The title of this post can seem strange to you. What mathematics has to do with magic? In my opinion, it depends on what feelings you took from math lessons at school, college or university. There are people who were frightened by math or bored by it. But there were also a lucky few who were able to spot something beautiful about math on their own or thanks to a good teacher. There’s another approach when math lessons cannot do the trick for you. Genrich Altshuller didn’t call it magic, but an encounter with a miracle. In the book How to become a genius. Life strategy of a creative person by Altshuller and Vertkin they mentioned that creative people at an early age encountered a miracle that heavily influenced their trajectory in life. If we take mathematics as an example, showing a kid that mathematics isn’t a boring, but actually interesting field to participate in can be such a decisive encounter with a miracle. And it seems to me the best way to appreciate the beauty of mathematics is by actually doing it.

To organize such an encounter for my daughter, I suggested her to create a YouTube Kids channel about mathematics for kids where she can upload videos on mathematics that kids can understand and relate too. She agreed and with a little help from me she was able to create two videos already. One about square numbers, that as their name suggest, have a shape of a square. Another one about a visual way of depicting the addition of two numbers that third graders are tasked to do at school. Let’s take for example, 111 + 37. It turns out that before kids start using column method addition it’s difficult for them to find the right answer. Using binary trees can visualize the composition of the two numbers and help a kid in summing these numbers together.

One interesting aspect of creating such short video clips is that it encourages a parent and a kid to work together to be ready to present the topic clearly in a way that a kid can understand what the video is about. Also, teaching something is one of the best ways to understand it yourself. One additional thing to mention is that the videos in the channel can cover topics that are not explained at school at all or taught only in upper grades, which provides a kid with an advantage of an early exposure to advanced and interesting mathematical topics.

This way of introducing kids to mathematics has a promise of removing boredom and rut repetition of solving similar exercises and shows kids that math has more to it than how it’s usually taught at school.

Previously, in one of my posts I wrote that I tend to read books as if I proof-read or edit them. There are a number of advantages in doing so. For example, reading books end to end ensures deeper understanding of the content. Attempting to solve each exercise in a book is also a positive thing to do, because even if you do not solve it, you can get a valuable insight just because you tried hard to solve an exercise. When you follow this advice, after a book or two this starts to happen automatically to you and you can’t help, but spot spelling mistakes, wrong diagrams used or mistakes in formulas etc.

To see a real example of the above suggestions let’s look at the All Things Being Equal book by John Mighton founder of the JUMP math teaching method. This book is not a math textbook, but a book about how math can be taught to kids. The main idea is that math can be taught to an average person and it doesn’t take a genius to like math and be productive in it. There are a number of exercises that John Mighton provides in the book to showcase his approach of teaching math at school. I’ve tried each one of them and lo and behold found a number of mistakes in the book. One mistake was spotted by my daughter. And this is exactly what I want to write about next.

Some concrete examples

I’ve got a softcover edition of the book, ISBN 9780735272903. On page 201 of this edition John Mighton provides an example of how a simplified Sudoku version can be accessible and enjoyable to kids. The regular Sudoku puzzle 9 by 9 cells looks as follows. Each row and column in the table below should contain all of the digits from 1 to 9, when no digit can appear more than once.

A simplified version is 4 by 4 cells, which is easier for kids to start to play with. Now, I’ll provide all four puzzles on the page 201 of the book.

Puzzle #1

This puzzle is solvable, so let’s move on to the next

Puzzle #2

This puzzle has a mistake in it. Try to find it yourself or jump to the end of the post for an answer.

Puzzle #3

It’s solvable. Move on.

Puzzle #4

This puzzle has a mistake in it and my daughter, who was in Grade 2 at the time, was able to spot the mistake.

Pay attention, that there cannot be a duplicate value on any row, column in the table.

Closure

When you spot and collect a certain amount of mistakes it’s good to inform an author of the book or a publisher about them. Usually, authors are grateful if you report mistakes to them and it could even result in a friendship or a nice communication with them.

It wasn’t the case with All Things Being Equal where there was no response when I sent the errata to the JUMP Math contact email. So I sent an email directly to the Vintage Canada who is the publisher of this book.

