Infinite Powers to explain

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

The books’ stack is changing

And so it continues…

This is a quick update on the status of the stack of the books I am reading. I am glad that it’s changing and worrying that its size remains the same overtime. The issue is, as I already mentioned in other posts, good books reference other good books and here we go. This time the culprit was Mind and the Cosmic Order book by Charles Pinter that mentioned the Selfish Gene and the ‘meme’ term created by Richard Dawkins. I have to say that I’ve heard about this book a long time ago, but never thought it was worth reading. But since a number of authors respected by me mentioned it I could no longer skip reading in. I should also mention that David Deutsch mentioned the meme term, coined by Dawkins, in his The Beginning of Infinity book. So did Jeff Hawkins in his recent A Thousand Brains book.

By the way, did you know that the preface to Jeff Hawkins’ book was written by Richard Dawkins?

The books that were on the stack physically or virtually since the last time

  • Mind and the Cosmic Order by Charles Pinter (A Book of Abstract Algebra brought me here)
  • The Right Kind Of Crazy by Adam Steltzner (well, have you heard about Curiosity and Perseverance?)
  • The Interstellar Age : Inside the Forty-Year Voyager Mission by Jim Bell (courtesy of watching a documentary on YouTube)
  • A Thousand Brains: A New Theory of Intelligence by Jeff Hawkins (following him since 2004)
  • Extraterrestrial: The First Sign of Intelligent Life Beyond Earth by Avi Loeb (a Google suggestion)
  • Unknown Quantity by John Derbyshire (I read the Prime Obsession so it was a natural continuation)
  • All Things Being Equal by John Mighton (was referenced by Anders Ericsson in his book on expertise )

Currently in progress

Infinite Powers by Steven Strogatz (well his text book on dynamic systems is a culprit). With regard to Steven Strogatz I want to mention his article he wrote for the Notices of the American Mathematical Society in 2014. In this article he descried tips on successful popular-science writing.

Writing about Math for the Perplexed and the Traumatized

Next to be popped from the stack

Selfish Gene by Richard Dawkins (mentioned by David Deutsch, Jeff Hawkins, Charles Pinter and others)

Rest of the stack

  • A Book of Abstract Algebra by Charles Pinter (Unknown Quantity sparked an interested in abstract algebra in me)
  • Number-Crunching by Paul Nahin (Paul Nahin’s book about Oliver Heaviside brought me here)
  • Discreet Mathematics by Lovasz, Pelikan and Vesztergombi (bought as a used book in 2014 while roaming in US, New Hampshire)
  • Applied mathematics by J. David Logan (who can resists math?)
  • Mathematical Modeling by Mark M. Meerschaert (the same as above)
  • The Mathematical Experience by Davis Hersh ( bought as a used book in 2014 while roaming in US, New Hampshire )

Why random reading could be useful

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.

Good books come in tuples

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.

Unknown Quantity is a math book to work through

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.

Prime Obsession with Math

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.

Mathematical Modeling

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

Math books Applied for Good

Math books and more books on math

Following the path of applied mathematics and popular science with math inclination I want to bring to your attention a couple of books that some of you may find helpful if not insightful.

Oliver Heaviside’s Maxwell’s Equations

Actually, I would rather start from a book which is an amalgam of history and mathematical physics in one and it’s a book about the self-taught mathematical physicist Oliver Heaviside who brought to you the so called Four Maxwell’s equations.

equations

book_heaviside

The book is Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age written by Paul J. Nahin an emeritus prof. of Electrical Engineering in University Of New Hampshire which we’ll return to later in the post.  What is interesting about the books is that it has a right amount of math for readers who are interested not only to know who Oliver Heaviside was,  but also what he did as a physicist and engineer.

 

 

 

 

Okay, the books

While reading very interesting Applied Mathematics book by David J. Logan (3rd edition, Ch. 4.4 Green’s Functions, p. 253)

step_function

I was, as always, diverged by the mentioning of the Heaviside Step function in the text that I felt an urgent surge to check a biography of this incredible person and, lo an behold, I was able to find the Paul Nahin’s book mentioned above and also quite interesting and short  article in the Physics Today magazine Oliver Heaviside: A first-rate oddity

David J. Logan

Having mentioned, David Logan I should say that I am reading the 3rd edition of his book, which is available in Scribd if you have a membership there, and even for free for 30 days trial period. It is always possible to buy the 4th edition, but the price is, quite frankly, astronomical.

logan

Applied Mathematics 4th edition by David J. Logan. What I like about this book is the detailed examples that help you understand the content of the book better, but even more I like the way David Logan explains the physical rational behind the differential equations. It helps very much to know how and why this or that math technique is applied in practice. In addition, another applied mathematician Mark H. Holmes book’s is also mentioned by David Logan which you also may find useful.

