Useful tools for video and audio editing

If you have your own YouTube channel or just make video or audio recordings using your mobile phone, it is good to know that there are editing tools that can help you remove undesired artifacts from recordings. For example, in an audio recording you’d probably want to remove or at least reduce a background noise. In a low resolution video you’d possibly want to have a better picture quality by making resolution higher if possible. Also, when you find a YouTube video that you think could have a better audio or video quality and this video has a Creative Commons Attribution license then you could download it, edit and upload again to YouTube. This post is exactly about such editing tools, or at least tools that I use myself and find very helpful. Most of them are free open-source tools except for video editing software.

Audio editing

Suppose, you have voice recordings that have a background noise. It would be nice to reduce it as much as possible without affecting the overall quality of the recording. There is a free tool that can do this and much more. It is called Audacity. Audacity is free and open-source professional grade digital audio editor and recording application software, available for Windows, macOS, Linux, and other Unix-like operating systems. I personally, use it to record myself playing drums. For this purpose I use two microphones and a two channel Behringer sound interface. After the recording was done I use Audacity to compress the recording and export it as mp3 file. But one of the features that is relevant to this post is the Noise Reduction functionality.

Real Life Example – Removing humming noise from Hamming’s lecture

For example, I used the Noise Reduction functionality in Audacity to remove background humming noise from the Dr. Richard Hamming’s 1990 lecture at NPS SGL. 

To do this I

Video editing – Super-resolution

Sometime videos can have a very low resolution, especially when they were recorded with old recording hardware, like old fashioned video cameras etc. But there is a solution to this problem which is called technically a super-resolution or upscaling. It allows to improve the resolution of the video by smoothing the pixels based on surrounding pixels. There are a number of implementations for an upscaling algorithms. Some of them like video2x upscaling software uses Deep Learning based upscaling implementation, for instance, NCNN implementation of waifu2x converter. Check out the GitHub repository of the video2x to learn how to use it.

Real Life Example – Upscaling Alexander Stepanov’s talk

For example, I used the video2x upscaling software based on Deep Learning model to upscale Alexander Stepanov: STL and Its Design Principles lecture from 320×200 to 640×400 resolution.

To do this I

  • Downloaded the original video (Creative Commons Attribution license (reuse allowed)
  • Used Audacity to add right mono channel to the original audio track which had only left mono channel available.
  • Used video2x software ran on a PC with Nvidia GPU to upscale the video from 320×200 to 640×400 resolution. Which took about 14 hours to process on that PC)
  • Used Movavi Video Editor to combine fixed audio track with the upscaled video
  • Uploaded upscaled video to YouTube.

You can try to play with the waifu2x Deep Learning powered upscaling website by uploading low resolution images and seeing the result by yourself.

Video editing

There are a number of free video editing tools out there, but from what I’ve seen the most useful ones that provide you with all required editing functionality are paid. And there is no workaround it. So I found this relatively inexpensive Movavi Video Editor software that I bought and use for all my video editing. Since I use mostly basic video editing, this tool suits me good. But if you are looking for more advanced capabilities, than you should check other versions of Movavi products or a different editor altogether.

How to download YouTube video for editing

If the YoutTube video has a Creative Commons Attribution license (reuse allowed) license you can use the video and edit it. There are non criminal ways to download such videos from YouTube.

Reading hundreds of books makes you a different person

This post is a summary of the hard copies of the books I read and recommend to read if you have similar interest as I do.

Exceptionally good books about Apollo Lunar Program

  1. Flight – Chris Kraft
  2. Left Brains for the Right Stuff – Hugh Blair-Smith
  3. Sunburst and Luminary – Don Eylse
  4. Apollo 8 – Jeffery Kluger
  5. Apollo 13 – Jim Lovell and Jeffery Kluger
  6. and much more here

Exceptionally good books about Deep Space Exploration

  1. Interstellar Age – Jim Bell
  2. Chasing New Horizons – Alan Stern and David Grinspoon
  3. Mars Rover Curiosity – Rob Manning
  4. The Right Kind of Crazy – Adam Steltzner
  5. and much more here

Exceptionally good books about Aviation

  1. Skunk Works – Ben R. Rich
  2. Have Blue and the F-117: Evolution of the “Stealth Fighter” – Albertt C. Piccirillo and David C. Aronstein
  3. The Power To Fly – Brian H. Rowe
  4. Herman The German – Gerhard Neumann
  5. and much more here

  1. The beginning of Infinity – David Deutsch
  2. The Music of the Primes – Marcus du Sautoy
  3. Prime Obsession – John Derbyshire
  4. Unknown Quantity – John Derbyshire
  5. Infinite Powers – Steven Strogatz
  6. The Joy of x – Steven Strogatz
  7. and much here and here and here

Exceptionally good books about Neuroscience

  1. Mind and The Cosmic Order – Charles Pinter
  2. On Intelligence – Jeff Hawkins

It’s a kind of magic. How kids can have fun with math.

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.

Kids can spot mistakes in math books too

Read as if you edit

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.


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.

Answer to the Puzzle #2

Chasing New Horizons is the book you’ve never heard about

Have you ever heard about New Horizons spacecraft? Did you know that it flew by dwarf planet Pluto in 2015, which was never been done before? Did you know that in 2018 it visited a Kuiper Belt Object Ultima Thule, now officially known as Arrokoth? If you answered no to any of these questions and you are interested in deep space exploration then you may find this post interesting.

The post is a short review of the book Chasing New Horizons : inside the epic first mission to Pluto. If you already read other books on the subject, then this one could resemble to you The Interstellar Age by Jim Bell or The Right Kind of Crazy by Adam Steltzner. If not, then buckle up and lift off!

Back in 2015 Pluto was still a dwarf planet that little was known about except its orbit, its approximate mass and volume, and composition of its atmosphere. No space mission had visited it before, though one of the Voyager probes was planned to visit it, but it didn’t happen. In 2015 with a flyby of the New Horizons spacecraft, Pluto has revealed its secrets and new exciting data became available to scientists and a larger audience.

The book Chasing New Horizons is all about telling the story of how this flyby became a reality and how dedication and perseverance of a group of relentless planetary scientists, engineers and space enthusiasts put their careers on a line to make this happen. It was written by Dr. Alan Stern who was a Principal Investigator (PI) behind New Horizons mission and Dr. David Grinspoon, an astrobiologist, who also took part in the mission. It tells the story of how the mission was conceived back in late 80’s of the previous century, how it took about 27 years from an idea to its implementation and what obstacles the team had to overcome to make it a reality.

What I liked about the book

As I’ve already mentioned in my other post, I find the that the most interesting books are books written not by journalists, but by actual scientists, engineers, project managers and others, who were there, who made the decisions, who first hand experienced what happened before their own eyes.

This book stands out in comparison to similar ones about space, since it is able to engage readers in an exciting story of exploration of new horizons despite the hurdles emerging almost daily along the way, that would prevent other people from proceeding forward. I like how the NASA’s inner politics, engineering tradeoffs and solutions to emerging problems were described in detail in the book. This way a reader gets a better context of how the events unfolded and why.

Significant part of the books is also dedicated to describing day-to-day activities, such as mission planning, spacecraft housekeeping that were carried out to support the ongoing New Horizons journey to Pluto. By providing these details authors made it feel like you actually were there in mission control room observing what had happened in a real time.

All in all, books like this make you appreciate what we people are capable of when we are driven by high goals of exploration, knowledge advancement and pure joy of adventures. And such books make you crave for more.


Programming and a school algebra formula finally deciphered

Explanation is due

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:

a2 – b2 = (a – b) * (a + b) = a2 + ab – ba – b2 = a2 – b2 , (1)

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

a3 – b3 = (a – b) * (a2 + ab + b2) = a3 + a2b + ab2 – ba2 – ab2 – b3 = a3 – b3, (2)

So, I think you get it and the next formula is for a4 – b4 ,

a4 – b4 = (a – b) * (a3 + a2b + b2a + b3) = a4 + a3b + a2b2 + ab3 – ba3 – a2b2 – b3a – b4 = a4 – b4 , (3)

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

an – bn = (a – b) * (an−1 + an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1) , (4)

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.

So, the second braces in formula (1) have

(a + b)

the second braces in formula (2) have

(a2 + ab + b2)

the second braces in formula (3) have

(a3 + a2b + ab2 + b3)

the second braces in formula (4) have

(an−1 + an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1)

Do you see it? When there is a2 on the left side there is a corresponding b2 on the right, when there is a2b on the left, there is a corresponding b2a 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) = a1 + b1 , i.e. the powers are 1, 1

(a2 + ab + b2) = a2 + a1b1 + b2 , i.e. the powers are 2, 1 + 1, 2

(a3 + a2b + b2a + b3) = (a3 + a2b1 + a1b2 + b3) , i.e. the powers are 3, 2 + 1, 1 + 2, 3

(an−1+ an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−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 = a0 or 1 = b0, and let’s look again at the expressions above

(a + b) = a1b0 + a0b1 ,

(a2 + ab + b2) = a2b0 + a1b1 + a0b2 ,

(a3 + a2b + b2a + b3) = (a3b0 + a2b1 + a1b2 + a0b3),

(an−1+ an−2b + an-3b2 + … + a2bn-3 + abn−2 + bn−1)

= (an−1b0+ an−2b1 + an-3b2 + … + a2bn-3 + a1bn−2 + a0bn−1),

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 a2 – b2; three for a3 – b3 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 a8 – b8.