So take care and report mistakes, who knows where it will take you.

I guess you also could have had the same feeling when you learnt algebra at school. Some formulas were clear and understandable, but some were cryptic and it was unclear how would anyone derive them. And then the only way to master it is to memorize it. For example, there is this known formula for a difference of squares:

a^{2} – b^{2} = (a – b) * (a + b) = a^{2} + ab – ba – b^{2} = a^{2} – b^{2} , (1)

Then, there was a little bit more cryptic formula for a difference of cubes, which is not that obvious for a regular student:

a^{3} – b^{3} = (a – b) * (a^{2} + ab + b^{2}) = a^{3} + a^{2}b + ab^{2} – ba^{2} – ab^{2} – b^{3} = a^{3} – b^{3}, (2)

So, I think you get it and the next formula is for a^{4} – b^{4} ,

And finally, we get to the most cryptic formula that could be frustrating in a school algebra lesson, the formula for a difference of two positive whole numbers (integers) of power of n

Now, the last formula seems frightening, and most interestingly one could ask, how did in the world anyone derive it? Also, how do you use it correctly?

Take it slow

Let’s look at it in a slow motion. If we look at how we get from formula (1) to formula (4) we can notice that there is some symmetry in the numbers in the second braces in each of the formula.

Do you see it? When there is a^{2} on the left side there is a corresponding b^{2} on the right, when there is a^{2}b on the left, there is a corresponding b^{2}a on the right side, etc. So this is the symmetry I am talking about. The general formula is actually a factorization of a polynomial formula. But we can look at it in a different manner, just to understand how to use it properly. The derivation of the general formula is a little bit more complex and can be found here.

One interesting thing to notice is that the sum of powers of each a, b or there multiplication ab in the second braces is always n – 1.

(a + b) = a^{1} + b^{1} , i.e. the powers are 1, 1

(a^{2} + ab + b^{2}) = a^{2} + a^{1}b^{1} + b^{2} , i.e. the powers are 2, 1 + 1, 2

(a^{3} + a^{2}b + b^{2}a + b^{3}) = (a^{3} + a^{2}b^{1} + a^{1}b^{2} + b^{3}) , i.e. the powers are 3, 2 + 1, 1 + 2, 3

(a^{n−1}+ a^{n−2}b + a^{n-3}b^{2} + … + a^{2}b^{n-3} + ab^{n−2} + b^{n−1}), i.e. the powers are n – 1, n – 2 + 1 = n – 1 , n – 3 + 2 = n – 1, 2 + n – 3 = n – 1, etc.

Now, also let’s pay attention that we can treat 1 as 1 = a^{0} or 1 = b^{0}, and let’s look again at the expressions above

I hope you can see that there is a systematic pattern which is going on here.

Rules of the game

Rule 1: The number of members in the second braces is always as the power of the initial expression, say two for a^{2 }– b^{2}; three for a^{3} – b^{3} etc.

Rule 2: The sum of the powers of each member in the second braces is n – 1, which was already shown in the previous examples.

Pay attention that this can be also proven by mathematical induction. But I leave it as an exercise for you.

How to use this formula and how to zip it

Now that we’ve noticed there is a pattern this pattern show us how to use the formula in a simple way without the need in rote memorization or blindly using someone else derivation.

The only thing is to remember that the first braces always have (a – b) and in the second braces the sum of the powers of each member is n -1. Let’s look at the concrete example of a^{8} – b^{8}.

Let’s start from the second braces, and write each member without powers in accordance to Rule 1. We know there should be n, i.e. 8 such members.

(ab + ab + ab + ab + ab + ab + ab + ab)

Now, let’s use the Rule 2 and add powers to each member in the second braces, remembering that for a‘s, powers start from n – 1 and decrement by 1 for each consecutive a, and for b‘s powers start from 0 power and increment by 1 for each b until n – 1. Applied to our example,

for a‘s: a^{7}, a^{6}, a^{5}, a^{4}, a^{3}, a^{2}, a^{1}, a^{0}

and b‘s: b^{0}, b^{1}, b^{2}, b^{3}, b^{4}, b^{5}, b^{6}, b^{7}

Now, putting these together in the formula we get,

= (a – b) * (a^{7} + a^{6}b + a^{5}b^{2} + a^{4}b^{3} + a^{3}b^{4} + a^{2}b^{5} + ab^{6} + b^{7}).

