February 28, 2017 Tuesday

Bedtime Story 

The Babylonians Had Set Our Time

So in essence, the problem of writing higher numbers boiled down to the issue of either reusing or repeating the digits.

This may seem extremely trivial now mon ami, but believe me, it took us apes seriously long time to arrive at the notation that we use currently.

As the history will go on to show us, even after having discovered this novel form of notation, it was lost yet again for 3000 years before it was rediscovered.

It seems the Babylonians (people of Mesopotamia in around 2000 BC or so) were the first to have arrived at the idea of positional notation of numbers.

They worked on a type of writing that goes by the name of Cuneiform script which they had inherited from the Sumerians (people in the region of Mesopotamia, modern Iraq in the period of 3000 B.C.)

The Cuneiform script in turn developed from pictographic pro-writing of the 4000 B.C. and 5000 B.C.

How do we know this?

Well, some men love to dig.

They go by the fancy name of archaeologists.

And these guys in 1850s dug up some half a million (500,000) baked and preserved clay tablets of which 400 of them had mathematics in them.

These belonged to the Babylonians of Mesopotamia.

400 tablets out of 500,000 containing mathematics really does seem to be small fraction, does it not?

Yet this fraction is slightly higher than the content of mathematical websites to all the websites across the world today.

Mesopotamians had a symbol for each power of ten and each digit was separated by a space.

Instead of base 10, they used the sexagesimal system (base 60) that we still use today to measure time, angles and the geographic coordinates.

60 is a superiorly highly composite number that has twelve factors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Of these 2, 3, and 5 are prime numbers.

Having so many factors allows 60 to be divided in so many ways that makes it a great choice for becoming a tool of time keeper.

As history as shown, 60 turned out to be the perfect watch maker.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 27, 2017 Monday

Bedtime Story 

Arithmetic, Geometry and Logic: The Pillars of Mathematics

To my mind, much before algebra must have come logic.

As long as there were humans, there must have been arguments for we can agree to nothing.

Disagreement is fundamentally in our nature even towards the most common sensibilities.

In this democratic times, and even not-so democratic times, each and individual considers his birthright to disagree.

Yet merely disagreeing and arguing is not enough.

Our ancestors long back realized that argument has to be based on logic.

Logic is the codification or formalization of arguments.

Now that we have the three early branches of mathematics, namely arithmetic, geometry and logic let us what kind of notations would have been used for each.

One thing that is common to both the written language and the mathematical notation is that both use two-dimensional strings of structures to convey messages that cross the two dimensions both spatially and temporally.

In fact, mathematical notation can be considered a branch of linguistics though as far as I know, no linguist has seriously taken up the study of mathematical notation (their primary object of investigation being languages, wither written or spoken).

As we saw in the case of the Ishango bone, the easiest way to represent numbers is unary system.

You represent one by one stroke or one scratch and then repeat them as many times as you wish to convey the number.

This unary system is also known as tally marking and was the beginning of mathematical notation.

Surely it did not take a genius to construct the tally marking and many civilizations independently of each other had figured this system out.

What happened next was more complex and far more diverse.

One way of understanding how mathematical notation evolved next is to imagine what was the fundamental problem the humans faced once they had the tally marks.

How to represent higher numbers?

Speaking more generally, the early societies would have wondered how to correlate the numbers they had in their own local spoken language with mathematical symbols.

If you see mon ami, most primitive civilization would have had words for one, ten, twenty, hundred, thousand, ten thousand and so on.

But they would have pondered how to get them down in mathematical symbols.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 26, 2017 Sunday

Bedtime Story 

To Understand Formalization of Mathematics One Needs to Understand How Mathematical Notation Came to Be

The story of mathematical notation is as pleasurable a narrative as the story of mathematics itself.

Obviously they are both interconnected.

The current mathematical notation that is used internationally is anywhere between one hundred to five hundred years old.

Most popular books on science and mathematics convey the feeling that the modern world of science and mathematics started from the Greeks.

I am not sure why such a strong bias exists towards this, but may be the entire Western world subconsciously holds the belief that they owe their supremacy to the Greeks.

This belief that the Greeks are the originators and pioneers of the modern philosophy is obviously far from true since the first prototype humans existed in Africa and lot of civilizations existed from that time since the Greeks.

We are talking of a time span of at least 100,000 years and maybe more.

