July 31, 2018 Tuesday

Bedtime Story 

Derivation of the Roots of Depressed Cubic Equation

Today we shall study the so called Cardano’s method (I say so called because it was Scipione del Ferro who was the real brain behind the solution) for arriving at the roots of the depressed cubic equation

           t³ + pt + q = 0    

The very first step of del Ferro in the solution was to introduce two variables u and v such that

 u + v = t

Now let us see what happens when we introduce the two variables u and v in the depressed cubic equation.

It gives us

      u³ + v³ + (3uv + p)(u + v) + q = 0

Here Cardano introduced a second condition on the variables u and v so that 3uv + p = 0

Imposing this condition on the above equation will give us

     u³ + v³ = - q

From 3uv + p = 0 we will also derive

      u³v³ = - p³/27

There two above equations are the sum and product of u³ and v³ and hence once can derive a quadratic equation from them.

The quadratic equation to which u³ and v³ are the solutions or roots are:

                    z² + pz – p³/27 = 0

Solving the quadratic equation gives

      u³ = -q/2 + √(q²/4 + p³/27)

      v³ = -q/2 - √(q²/4 + p³/27)

Since t = u + v, then

      t = cubic root of (-q/2 + √(q²/4 + p³/27) + cubic root of (-q/2 - √(q²/4 + p³/27)

This is the Cardano’s solution for the depressed cubic equation and you can see even it required a high level of ingenuity and three brilliant mathematical minds to arrive at the solution.

Now lastly, we shall look into the quartic function which has the form

   f(x) = ax⁴ + bx³ + cx² + dx + e where a ≠ 0

A quartic function or a quartic polynomial can be converted to a quartic equation by equating it to zero.

Thus a quartic equation looks like

      ax⁴ + bx³ + cx² + dx + e = 0 where a ≠ 0

The derivative of a quartic function is a cubic function.

The solution of the quartic equation was published in the same book (Ars Magna) of Gerolamo Cardano in 1545 but its solution to Cardano was aided by his bright servant Lodovico de Ferrari.

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

                              

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

July 30, 2018 Monday

Bedtime Story 

Scipione del Ferro, Niccolo Fontana Tartaglia and Gerolamo Cardano Present to Us Solution to Depressed Cubic

It was only somewhere in 1540 or 1550 that two men from Mediterranean Europe, the next power house of mathematics after the Islamic World, that the two men, one a Venetian by the name of Niccolo Fontana Tartaglia (the stammerer) and Italian by the name of Gerolamo Cardano (an accomplished gambler and chess player in addition to being a perpetually poor doctor) provided to the world the general formula for the solution to cubic equations.

Even before them, there was another Italian mathematician by the name of Scipione del Ferro who had arrived at a solution to what is known as depressed cubic equation or simplified cubic equation.

He never published his work because strangely enough in those times there was this concept of public mathematical duel where mathematicians publicly posed challenges to each other.

What is most surprising that the loser in such public mathematical duels was vulnerable to the loss of his funding or position in the University if he happened to occupy.

Scipione del Ferro was a lecturer in arithmetic and geometry at the University of Bologna and because of his fear of being challenged he kept his mathematical works private and only wrote them down in his personal notebooks that was shared with a group of very selected people close to him.

A general depressed cubic equation looks something like this:

             x³ + px = q 

It is known exactly how del Ferro came to his solution for the depressed cubic equation as only one note book of his had survived that was inherited by his son-in-law who in turn had happened to share it with Girolamo Cardano.

Cardano in his turn published the solution in his Ars Magna or “The Great Art” in 1545.

The first edition of the book came in three volumes and consisted of forty chapters dedicated specifically to algebra.

It was first published in 1545 and is considered as one of the three greatest scientific work of the Renaissance along with Nicolaus Copernicus’ ‘On the Revolutions of the Heavenly Spheres’ (1543) and Andreas Vesalius’ ‘On the fabric of the human body in seven books’ (1543).

If I were to provide you even today the general solution of the cubic equation, you will understand why it took so many hundreds of years before the human apes arrived at its solution.

So here goes, let me show you the solution as addressed by Gerolamo Cardano in his book Ars Magna, but surely helped in a major way by both Scipione del Ferro and Tartaglia.

Cardano was a gentleman and in this book explicitly acknowledges the formula for solving the cubic equation was provided to him by Tartaglia.

Once again it needs to be stressed that this Cardano’s method is only applicable to the depressed cubic equation

           t³ + pt + q = 0     

We shall take a closer look at the Cardano’s method of the solution to the general depressed cubic equation in the nights to come.

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

                              

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

July 29, 2018 Sunday

Bedtime Story 

Quadratic Equation

Last night we had ventured into quadratic equations which has the general form

                 ax² + bx + c = 0

In such kind of an equation, x is the unknown whereas ‘a’, ‘b’ and ‘c’ are known numbers wherein ‘a’ cannot be equal to zero.

This is so because if ‘a’ was to be zero, then the equation would no more be quadratic but a linear one.

The numbers ‘a’, ‘b’ and ‘c’ are called the coefficients of the equation and they can be distinguished from each other by calling ‘a’ the quadratic coefficient, ‘b’ the linear coefficient and ‘c’ as a constant or the free term.