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: a7, a6, a5, a4, a3, a2, a1, a0

and b‘s: b0, b1, b2, b3, b4, b5, b6, b7

Now, putting these together in the formula we get,

a8 – b8 = (a – b) * (a7b0 + a6b1 + a5b2 + a4b3 + a3b4 + a2b5 + a1b6 + a0b7)

= (a – b) * (a7 + a6b + a5b2 + a4b3 + a3b4 + a2b5 + ab6 + b7).

Zip it

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

a^n - b^n=(a - b)\sum^{n - 1}_{k = 0} a^{(n - 1) - k}{b^k}

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

  1. First vs. last letter
  2. Second vs. one before last
  3. etc
  4. 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;

Let’s it. Take care.

Get Back for More Than 50 years. Thoughts on The Beatles: Get Back documentary

Back to the… past

Have you heard about a new documentary called The Beatles: Get Back. It was created from more than 150 hours of audio and 60 hours of video while The Beatles were working on their new live performance back in January 1969. I header about this film while checking news on the BBC website. The trailer of the Get Back got me curious. The video was crisp as though it was filmed just a couple of days ago. The sound quality was good and the footage was interesting.

It turned out that the documentary was available only at Disney+ channel, so I revived my membership there just to watch this series. The Get Back consists of three episodes, each more than 2 hours long. So be ready to dedicate a good chunk of your day or even two days to be able to watch it in full, unless you intend to skim through it.


What is interesting about this series of films that you can watch it from a number of viewpoints. First, of all there is a historical viewpoint, since it was filmed in January 1969 which is almost 53 years away from now. Back in 1969, Apollo 11 landed on the Moon, Vietnam war was in its height, the Hippie movement was on the rise and solid state transistors started to make their way into electronic music equipment. So it was a hectic time to say the least. The Beatles at this period found themselves at uncertain times when the band’s future was questionable. It is also interesting to notice the fashion, cultural norms, food and other things that changed since then. For example, it is no longer acceptable to smoke cigarettes in closed public places.

At this backdrop the Get Back documentary is unwinding in its full beauty.

Episode 1

First part of the documentary starts at the Twickenham Film Studios were the Beatles is trying to come up with more than a dozen new songs for the upcoming live performance, possibly a TV show. Some of these songs ended to be the songs for the Let It Be album. But the band finds very quickly that the place is not that good for creative process. What I find interesting, that most of the band members and their spouses smoked cigarettes inside the pavilion, threw cigarette buds on the floor which we can hardly imagine happing these days.

Looking at internal dynamics inside the band and how the creative process unfolded it was interesting to see interactions between Paul McCartney and John Lennon, some tension between McCartney and George Harrison and a calmness and wisdom of Ringo Star. What I liked to see is how the band worked together on lyrics for the songs. What I find surprising is how Ringo the drummer is able to come up with grooves on the fly without the band members telling him what to play. If we look around we see that during the rehearsal sessions musicians ate breed and butter with honey and drink tea with milk, beers and vine. What you wouldn’t notice our nowadays omnipresent mobile phones.

Speaking from the musical viewpoint, more exactly, Ringo’s drum kit I find it to be very basic 5 piece kit. It had a snare, two rack toms, one floor tom. Hi-hat, crash and ride cymbals. At the very end of episode one it can be seen that Ringo also used a splash cymbal. As for his throne he had a simple chair with a back rest. Since, apparently, there were no Drumtacks mufflers back then he used some kind of fabric that he covered the snare and the floor tom with to muffle the sound. As for the mics the kit was miced with a kick drum mic, there was also a microphone on the right side and one microphone above the kit.

Episode 2

In the part two the musicians moved to the basement studio in the Apple building were they spent the rest of the documentary.

Turning to the Public Address (PA) equipment it is interesting to notice the vertical Fender Solid State sound speakers.

Episode 3

I find the third part most interesting since it is in this part that the band decides it will be fun to have a live performance on the Apple Studio’s roof top. And as it turned out this decision brought with it a lot of fun for most of the people on London streets, while driving some people mad, including police officers. Also in this episode you may find how the Octopus’s Garden song came to life.

All in all, if you interested in The Beatles’ history and also want to feel how a real creative process is happening within the band during rehearsals then this documentary is mandatory for you.

When math powers algorithms it’s entertaining

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 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.

Unboxing inventions and innovations

Photo by Kelli McClintock on Unsplash

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.

Levels of understanding or how good explanations matter

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

2^n – 1 = (2^a – 1) * ( 2^a*(b-1) + 2^a*(b-2) + … + 2^a + 1).

It turns out that to derive it there is need to remember what polynomial factorization is, or remember how to divide one polynomial by another, or to know what Polynomial remainder theorem is. Also it’s good to know for a start what is a polynomial.

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.