you are reading this, chances are you have developed a keen interest in the Reimann Hypothesis. Maybe you read John Derbyshire’s excellent book Prime Obsession. Or perhaps you read that the Riemann Hypothesis is one of the seven Millennium Prize Problems, with a $1 million prize for its proof.
To advance your knowledge substantially beyond Derbyshire’s book, you must have (or develop) a good understanding of the field of complex analysis (we will describe that as knowledge at the “hobbyist” level). So, this book is probably not for you unless you are at least at the hobbyist level.

Riemann Zeta Function

The book was written by H.M. Edwards in 1974. I’d rather describe it by continuing the citation from the previous book by Terrence P. Murthy:

After developing an interest in the Riemann Hypothesis, the first stopping point for many is Edwards’ excellent book Riemann’s Zeta Function. The Edwards book provides a wealth of information and insight on the zeta function, the Prime Number Theorem and the Riemann Hypothesis. And that brings us to the next group of people who do not need this book. If you eat, sleep and breath complex analysis, we will say you are at the “guru” level. In that case, the Edwards book will be easy reading and will provide you with the information you need to substantially advance your knowledge of Riemann’s Paper and the Riemann Hypothesis.

As you may tell, “guru” level in math is required to fully digest this book. So it want be easy to say the least.

A good introductory paper on the subject

If you are interested in a short, but engaging introduction into what are Prime Number Theorem and the Riemann Hypothesis I recommend to read Don Zagier’s The First 50 Millions Prime Numbers paper, published in New Mathematical Intelligencer (1977) 1-19.

If you know Russian you can read the same paper that was published in Russian only in 1984 in the Uspekhi Matematicheskikh Nauk journal.

Parting words

All in all, these five books can take a good chunk of a full year to work through or possibly even more, especially the last two. So what are you waiting for? Life is too short to waste it on watching TV series or YouTube nonsense. The treasures of math and deeper understanding of the world are awaiting for ones who know where to look for.

This post continues a series of post were I provide my thoughts on books that I deem worth reading.

This time it is the Infinite Powers book by Steven Strogatz that takes the reader into a realm of taming infinity to grasp nature’s secrets. The title of the book ambiguously plays on a method of using power series to approximate curves and powers which such a method, when exercised skillfully, brought to humankind. The book artfully describes how deferential and integral calculus was developed from Archimedes efforts to measure quadrature of curves through Descartes and Fermat, culminating in calculus invented by Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany working independently. Lots of examples are provided showing how calculus is essential in many of inventions that are an important part of the modern civilization, be it GPS navigation, microwave ovens or the development of effective treatments to viruses induced diseases.

The power of insight

What I liked about the book is how the author is capable of explaining mathematical concepts, that usually require a solid mathematical background, mostly using analogies and down to earth explanations. Though, what I also liked that a mathematically inclined readers were not ignored, since what could, possibly, explained using proper mathematical notation was described as such. It is difficult to appreciate the beauty of calculus without using the mathematical symbols standing for derivative (differentials) and integral, that are so familiar to many people. I should say any person who studied at school should be familiar with them, and if not, the book provides a gentle and very intuitive explanation of what derivative stands for and why it is required, the same goes for integral.

As an example of a good explanation, I want to emphasize a Pizza Proof that is used to find an area of a circle. Since I recalled the formula for it being A = pi * R², it was very interesting to see how the Pizza Proof showed clearly that A = R*C/2, where R is a radius of a circle and C its length. I think since school time I was curious where the power of two came in the area formula that I remembered. So, using the Pizza Proof result and substituting the C in it with the known formula for the length of the circle which is C = pi * 2R (which derivation was also explained in the book ) we get A = pi * R². It was a nice insight, first one of many that the book provided.

I also liked how the method used by Fermat to find the maximum value of a curve, using a smart approach of double intersection, provides a correct result similar to what using derivatives would give. Another example, that was also insightful showed how the concept of derivative and curve interpolation could be used to find patterns in the seasonal changes of day length compared to the rate of change of day length, which both could be approximated by a sinusoidal function with a quarter cycle shift (pi/4 phase shift).

One can’t explain math without using it

Importantly, the concept of derivative was developed and shown very clearly using proper mathematical notation which should be clear even to readers coming from non-mathematical background, since the explanations gradually and systematically build up from simple to more advanced, as a reader progresses through the book (which means that the book should be read continuously). Then the concept of integral is shown quite remarkably well and the two concepts combined to showcase the Fundamental Theorem of Calculus about the duality of derivatives and integrals.

As I always mention, this book passes the test of providing references to other resources on the subject, like original papers of Newton or The Archimedes Palimpsest, which I was unaware of before reading the book.

Summary

It could be that reading Infinite Powers would provide your with appreciation of how calculus is essential in our day to day life and understanding of the world around us. And, maybe, show why mathematics could be beautiful in its own way and also applicable and useful, which could be an unexpected revelation to some.

Other resources

• Unpublished Appendixes to the book
  • How Fermat Integrated x²
  • Newton Schools Leibniz about Power Series
• To get an overview of the book by Steven Strogatz check these two accompanying lectures
  • Infinite Powers
  • The Beauty of Calculus
• If you are curious how Steven is able to explain concepts easily check his essay
  • Writing about Math for the Perplexed and the Traumatized
• And if you liked Infinite Powers consider his other book
  • The Joy of x
• Lastly, if you have a math background and are interesting in dynamical systems consider reading (I should say studying)
  • Nonlinear Dynamics and Chaos

Random thoughts on a reading process

I have to confess I am an obsessive reader. I like books, I like to read them a lot, I like to read them daily. It seems like the most efficient way of reading books or doing other tasks is doing it in a sequential way, where each book completed before the next one is read. The issue is that I cannot help, but defy this approach. I can read a number of books in parallel, jumping from one to another and returning back again. I also can be distracted by a reference to a different book, and so it goes.

Now, we may ask is there any point is such haphazard reading, where the focus is constantly lost, things and thoughts are getting mixed? Personally, I do not find this confusing or disorganizing, but actually, I see some merit in this approach. First, you do not get bored and have some fresh point of view when you return to a book (if you remember where you’ve left last time). Second, since good books are just like candies, it’s difficult to decide where to start, what have next and when to finish.

Stack overflow of the books

Having described my non-linear approach to reading I should mention that nevertheless, on average, I usually able to read 1.5 books a month. This is nice, but there are a couple of books that are still in the stack and they tend to overflow it. There is this book What’s Math Got Do With It by Jo Boaler, then there is The Fabric Of Reality by David Deutsch, underneath is the Unknown Quantity by John Derbyshire. Further below is All Things Being Equal by John Mighton, traveling by US post is a Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction by Paul Nahin, and last but not least is infinite powers by Steven Strogatz.

Speaking of Steven Strogatz book. Last year I was looking for a good book on applied mathematics and stumbled upon Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering which is very good. Then I checked other books Steven wrote and found that in addition to textbooks he also wrote a number of popular science books. One of them was infinite powers. It is enough to read just a few pages of this book to understand that it’s a pure gem in the world of popular science books and if you’d like to really get a good understanding of what differentiation and integration is without going directly to a calculus course then this book may be of interest to you. When I finish reading it, I’ll write a more extensive review of the book.

Random walking

For now, keep reading and try reading sequentially, otherwise start a random walk. Who knows what you stumble upon and where it take you.

This post continues a number of post were I kind of reviewed books that I read and thought it would be helpful to share them with other readers. All Things Being Equal: Why Math Is the Key to a Better World by John Mighton is such a book that deserves to be shared and read by people who care about math education, their children’s math education and math in general. The title of this post is not a mere gimmick, but it means that good books always mention or reference other authors or books that worth reading. This is what exactly happened when I read the book by Anders Ericsson that I mentioned in the previous post. In the Peak Ericsson mentioned John Mighton a Canadian mathematician that incorporated elements of deliberate practice with clear goals and problems that had increasing level of difficulty to teach math to children. This approach is now known as JUMP Math and it is taught to thousands of kids helping them master mathematics while enjoying the subject, unlike in the usual way math is taught in schools.

What is so interesting about this book?

I am past 1/3 of the book, and so far I wasn’t disappointed. The book itself is not only about teaching math to kids. John Mighton discussed also psychological approaches, such as a research into Expertise that plays important role in education in general and in math in particular. He also provides us with an interesting observation that usual math education results in the same distribution of grades among pupils of public schools and among pupils in private schools. It worth mentioning, intellectual poverty a term he coined to emphasize that even though there is a research in to expertise that resulted in clear guidance on how to effectively approach teaching, we as a society still do not incorporate this approach, and what we get is a suboptimal outcome, where kids dislike math, since they think they are not good at it, they have no innate ability or inclination towards it.

Apart from this, John Mighton incorporates a number of examples from math lessons at schools, where he shed light on some of the arithmetic operations that are usually taught as a mere algorithms, without explaining how they work and why. For example, he provides a neat explanation why one could substitute a division of a number by a fraction, by a multiplication of the inverse of that fraction.

Overall

The book is worth reading, since it provides a fresh approach to teaching math to kids and adults alike, in an engaging and exciting way, where kids are gently guided by discovering math step by step, building on the knowledge they gain at a previous step, facing gradually increasing challenges along the way.

This post is similar to my other posts on books I read or am in the process of reading. This time it is second book by John Derbyshire I read on mathematics. The previous book Prime Obsession was an inspiring, interesting and a pleasure to read, since it was all about the Riemann hypothesis. It took me though a little effort to not only read it, but also work through author’s explanations.

So this second book is called Unknown Quantity and it is as captivating as the Prime Obsession was. What is different about the Unknown Quantity that it has more of a historical context on how algebra developed from ancient Mesopotamia to our days.

What I like the most about how John explains mathematical topics in his books is the way he is capable of explaining mathematics the way I never experienced in a school or later in a college. Most of the time math was taught as a given, without trying to convey the essence of the subject, why this formula such and such, how it was conceived and developed. In my opinion, these are very important questions, if not the most important in mathematics. Questioning and curiosity are crucial in mathematical research.

For example, in the Unknown Quantity John shows with enough details how general solutions to second, third and forth degree equations were developed. Why determinant is useful in solving systems of linear equations and why it is important in matrices. These are only some examples, since I haven’t yet finished reading the book.

In short, if you are curious about algebra, and want to know how it evolved historically, and also get some new insight about math you were taught, but never really understood, then the Unknown Quantity is the book for you.

Meet the math book you’re were craving for so long

If you are interested in math and like to have your hands dirty in nitty gritty calculations then the Prime Obsession book by John Derbyshire is just for you. Unlike other popular books on mathematics it provides a gentle and powerful introduction to all math you need to know to understand the Riemann Hypothesis (RH). Reading, I should say, working through the book you’ll learn about interesting properties of Prime Numbers, meet the Prime Number Theorem (PNT) and really understand what the Riemann Hypothesis is all about. In this book you’ll meet Gauss, Euler, Riemann, Hilbert and other renowned mathematicians that influenced the development of mathematics.

What I find most useful about this book

There are books that require a discipline to read through, there are books that are plain boring, and there are books that excite you and your imagination, books that you can’t help, but continue reading more and more. The Prime Obsession is of the latter kind.

What I most like about the book is the historical context John Derbyshire provides throughout the book in addition to his sense of humor and his ability to explain required math in a way that each mathematically inclined person can get fast. I should mention, that having an engineering degree could speed up you understanding significantly, but strictly speaking, it is not required.

In addition, the references to other books on mathematics that John provides are very useful and may provide you with additional materials to digest, like the Hardy’s A Course of Pure Mathematics.

Where to get the book?

Surely, the easiest way to get the book is to buy a copy of it in a Kindle format or a print one. I bought a used one quite cheaply, for less than a Kindle book, which generally cheaper than a hard copy. An old fashion way would be to go to a nearest library and fish for the book their.

A new math book each blog-post 

There are quite a few books on mathematical modeling available out there, but I want to literally and figuratively focus  on a single one, which is Mathematical Modeling by Mark M. Meerschaert.

First, a number of details about the author of the book. Mark Meerschaert is  a University Distinguished Professor in the Department of Statistics and Probability at Michigan State University. He authored a number of books among them the Mathematical Modeling.

What is special about the book?

I have a third edition of the book and I want to provide some thoughts about it. Personally, I like books that provide detailed explanations and ample of examples accompanying the theoretical parts of the book. In my opinion, author’s own view on the subject phrased in his own words, instead of strict adherence to formal definitions is a valuable aid in comprehending mathematical theory.

As for the content of the book, it is divided in three parts which reflects the fact that most of the mathematical models fall into three types 

• Optimization Models
• Dynamic Models 
• Probability Models

Each chapter in the book has detailed examples and quite a few exercises for the reader to tackle. What is also nice that the book is quite practical and have examples from various fields of science and engineering.

References

• Mark Meerschaert in his own words.
• Mark’s page at Michigan State University