As much as we are indebted to the Greeks and the Hindus and Arabs for what they gifted the modern world, the Greeks in turn too must have been thankful to the civilizations that came before them, namely Egyptians and the Phoenicians (of the fertile crescent) who rose and fell before them.

In fact, certain rocks that go back dating to 70,000 years ago found in the Blombos Cave of South Africa have engravings of geometric patterns in them.

It is hard to say what kind of mathematical thinking a Ramanujan or a Gauss of that period would have had, but there certainly something was going on mathematically speaking.

Somewhere around 35,000 years ago there is evidence to show that humans in Africa made attempts to quantify time.

Again in a bone that is a fibula of a baboon that was found in Ishango, somewhere close to the border of modern Uganda and Congo, there are columns of notches that was probably a primitive form of numerical system.

It has been dated to about 20,000 years.   

Some claim it has an early reference to prime numbers though that seems debatable and vehemently arguable.

It, most likely, had to be some kind of a record of a natural phenomenon occurring periodically, may be it be the lunar months or menstrual cycles of a specific woman.

It was only in 3400 BC in Mesopotamia that we come across firm evidence of established numerical system.

By this time and gradually later arithmetic must have become a necessity as a tool in trading to count money and basic book-keeping.

Geometry would have become essential to the Mesopotamians, the Sumerians, the later the Hindus and Egyptians for the purpose of land surveying.  

So the early arithmetic and geometry arouse out of sheer need to solve the real world problems. 

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   The Ishango Bone is more than 20,000 years old found near the Semliki River near the border of Congo and Uganda

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 25, 2017 Saturday

Bedtime Story 

Why the Need To Formalize Mathematics?

One of the reasons of formalizing mathematics by the intellectual giants of the human apes is to understand what lies beneath the entire edifice of mathematics.

You see mon ami, before the Greeks, in Mesopotamia and in other civilizations, mathematics served as a real useful tool in the real world of daily affairs.

You needed mathematics for business, for division of land in agriculture, in the markets where trading was done and by the royalties for bargaining at the times of war and matrimonial alliances.

Euclid for the first time (at least what we are aware of), decided to detach mathematics from all the worldly affairs by formalizing it using a set of basic axioms.

His basic definitions about points and lines were not really concerned with the world we live in.

This was one method of formalizing mathematics.

There was yet another type that is well less known as it is well less spoken about.

That was the formalism of logic by Aristotle.

As I told you in my previous bedtime stories, both the Greek school of logic and the Hindu Nyaya School of logic developed pretty much nearly at the same time.

What was common to both was that even though both the schools tried to bring rigor into logic, they both failed to connect mathematics with logic or bring mathematics into logic.

It was only in the 1700s and 1800s when men in Europe and England such as Gottlob Frege, George Boole, George Peacock and Augustus De Morgan began to seriously apply mathematics and mathematical rigor to logic.

Bertrand Russell was like David Hilbert, another one those brilliant optimists who thought that by tightening up all the basic and relevant definitions using logic, he could rigorously derive every known aspect of mathematics.

He believed that even elusive topics such as infinites and liar’s paradox could be tackled if sufficient logical rigor was applied at the very fundamental level.

Yet he had doubts.

For while examining the liar’s paradox he came across yet another inherent inconsistency; the paradox of self-reference.

Is it possible for a set of all sets that do not contain themselves to in fact contain itself?

To solve this great conundrum, Russell introduced to mathematical logic the idea of types of sets.

This was not enough though.

He along with Whitehead decided to completely formalize mathematics by writing it down with a completely new set of symbols.

It was well known even much earlier that all the languages that had been invented by the human apes were to imperfect for mathematics.

Even as early as in the late 1600s, Gottfried Leibniz had brought in the idea of introducing a completely new notation for mathematics.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 24, 2017 Friday

Bedtime Story 

What Bothered David Hilbert also Concerned Two Englishmen Alfred North Whitehead and Bertrand Russell

David Hilbert had spoken these famous two lines on September 8, 1930 at the annual meeting of the Society of German Scientists and Physicians.

These words reflected his optimism of generating that perfect self-contained, consistent and complete formal mathematical system bereft of any flaws, paradoxes and indecisiveness.

One of the first serious attempt to create such a perfect and comprehensive mathematical system was the 1910 three-volume treatise by Alfred North Whitehead and Bertrand Russell.

Agreed that Giuseppe Peano did it before them, but his work was neither as voluminous nor as comprehensive as theirs though it must be said that Peano’s work was one of the inspirations for these men to work on this book.  