The solution to such types of second-degree polynomial equations can be expressed in terms of its coefficients, using only addition, subtraction, multiplication, division and square roots.

The solution should be familiar to most of us even if we may not recall it perfectly well.

Its roots or the value of x is given by the general formula

-b  √(b² – 4ac)/2a  

We surely would have, some time in our childhood, forced to use this formula without been given any appreciation of the beauty of it.

That is understandable since the knowledge that has accumulated by now is so immense that it takes longer and longer for each child of the next generation just to get familiar even with the very surface of these progressive developments.

We have a beautiful generalized formula expressed in just the coefficients of a quadratic equation.

The trouble with the mathematicians is, as I must have stated before, that they never stop and they continue to stretch their imagination and keep asking ‘what if we go one step further?’

After solving the quadratic equation there came the desire to have a general solution to the cubic equation which has this following general form:

      ax³ + bx² + cx + d = 0

Such algebraic equations can also be treated as function in the form:

      f(x) = ax³ + bx² + cx + d

Such a type of equation or equations of these types was familiar to almost all the ancient civilizations including the Babylonians, Greeks, Chinese, Hindus and Egyptians.

The problem of doubling the cube is a very old problem and the solution to it involves solving of a cubic equation, albeit a very simple and primitive one.

Almost all the civilizations (just remember, the mathematician who did in each civilization remains unknown and that is the fate of most of the geniuses) had found a way to solve the cubic equations though as far as we know none had come to general formula for it.

We shall continue with our story on the polynomials in the nights to come.

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

                              

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

July 28, 2018 Saturday

Bedtime Story

On Education

Any subject that is taught today is completely bereft of its historical development simply because of lack of time and the tremendous amount of knowledge we humans have amassed thanks to our ability to record information and develop over it.

Each generation has to spend more and more time just to catch up what has already been discovered, invented, documented and published.

More than that, the best years of the minds are perhaps wasted in the preparation for examination most of which demand rote learning and vomiting rather than critical thinking and inventiveness.

It is my personal theory that the chief reason for such kind of modern education system is overpopulation (which is a very controversial subject since most human apes are creatures of emotions and driven by their biological imperative rather than rational thinking).

To accept that too many of us is the reason for most of our troubles is first of all hard to digest and the very first reaction is to brush it aside; and even if acknowledged, it is virtually impossible to alter our behavior to any meaningful effect; apes addicted to tobacco and nicotine will easily get my argument.

We are, after all, biologically programmed to generate families around us both for our physical and psychological needs.

Because of overabundance of average apes, our education system in general and examinations in particular have become a technique of elimination rather than education or selection.

And of course, the primary role of education of differentiating right from wrong, good from evil and generation of curiosity and critical thinking I can say for sure is absolutely lost since neither the educators nor the parents would hardly inclined into making these a priority in the education of their children.

The entire focus is on employability and amazing of wealth.    

And so with this kind of milieu around us subjects are introduced to us and our children in neatly arranged and crisp format.

Just like everything, algebra is taught in a very neat and sterile manner starting from its axiomatic definitions of its structure and then going on to establish their properties.

Yet, historically no science and specially mathematics develops in this manner.

Most of the current algebra developed very disparately and then later acquired a common theme and then presented to current students in a refined manner as if everything was tailor made from them.

One of the most important threads, as I said earlier, was the quest to the solutions of polynomial equations higher than four.

Most educated people with basic education of mathematics are pretty apt at solving linear equations of the first degree.

The algebraic equations of second degree are known as quadratic equations and come in forms such as

   ax² + bx + c = 0 where a ≠ 0 

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

                              

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

July 27, 2018 Friday

Bedtime Story 

It started with Useful Linear Equations 

As I was saying last night that almost anybody who cares to read my bedtime stories will be very much familiar with both the linear and quadratic equations.

We have also been taught to tackle system of linear equations in the form of simultaneous equations.

Take for instance the following two equations:

         2x + 3y = 6

         4x + 9y = 15

I am positive this would not frighten most of us and even with our average algebraical skills we should be able to obtain the value of the two unknowns.

In the language of mathematics, the solution to the variables is arrived by the technique that is known as elimination of variables.

Computers can be taught or rather programmed with the algorithm of the elimination of variables to solve a system of several such equations with not just two but multiple variables.  

Keep in mind that the coefficients of these equations can belong to any field and not just natural numbers or rational numbers.  

Such algorithms have a great many practical uses in the field of engineering, physics, chemistry, computer science and economics.

Pure mathematicians though do not really care about the practicality of it though some of them may as there is no consistency with us apes, not even mathematicians.

Since these kinds of algebraic equations are useful and come in handy in markets, economy, and architecture and so on, they are not abstract and have been used since many centuries.

Abstract algebra too arose from such need-based mathematical imperatives.

One of such attempts that was being seriously investigated by some mathematical minds was the solutions to the general polynomial equations of higher degree.

As is self-evident, much of normal life, economics and markets would be able to run along fine with the so called elementary algebra but mathematicians are never satisfied with that.

They are extremists not in the common derogatory sense of the word but always try to extrapolate the matter further and further away towards its extreme.

So once the solutions to linear equations were known, their site was set to equations of higher degree and with more variables or unknowns.

It is here that we introduce the Diophantine equations.

Thus the target of such similar studies was the arithmetical investigations of quadratic and higher degree Diophantine equations, meaning polynomial equation with two or more unknowns wherein only the integer solutions are sought.

An example of Diophantine equation would be:

   aⁿ + bⁿ = cⁿ

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

                              

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

July 26, 2018 Thursday

Bedtime Story 

"Evidence" for the Martians

The general idea (humorous of course) was that these brilliant Hungarians were actually the descendants of a Martian scout force that had chanced to land in Budapest somewhere in 1890s or even 1900s.

But finding the place and the planet either hostile or not suitable to their needs they departed but did so only after reproducing with some earthling women.

The children who were born out of this chancy mating came to be known as the Martians and they all emigrated to the United States.

Von Neumann makes a claim that these Martians were born geographically very close to each other and showed remarkable similarity in their academic careers.

Their academic interest usually started with chemistry which would sooner or later take them to prominent German universities where they would be drawn towards physics or mathematics.

The career in physics would eventually take them to the United States (probably because fundamental research in pure physics or astrophysics was getting costlier affair with each passing day requiring either expensive telescopes or particle colliders that require a large team of physicists and technicians to get them going smoothly).

György Marx had more convincing evidence about the Martians.

He said that the names of three extraordinary Martians John von Neumann, Leo Szilard and Theodore von Kármán (you might recall that he featured in one of my story on the Kármán line many nights ago) are not to be found in the maps of either Budapest or Hungary but on the Moon as craters bearing their names.

So that was the story about the Martians of the United States.

Now let us return back to the story of mathematical education, or rather the educators of mathematics.

As I was making a claim that these days for any person with a decent understanding of mathematics and possessing good mathematical skills can easily make a decent living for himself as these skills are as needed in modern society as those of a skilled surgeon.        

But if one were to go a century back, things were very difficult for mathematicians as the demand in the market for such a skill was quite low.

Perhaps the best reward for an able mathematician would have been access to or appointment in a senior faculty position of some famous university whose salaries at the very best would be fairly modest.

So let us hope we get to understand the workings of the great mathematical minds in parallel with their work and the implications to understanding of nature and reality.

Now I shall concentrate on real algebra starting from elementary one and then slowly advancing to more difficult and try to link it with our ongoing story of group theory and mathematics of symmetry.

We have all been taught how to solve linear equations, simultaneous equations and then quadratic equations in our high schools so much so that these are perhaps one of the few things in mathematics and algebra that are not only familiar but also seem to be friendly.

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

                              

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

July 25, 2018 Wednesday

Bedtime Story 

Leo Szilard and the Martians

We shall continue tonight with the abstract from Marx’s “The Martians”:

“Civilization, science, and technology will follow.

Then, yearning for fresh worlds, they will travel to neighboring planets, and later to planets of nearby stars.

Eventually they should spread out all over the Galaxy.

These highly exceptional and talented people could hardly overlook such a beautiful place as our Earth.

- “And so,” – Femi came to the overwhelming question, - “if all his has been happening, they should have arrived here by now, so where are they?”

It was Leo Szilard, a man with an impish sense of humor, who supplied the perfect reply to the Fermi Paradox:

“They are among us,” – he said - , “But they call themselves Hungarians.””

Just to elaborate on the humor of Leo Szilard, a Hungarian-German-American physicist who conceived of the idea of nuclear chain reaction (that was dismissed rudely aside by Rutherford) and thus played a key role in the development and fruition of the Manhattan Project, I shall briefly decipher the cryptic statement of Szilard.

Somewhere around the World War II, both before and after, many scientists and mathematicians both from the Western and the Central Europe emigrated to the United States to escape the fascism that was gulfing Europe.

The task of the Nazis after the war was taken over with great zeal by Stalin’s communism and perverted Marxism behind the iron curtain against which the Hungarians vehemently rebelled but as is well known Soviet dictators would brutally crush down any such type of uprising with an iron hand.

Among those who emigrated to the United States a substantial and significant portion was Hungarian of Jewish decent that included John von Neumann, Eugene Wigner, George Pólya, Paul Erdös, Leo Szilard, Edward Teller (“father of the Hydrogen Bomb”) among others.   

The list is very long and their contributions to the American science are remarkably highly out of proportion with their relative number to the total number of émigrés from Europe.

What was common to all these Hungarians that like Hindus they spoke English with a strange-sounding strong accent that made them stand out in the society and perhaps made them difficult to assimilate into the general American society (within the Academia and Universities such accents are irrelevant and perhaps even welcomed).

With their strong accent, superhuman intelligence and origins from an obscure Eastern Bloc country it did not take much time nor imagination for them to be christened as the Martians.

It may have offended them initially for all I know but they also found it humorous and hence they adopted it themselves.

People like John Von Neumann not only adopted the name for themselves but also came out with humorous evidence and anecdotes to justify the christened name.

We shall continue with the story of the Martians in the nights to come.

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

                              

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