Zip it

So, now we ready to zip this formula using the math notation for the sum:

where k increments from 0 to n – 1, i.e. 0, 1 , 2, …, n – 1.

An interesting turn of events

What is nice about this formula is the fact that it’s actually a concise description of an algorithm that checks whether a certain string is a palindrome.

The main idea is to take a sequence of letters (an array of characters in programming speak), and then start comparing

First vs. last letter

Second vs. one before last

etc

For each such case above check whether letters are the same. If there is at least one instance when they are not the same, then it’s not a palindrome.

In Java programming language this algorithm could be implemented as follows (run this code in online Java compiler)

public class Main {
public static void main (String[]args){
String word = "TENET";
System.out.println (isPalindrome(word));
}
static boolean isPalindrome (String word){
char[] charArray = word.toCharArray();
int n = charArray.length;
for (int k = 0; k <= n - 1; k++){
if (charArray[k] != charArray[(n - 1) - k]){
return false;
}
}
return true;
}
}

I think that I already wrote previously that a couple of years ago I bought the Elements of Programming book by Alexander Stepanov and Paul McJones. The issue was that the book content was hard for me to grasp at the time. I can hardly say that I now understand it better, but now I got where the rationale for that book came from and why it was written the way it was. It turns out the Alexander Stepanov as a mathematician was influenced deeply by Abstract Algebra, Group Theory and Number Theory. The elements of these fields of mathematics can be traced in the Elements of Programming clearly. For example, chapter 5 is called Ordered Algebraic Structures and it mentions among other things semigroup, monoid and group, which are elements in Group Theory. Overall, the book is structured somewhat like Euclid’s Elements, since the book starts from definitions, that are later used to build gradually upon in other chapters of the book.

Which brings me to the main topic of this post. By the way, the post is about a different book Alexander Stepanov wrote with Daniel Rose and that book was created by refining the notes for the Four Algorithmic Journeys course that Stepanov taught in 2012 at A9 company (subsidiary of Amazon). The course is available in YouTube and it consists of three parts each having a number of videos and the Epilogue part.

I highly recommend to watch it to anyone who is curious about programming, mathematics and science in general. The course is entertaining and it talks about how programming, or more exactly algorithms that are used in programming, are based on algorithms that were already known thousands of years ago in Egypt, Babylon etc. Alexander Stepanov has a peculiar way of lecturing and I find this way of presentation funny. The slides for the course and the notes that were aggregated in the Three Algorithmic Journeys book draft are freely available at Alexander Stepanov’s site.

So the book which I want to mention is From Mathematics to Generic Programming which was published in 2014 and is a reworked version of the Three Algorithmic Journeys draft. This is how Daniel Rose describes this in the Authors’ Note of the book.

The book you are about to read is based on notes from an “Algorithmic Journeys” course taught by Alex Stepanov at A9.com during 2012. But as Alex and I worked together to transform the material into book form, we realized that there was a stronger story we could tell, one that centered on generic programming and its mathematical foundations. This led to a major reorganization of the topics, and removal of the entire section on set theory and logic, which did not seem to be part of the same story. At the same time, we added and removed details to create a more coherent reading experience and to make the material more accessible to less mathematically advanced readers.

My verdict

As authors mentioned the book is geared towards Generic Programming, but I recommend to read both of them in parallel, since each one complements the other. I think that the Three Algorithmic Journeys is even better than the From Mathematics to Generic Programming (FM2GP). First, it’s free and second, ironically, it’s more generic than the FM2GP book.

It seems like there is hardly a person who didn’t hear the phrase “Thinking outside of the box”. As Wikipedia entry says it’s “a metaphor that means to think differently, unconventionally, or from a new perspective.” While it sounds good in theory, it is unclear what one should do to think unconventionally, differently, creatively etc. Only demanding from someone to think outside of the box, doesn’t provide clear guidance on how to achieve this goal.

The same issue happens in education, when a student is taught any subject that requires thinking beyond what was taught in a lesson or a lecture. There are people who can do better than others in such situations and we tend to label them as creative, smart and sometimes genius. But the psychological research into what makes experts experts, for example done by Anders K. Ericsson et al, shows that this has to do more with the way an expert practiced, and not the innate cognitive abilities.

So what makes us creative and can it be taught and learned? The short answer is yes and the rest of this post will try to justify this answer. The question of creative thinking is relevant in most fields of daily life where problems arise and when there is no obvious way of how to solve them. Here we go into realm of innovation and invention. There are many definitions of these two terms, so let me quote one from Merriam-Webster on the difference between invention and innovation

What is the difference between innovation and invention? The words innovation and invention overlap semantically but are really quite distinct.

Invention can refer to a type of musical composition, a falsehood, a discovery, or any product of the imagination. The sense of invention most likely to be confused with innovation is “a device, contrivance, or process originated after study and experiment,” usually something which has not previously been in existence.

Innovation, for its part, can refer to something new or to a change made to an existing product, idea, or field. One might say that the first telephone was an invention, the first cellular telephone either an invention or an innovation, and the first smartphone an innovation.

Chuck Swoboda, in his The Innovator’s Spirit book also provides detentions for an innovation and an invention that will be discussed in this post and they are

An invention, by definition, is something new—something that’s never been seen before. An innovation, on the other hand, especially a disruptive one, is something new that also creates enormous value by addressing an important problem.

While I do not have any objection to his definition of an innovation, I don’t agree with the definition of an invention. Saying that invention “is something new that’s never been seen before” is too vague a definition to be practical. It takes a quick look into submitted patents to see that there are lots of similar, if not outright identical patents issued for inventions. Which means the definition of invention being something never seen before fails to capture this. Also by the same token invention “being something new” fails too.

But it turns out there is quite precise definition, that exits since 1956, of a technical invention, which was provided by Genrich Altshuller and Rafael Schapiro in a paper About the Psychology of Inventive Creativity (available in Russian) published in Psychology Issues, No. 6, 1956. – p. 37-49. In the paper they mentioned that as a technical system evolves there could arise contradictory requirements between parts of the system. For example, lots of people use mobile phones to browse the internet. To be able to comfortably see the content on the screen of the phone, the screen should be as big as possible, but this requirement clashes (contradicts) with the size of the mobile phone, which should be small enough to be able to hold it comfortably in a hand or carry it in a pocket.

Altshuller and Shapiro defined the invention as a resolution of the contradictory requirements between parts of the system, without having to trade off requirements to achieve the solution. This definition of invention allows to talk precisely about what can be thought as invention and what can’t. Generally speaking, contradictory requirements can be resolved in space, time or structure. For example, returning to the mobile phone example, to resolve the contradiction in structure of the phone, between the size of the screen and the size of the phone there is a functionality that was introduced in mobile phones that allows to screencast the video and audio from a phone to a TV screen using Wi-Fi radio signal. YouTube application on Android phones supports this functionality.

Altshuller wrote a number of books on the subject of creative thinking, particularly books that developed the Theory of Inventive Problem Solving (abbreviated as TRIZ in Russian). In these books the ideas about a contradiction, an invention and an algorithmic approach (ARIZ) to how to invent by solving contradictions in technical problems are elaborated. To name just a few books in chronological order, written by Altshuller

How to learn to invent (“Как научиться изобретать”), 1961

Algorithm of Invention(“АЛГОРИТМ изобретения”), 1969

Creativity as an Exact Science: Theory of Inventive Problem Solving (“ТВОРЧЕСТВО как точная наука: Теория решения изобретательских задач”), 1979

What is important to mention about the books is that they contain systematic, detailed and step by step explanations of how to invent using an algorithm. Lots of examples and exercises for self-study included in them. The books by Altshuller somewhat resemble in their content and in a way of presenting the material books written by George Polya.

Polya being a productive mathematician was also interested in how to convey his ideas in a way that could be easily understood by other people. To this end he wrote a number of books directed to pupils, students, teachers and general audience.

For example, his book How To Solve it first published in 1945 is a step by step instruction set on how to approach mathematical problems in a systematic way, using heuristics that mathematicians accumulated doing math for thousands of years. It very much resembles to me the structure and approach taken in Altshuller’s How to learn to event. Later, Polya wrote two additional books on how mathematicians think and how they arrive to mathematical theories. Each of the books consist of two volumes and they are

What is interesting to mention is that the books written by Polya and Altshuller more than fifty years ago contained very insightful ideas and heuristics to tackle math and inventive problems. But today it’s still difficult to find a widespread adoption of these ideas in education, industry or elsewhere. For example, The Princeton Companion to Applied Mathematics book from 2015 mentions only a rudimentary number of math Tricks and Techniques in the chapter I, Introduction to Applied Mathematics, on pages 39-40, out of 1031 pages.

As well as the general ideas and principles described in this article, applied mathematicians have at their disposal their own bags of tricks and techniques, which they bring into play when experience suggests they might be useful. Some will work only on very specific problems. Others might be nonrigorous but able to give useful insight. George Pólya is quoted as saying, “A trick used three times becomes a standard technique.” Here are a few examples of tricks and techniques that prove useful on many different occasions, along with a very simple example in each case.

– Use symmetry… – Add and subtract a term, or multiply and divide by a term…. – Consider special cases… – Transform the problem… – Proof by contradiction… – Going into the complex plane…

As a summary, if you are curious whether it’s possible to learn how to be more creative, inventive or, in general, approach problems in a systematic way, then check the books by Genrich Altshuller and George Polya. They may provide you with just the tools that you were looking for, but didn’t know where to find.

In this post I want to talk about why providing good and detailed explanations can be a key in deep understanding of things in different fields of life. Particularly, I want to talk about detailed proofs in mathematics that have good step by step explanations of how the proof was constructed. Recently, I’ve started to read the books on math that are piling on my table just as the image above depicts.

I want to point your attention to the second book at the top of the pile, which is Prime Numbers and the Riemann Hypothesis book from 2016 written by Barry Mazur and William Stein. This book talks about Riemann Hypothesis by starting from ‘simple’ math and gradually moving to details about Riemann Hypothesis that require more advanced math background.

So what levels of understanding and well explained proofs have to do with the content of the book. Well, you see in the first part of the book, which authors claim requires some minimal math background a read sees this

Here are two exercises that you might try to do, if this is your first encounter with primes that differ from a power of 2 by 1:

1. Show that if a number of the form M = 2^n – 1 is prime, then the exponent n is also prime. [Hint: This is equivalent to proving that if n is composite, then 2^n -1 is also composite.] For example: 2^2 – 1 = 3, 2^3 -1 = 7 are primes, but 2^4 – 1 = 15 is not. So Mersenne primes are numbers that are

– of the form 2^ prime number – 1, and

– are themselves prime numbers

Some context for the quote

Here comes a little bit of a context about this quote. The quote comes from, part 1, chapter 3: ‘”Named” Prime Numbers’, on page 11. The chapter describes what are Mersenne Primes, which are prime numbers that are one less than a power of two:

M = 2^n – 1

Also, it’s good to know that a prime number is a whole number (positive integer) that can be divided only by itself or 1. For example,

2, 3, 5, 7, 11 … are prime numbers since they can only be divided by themselves and 1.

Now, that we know what prime numbers are, I want to draw your attention to the point in the quote where it says, that the exercises are good for ‘your first encounter with primes that differ from a power of 2 by 1‘. Well, in my opinion these exercises are good only for readers who have at least a BSc in Math or possibly an engineering degree. In a couple of sentences we’ll see why I think so. Since I consider myself as a person who is interested in math and have a BSc degree in Electronics, I think the first part of the book which is indented for a layman person is just for me. Were you able to arrive at the proof for that simple exercises above?

Frankly, I wasn’t able to prove it. So, I went and looked for a proof on the internet. As always, Wikipedia is at the top of the Google search when it comes to math topics. Lo and behold there is a page in the wiki about Mersenne primes, what they are and a number of proofs related to them. One of the proofs was exactly the solution to the exercise in the quote above. Here it comes:

Note: Since, I had no time to learn how to use LaTeX properly in the WordPress, and believe me WordPress doesn’t make blogger’s life easy I write the proofs and picture them.

Does this proof looks like a piece of cake to you? Is it intuitive and easy to grasp? In my opinion, it’s not and also this proof shows why Wikipedia gets a good portion of criticism about its content.

What I find not so obvious about that proof is how the right hand side of the equation came about. Especially, the part that has powers of a * (b – 1) , a* (b – 2), … , a, 1. And then the statement that says ‘By contrapositive, if 2^p -1 is prime then p prime‘. In this particular statement, if you had no courses on mathematical proof, logic or Abstract Algebra, then the words composite and contrapositive can be a little bit mysterious.

I thought to myself, well, the proof looks kind of unclear to me, so I searched better using the internet and also checked the books I own. I was able to find a couple of proofs that were easier to understand and also was able to find a proof in the Book of Proof by Richard Hammack that you can see in the title image for the post. It’s the blue one and it comes third from the top of the pile.

Let’s start with the proof from the Book of Proof, which sounds like a good place to begin with. The proof is a solution to the exercise 25 from Chapter 5 in the book.

If we look carefully at this proof it look almost exactly as the proof in the Wikipedia. It even look less clear. But one important thing to notice is that this proof shows where the ‘1’ comes from on the right hand side of the equation, in comparison to the proof in the wiki. I think so, since 2^(ab-ab) which equals 1, provides more information than a simple number 1. This is because it provides some clues on how the proof was constructed. But for a reader who does not remember math, both of the proofs are not that helpful. Also, we need to take into consideration, that the Book of Proof intended audience is undergraduate students of exact sciences. So the book presupposes some math background.

Some math background

We already mentioned that a prime number is a positive integer that can be only divided by itself or number 1. All other integers, are composite, since any of them can be composed by multiplying prime numbers. This is where composite comes from. As for contrapositive the Merriam-Webster site provides this definition

a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition or theorem and interchanging them

if not-B then not-A ” is the contrapositive of “if A then B ”

For example,

If it was raining then it is wet.

Then contrapositive statement would be

If it is not wet then it wasn’t raining.

Returning back to the original exercise from the Prime Numbers and the Riemann Hypothesis book, now, it becomes a little bit clear, what the hint [Hint: This is equivalent to proving that if n is composite, then 2^n -1 is also composite.] there was for. So the proofs, went to prove that

If n is composite then 2^n – 1 number is composite

and by contrapositive

If 2^n – 1 number is not composite(i.e. prime) then n is not composite (i.e. prime)

Polynomial factorization

Now, that we know what contrapositive proof is that’s turn to the right hand side part of the equation which is

Since this post is becoming to long I need to make it shorter, which defeats the point of providing detailed explanations 😦

But I’ll provide some hints on how the right hand side was derived.

There is a known math formula to compute the following expression a^n – b^n

It turns out that the equation in the proofs that were mentioned in this post has the same structure as the formula to derive a^n – b^n. But the most interesting part is this. If you look at the last equation the Roman numeral I stands for the initial composite number 2^n – 1, and it is composed by multiplying Part II by Part III.

Part I = Part II * Part III

What is interesting about this composite number and that me use A, B and C instead of using I, II and III

A = B * C

is that B and C are factors of A, or alternatively B is a divisor and C is a quotient.

So to summarize the initial number 2^n -1 is composite since it is a product of two other numbers.

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.

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.

Have you heard about a new book Thinking Better: The Art of the Shortcut in Math and Life by Marcus du Sautoy? If not, you may consider given it a try, since this book is very engaging. The book is about various topics in mathematics and how they provide shortcuts that can be used in different fields of daily life to find solutions to seemingly daunting problems. These shortcuts speed up the computations and free time for you to do other things. Each chapter of the book starts with a puzzle that the reader is asked to solve. Then the main content of the chapter is related to the puzzle, while the solution is provided in the end.

I’ve already read The Music of the Primes that was also written by Marcus and thoroughly enjoyed it. What makes this book to stand out in comparison to The Music of The Primes is that it’s focused on practical advise that could be actioned by readers. But what I find most compelling about this book is that reading it is not enough to get the most out of the content, there is also a need to play with the content of each chapter, carefully examining it. Otherwise the subtle details and insight could be lost and not understood properly.

Marcus tries to present his ideas in a way that is accessible to a reader who is not supposed to be a math expert. That is why it seems his approach is to use as little of math terminology and formulas as possible. Even the word ‘formula’ doesn’t show up until page 20 into the book. This approach to writing popular science books is not new and it tries to achieve a trade off between a number of readers who may be frightened by the mathematical notation, and the number of readers who could be disappointed by the lack of it. Which brings me to the main point of this post. Even though Marcus doesn’t shy away from writing down some equations to exactly convey his ideas, there are places in the book where additional mathematical details could clarify his point even better. I also suggest to name the math objects as they are used in mathematics. There is no harm in doing that just like Steven Strogatz did in The Joy of x or John Derbyshire in the Unknown Quantity books. Proper math notation doesn’t frighten, but could actually help readers understand concepts better.

For example, in the first chapter of the book, on page 20 the following sequence is introduced

1, 3, 6, 10, 15, 21 …

Which as Marcus explains is a sequence of triangular numbers and the general formula to get the nth triangular number is

1/2 * n * (n + 1)

What I think is missing here is the fact that Marcus stops short of explaining that there is a general formula to find the sum of members of a finite arithmetic sequence. This is important, since each nth number in the triangular numbers sequence is a sum of the nth numbers of the arithmetic sequence that Marcus mentions on page 2, talking about Gauss’s school lesson

1, 2, 3, 4, 5, 6 …

Notice that this sequence is arithmetic, since the difference between each consecutive member and the previous one is constant. In this particular case it equals to 1.

For example, that general formula is

Sum_{n} = 1/2 * n * (a_{1} + a_{n}), where a_{1} is the first member, which in the case of natural number sequence is 1; a_{n} stands for the last number until which we want to sum and n stands for the number of members starting from the first to the last we want to sum. Then substituting these values into the general formula we arrive at the formula Marcus mentioned in the book

nth_{triangular}_{number} = 1/2 * n * (1 + n)

What this gives a reader is that now it’s possible to use this general formula to calculate sums of other arithmetic sequences, for example the sequence of odd numbers.

1, 3, 5, 7, 9, 11, 13, 15 …

Let’s calculate the sum of first 8 members of this sequence, a_{1} = 1, a_{n} = 15, n = 8, then

One additional, example where this general formula could have been mentioned by Marcus was on page 28, where he explains how to calculate the number of handshakes in a population of a city having N people. It goes like this. Imagine these N people standing in a line. Then the first person can make N – 1 handshakes ( minus one, since he cannot shake his own hands). Then the second one can make N – 2 handshakes, continuing in similar fashion until the last person who can make no handshakes, since everyone already handshaked him. Then Marcus mentions that the sum of handshakes is the sum from 1 to N – 1, which is the sum that Gauss was asked to perform in his math class:

1/2 * (N – 1) * N

It seems to me a reader can be very confused at this point, since it’s not that clear that this is the sum Gauss was tasked to perform. This is because Gauss was asked to calculate the sum of the first nth natural numbers,

1, 2, 3, 4, 5, 6 …

As was mentioned earlier the general formula for this sum is Sum_{n} = 1/2 * n * (a_{1} + a_{n}) and in this particular case it’s

Sum_{n} = 1/2 * n * (1 + n), Then, one may ask, where does the 1/2 * (N – 1) * N comes from?

To get out of the confusion state, two things could be helpful. First, it’s a diagram of people standing in line waiting to be handshaked, and a little bit more details on how the 1/2 * (N – 1) * N formula was derived.

Looking at the diagram with four people in line we can see that the total number of handshakes is 3 + 2 + 1 + 0 = 6. Now, if we take N people in line as in the book’s example, then the first person in line is a_{1} = N – 1 , and the last person in line a_{n} = N – N, and the total number of people in line is N. Then substituting these numbers into the general formula for the sum of arithmetic sequence we got:

Sum_{n} = 1/2 * n * (a_{1} + a_{n}) = 1/2 * (N – 1 + N – N) * N = 1/2 * (N – 1) * N

This is how the formula in the book was derived.

This post is not the last one, since I have not finished reading the book yet. There are more to come.

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 * R^{2}, 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 * R^{2}. 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.