Both these gentlemen were English, mathematicians, logicians and philosophers, all in one.

Russell was the younger of the two and in fact was a student of Whitehead at the Trinity College in the 1890s.

This three-volume work of these two great minds goes by the name of Principia Mathematica and is considered one of the greatest intellectual feats of the twentieth century.

Modern Library, the American publishing company ranks Principia Mathematica 23^rd in the list of top 100 English-language non-fiction books.

In a way, Principia Mathematica that I shall abbreviate to PM since it will come several times, was a product of extreme uneasiness and discomfort with the prevailing state of mathematics of those times (late 1800s and early 1900s) and whose (state of mathematics) most vocal proponent was David Hilbert.

It was essentially an attempt to reduce mathematics to pure symbolic logic.

That means even the set of axioms and inference rules were reduced to symbolic logic and starting from them, attempt was made to prove all of the known mathematics.

Just the nature of task that was attempted and carried out calls out for immense respect towards these gentlemen.

The work dealt only with the set theory, cardinal numbers, ordinal numbers and real numbers.

A fourth volume on the foundations of geometry was thought out but by the end of the three volumes, these two remarkable men were overwhelmed with intellectual exhaustion.

Yet they convinced the world that in principle, almost all of known mathematics could be derived from this formal symbolic logic.

This work is almost or is essentially deduction of mathematics using pure logic.

 One of the most memorable fact of this painstaking tedious work that is often spoken about is that it took more than eighty pages into volume 2 to prove that 1 + 1 = 2.

This is stated as a proposition and is followed with a comment:

 “This proposition is occasionally useful”.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 23, 2017 Thursday

Bedtime Story 

The Nazi Minister Asks David Hilbert a Question

This is what happens when political ideologies take over education and universities.

It is a monumental tragedy that keeps on repeating in all cultures and at all times.

There is nothing common between mathematics/sciences and religion/political ideologies.

Once in 1934 while attending a banquet, David Hilbert was made to sit next to Bernhard Rust, the new Minister of Science, Education and National Culture of the Third Reich.

It was this minister who had passed the decree that all students and teachers ought to greet each other with that famous Nazi salute.

Rust made it clear his goal; to make all the Germans aware of their so called “Aryan” ethnicity and cleanse all the universities of Jews.

Most tribes or societies show this tendency of generating social uniformity and racial conformity which can to some extent be explained by the gene-centric view of evolution from which follows the concept of kin selection and hence tribalism.  

This is what Bernhard Rust, the minister of Science and Education had to say about non-Aryan science and “Jewish physics”:

“The problems of science do not present themselves in the same way to all men (which is very true but in ways other than he intended to mean).

The Negro or the Jew will view the same world in a different light from the German investigator”.

So Hilbert being one of the top men of Göttingen was made to sit with this Nazi Minister of Science and Education at the banquet.

Sitting next to him, Bernhard Rust asked Hilbert the question:

“Did the Mathematical Institute really suffer so much from the departure of the Jews?”

To it, Hilbert replied:

“Suffered? It does not exist any longer, does it!”

This is what tribalism and nationalism can do to a society; eviscerate it!

Hilbert passed away in 1943, very fortunate not to have witnessed the grotesque evil and savagery that was already being unleashed in its chilling fury.

Such a great man and thinker and less than a dozen people came for his funeral.

I wonder how many will come for mine (assuming there will be one).

In his gravestone the elegy reads:

Wirmüssenwissen.

Wirwerdenwissen.

In English, this translated to:

We must know.

We will know.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  Mark the words written at the bottom of David Hilbert's tombstone at Gӧttingen: 
Wir műssen wissen
Wir wurden wissen

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 22, 2017 Wednesday

Bedtime Story 

Why Historical Perspective is Essential for a Meaningful Bedtime Story

To make my bedtime stories and the story of great accomplishments of the human apes in mathematics and science complete, it is pertinent that I do not dissociate these events from the prevalent socio-economic conditions of those times.

History and historical background surrounding any event is indispensable if one wishes to fully fathom the import of it.

Historical perspective if you would have noticed, is one the corner stones of my bedtime stories and I believe that is the way history must be taught.

Not as a pure undetached subject but in relation to the other significant events, the most important of which are sciences, mathematics and of course wars.

Wars are perhaps the most regular feature of human apes which essentially is a struggle for resources.