 

 

 

Paul J. Nahin

Now that’s get back to Paul Nahin. It turn’s out he produced a whole series of books on Physics, Mathematics, Electrical Engineering and Computer Science which can be called popular, but actually are an essays full of wonderful applied mathematics. Paul is able to explain things in engaging and easy to understand manner. As people like to say, I wish I had come across his books earlier in my life, but it is what it is and it’s good that I was able to find them. Thanks to the Scribd digital library I was able to glimpsed through all of his books available there and I’d recommend to math inclined readers to check the following books.

simple_physics

 In Praise of Simple Physics: The Science and Mathematics behind Everyday Questions will take you into the physics journey that you could have been missing since your school or collage years. Maybe, you weren’t able  to understand it back then or had no time, but this time it will be different thanks to Paul’s ability to explain physics in an easy to grasp way.

 

 

 

 

And one additional book that I find quite impressing 

crunchung numbers

Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction as all books by Paul J. Nahin this one draws examples from different areas of exact sciences and engineering that will keep you awake at night following the stories and trying to solve the puzzles yourself.

 

 

 

 

Mark H. Holmes

holmes

Remember, I’ve mentioned Mark H. Holmes so he also wrote a couple of books on applied math, and I’d recommend you to check his Introduction to Numerical Methods in Differential Equations which I find also very useful and a helper while reading aforementioned books on applied math. Unlike his Introduction to the Foundations of Applied Mathematics, which I find cryptic due to the lack of detailed examples, Introduction to Numerical Methods has quite a few of them. This makes the book kind of easy to digest.

 

 

 

Last, but not least

To make sense in this whole unfamiliar forest of applied mathematics there is a nice book that has all you need in one place classified and summarized to be your guidance on your quest to master the math and apply it for good. It is

all_of_it

The Princeton Companion To Applied Mathematics.

 

 

 

 

 

 

 

Dare think, keep on going, and be carried forward on wings of math muse.

References

Math is in the air

Start this year the right way

New year’s time is usually a time to make some new year’s resolutions. I won’t do it and instead this year for me at least will be solely focused on applied mathematics. Math topics interest  me for a long time. But I never took it seriously to invest quality time into studying advanced math topics with enough detail. This year will be different. The plan is to start from some quite general books on math that try to approach the topic in an engaging way like Measurement book by Paul Lockhart and slowly transitioning to more technical books for applied mathematics like Elements of Applied Mathematics by Zeldovich and Myskis and  Nonlinear Dynamics and Chaos by Steven H. Strogatz.

A little bit about the books

Why these three books you may wonder? Actually, there are four books I want to focus on. What is so specially about these books is the fact that they do not simply talk about one specific field in math with a very narrow focus on the subject, but like Donald Knuth’s The Art Of Computer Programming volumes approach the subject in a more general way without being fearful to delve in fields of physics, engineering, biology etc.

To summarize the books are:

measurement

Measurement by Paul Lockhart which tries to show math as an engaging activity that resembles arts, such as music, painting where there is a place for creativity,  a joy of new discoveries or a pain of being stuck trying to get a solution.

 

 

 

 

2020-01-01 16_03_56-Elements Of Applied Mathematics _ YA. B. Zeldovich, A. D. Myskis _ Free DownloadElements of Applied Mathematics by Zeldovich and Myskis which is an old book, but it’s still relevant today, at least in many parts of it, as it was back in 1972. As authors themselves put in the foreword of the book

So our advice is: read our book and study it. But even if there is not time enough to make a deep study, then merely read it it like a novel, and it may be just the thing you will need for solving some difficult problem.

 

 

math_arnoldMathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians by Vladimir Arnold. This book is a collection of applied math problems  that were drawn from various fields such as physics, engineering etc.

Note: If you’re capable of reading in Russian then this book is available in full for free here.

 

 

nonlinear-dynamics-and-chaosAnd finally there is the voluminous  Nonlinear Dynamics and Chaos by Steven H. Strogatz that makes me feel better by having hundreds of differential quotations.

Note: The older edition of this book is available for free, for example, here.

A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages.

 

Now is the best time to start

This paragraph is dedicated to myself and possibly you. Remember that now is the best time to start doing what you wanted, but postponed ad infinitum. So start by taking small steps to big results later. Or at least to having satisfaction from solving some non-trivial tasks and applying a new skill to real life problems.

Good mathematics books to read and work through

Math is entertaining when you try to play with it

If you have an interest in mathematics be it pure or applied there are plenty of books were written on the subject varying by the depth of the presented material and the need to know a certain level of math to be able to not only read the book but also gain some practical insights by working through the examples and tasks. Personally, I like more books that freely use math in the description of the examples and give tasks for a reader to accomplish. It seems like this is the only way to really understand what author tried to convey. It’s like reading a book on programming and trying right away the code samples, changing them.

Following is the description of a number of popular math books that I find very insightful, useful and entertaining, since reading them not only gives an appreciation of the beauty of math, but also makes one feel better when he or she is able to find a solution to tasks in the book.

A Mathematician’s Lament

lament-1

I want to start with the article that Paul Lockhart wrote, that later was expanded to a book with the same name which is A Mathematician’s Lament. The first part of the book is essentially the article itself. So if you read only the article you read half a book already. If you have an Amazon account you may buy a Kindle version of the book and it may take you a couple of hours to finish it. Then you may return it for refund and that’s it. You got the entire book for free.

The article and consequently the first part of the book presents readers with a very strange way that math is taught in schools using very clever analogy to how music might have been taught if it would be taught like math in most schools today. 

The second part of the book tries to show some solutions to the problems of how math is taught that were described in the first part of the book.

I recommend this book to all who disliked math and thought that it was boring and  disgusting. Maybe, you’ll change you thoughts on the subject.

From popular to more hands-on math books

primes

Recently, I’ve read the The Music of the Primes book by a mathematician Marcus du Sautoy on Riemann hypothesis. Previously I read a  book about primes and as part of my studing at college learned a thing or two about them, but I never appreciated how interesting it may be to follow the path along with mathematicians trying to prove Riemann hypothesis. This hypothesis is one of the seven problems that Clay Mathematics Institute thinks worth 1,000,000 USD for one who’s able to prove it. Though less money is given to one who will disprove it. 

 

 

The Riemann hypothesis states that all interesting solutions of the equation

ζ(s) = 0, where ζ(s) is a Riemann Zeta function, ζ(s) = 1 + 1/2s + 1/3s + 1/4s + …

 lie on a certain vertical straight line which is  Re(s) = 1/2,  Re(s) stands as for a real part of the argument s.

What I liked about The Music of the primes that Marcus wasn’t afraid to show a little bit of mathematics that was related to the saga of trying to prove the Riemann Hypothesis. He also was able to create an adventurous story that connected math and physics and such a mundane thing as RSA cryptosystems that was used in Internet secure web communications. In addition, Marcus du Sautoy mentions a large number of prominent mathematicians who deserve a separate book to be written about them.

 

Math is a pleasure to play with when it is presented in an interesting manner

The last book that I want to mention in this post is the book by Vladimir Arnold who was one of the distinguished Russian mathematician and the one who solved the Hilbert thirteenth’s problem in the age of twenty. Only a number of numerous books written by Vladimir Arnold were translated into English from Russian, but even the ones that were are still very exciting to read and include lots of tasks to be resolved by a reader. I should say that Arnold’s popular math books are actually a kind of math courses. If you’ll check one of his books you’ll understand what I mean by this. What I find appealing in the Arnold’s books is his ability to explain complex topics in a simple way that is entertaining and makes you long for more. By the way if you can read in Russian you may find all of his books and many others for free at the Moscow Center For Continuous Mathematical Education web site library.

If you like physics and applied math, then you gonna find Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians book by Vladimir Arnold very informative and entertaining at once. In it you’ll find a number of task and solutions to them drawn from various fields of physics along with a simple to grasp explanations that makes complex things seem beautiful.

There are additional Vladimir’s books that may be found in English so if you’ll find this book useful to you then there are others you can enjoy too.

It’s only the beginning

This post is only the first one in a series of post that will accompany me while I myself read and work through the popular math books and try to report on interesting gems I find in them.