To me it seems very ironic that it is always the powerful empires who are most endowed with resources that are ever ready to acquire more of land, humans and natural wealth such as minerals and water.

So it was with Germany after its humiliating defeat in the Word War I.   

While Germany was being fortified into a war machinery of unimaginable proportions, its intelligentsia was running helter-skelter for cover.

Hilbert along with Felix Klein had transformed the Göttingen University into the world class center of mathematics from 1895 onwards till 1930.

It was exactly what the Institute for Advanced Study at Princeton would soon go on to become after Gottingen in 1940s.

Göttingen under him had nurtured the most dazzling mathematicians of 20^th century such as Ernst Zermelo, John von Neumann, Alonzo Church, Hermann Weyl, Wilhelm Ackermann, Emmy Noether and many others.

So it was a tragedy when from 1933 onwards he had to watch the Nazi regime purging out all the talent out of Göttingen and even out of Germany into the welcoming arms of the United States of America.

As the German Jews and Jews from other adjoining nations were exiled, deported or mass murdered, Germany and Göttingen were laid skeleton bare of their intellect.

It was as if hyenas and vultures were ripping out the flesh and blood of a most robust personage and sucking out slowly the very life from this virile body. 

In front of the Nazi machinery, Hilbert could do little; he stood and watched helplessly watching his peer group slipping out of his hands and circle that he had so carefully raised and tended to.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 21, 2017 Tuesday

Bedtime Story 

Next Hyperoperation After Exponentiation is Tetration

Simple examples will clarify what hyperoperation means.

For example, what can be more primitive method of compounding a number x than adding a one to it and making it x + 1?

That would be a successor function.

The next primitive operator is addition.

Addition operation merely specifies the number of times 1 needs to be added to x to get the next value of x.

Multiplication follows next as it defines the number of times a number must be added to itself.

Exponentiation is the next in this hierarchy of hyperoperations that defines the number of times a number is to be multiplied to itself.

After exponentiation, the next hyperoperation is called tetration or hyper-4.  

Tetration is defined as iterated exponentiation.

Let us see how they can be beautifully represented with symbols without using so many useless words.

Succession:

a’ = a + 1

Addition:

a + n = a + (1 + 1 + 1 +…+ 1) where n copies of 1 are added to a.

Multiplication:

a . n  = a + a +…+ a  where n copies of a are combined by addition   

Exponentiation:

aⁿ = a x a x…x a where n copies of a are combined are combined by multiplication

Tetration:

ⁿa = a^a..a where n copies of a are combined by exponentiation, from right to left.

I could not write the “a”_s and the dots the way I wanted to, as each a and each dot should be written exponentially to the one left of it.

This above operation is the n^th tetration of a.  

Perhaps it will be better if I can you a picture of it.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

February 20, 2017 Monday

Bedtime Story 

Ackermann Function

Much to my surprise, it was that high school mathematics teacher Wilhelm Ackermann who had first come up with a total computable function that is not primitive recursive.

Now I have to tell you about primitive recursive functions.

Primitive recursive functions are a subclass of number-theoretic functions.

These functions map natural numbers to natural numbers.

This essentially means that both the argument (input value) and the function value (output value) of such a function would be a natural number.

So you see it that it is nothing great.

A simple function of addition would be primitive recursive.

So would be division, factorial and exponentiation.

If you ask why not subtraction and multiplication, this is simply because subtraction is negative addition and multiplication is addition multiple times.

Primitive recursive functions on the other hand are a subset of total μ-recursive (partial) function.

I will not go very deep into this stuff since the point is to give you a basic-level understanding of this concept.

Now let me show you what exactly Ackermann function is.

Ackerman denoted his function with the Greek letter phi or .

It was a three-argument function stated as (m,n,p).

The function was defined in a way such that for p with a value of 0, 1 and 2 it gave rise to the basic operations of additions, multiplication and exponentiation.

Let us how he put it mathematically.

It is not so daunting even for an average ape like me so it is worth writing them down:

(m, n, 0) = m + n,

(m, n, 1) = m x n,

(m, n, 2) = mⁿ

With the values of p > 2, it becomes even more interesting.

Ackermann function with values 2 enters the realms of what in mathematics is known as hyperoperations.

You can consider a hyperoperation as a way of compounding numbers with a singular point being that the increase is dependent upon the iteration of the previous hyperoperation.

So in that sense, even functions like successor, addition, multiplication and exponentiation are also hyperoperations, successor operation being the most primitive.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                   

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg