November 30, 2016 Wednesday

Bedtime Story 


How Algebra and Analysis Took Over Geometry

Euclidean space is a coordinate space (space with a coordinate system that forms the basis of constructing vector space or a manifold), that allows several real variables (n) to be treated as a single variable.

The idea of coordinate space or space is very interesting and has undergone a remarkable transformation from the times of Euclid when geometry ruled.

In the days of Euclid, the two-dimensional space was largely defined with the Euclidean plane and the three-dimensional space with postulates and built over it, theorems.

Geometry and the structures created out of it were used to define the real numbers.

Such was the hold of geometry over other branches of mathematics.

Then gradually, very gradually over the years, algebra and mathematical analysis developed and took reigns over mathematics.

Mathematical analysis studies limits using calculus (differentiation, integration), derivatives, continuity and so on.  

                                     

This is possibly the most famous limiting equation often stated as limit of (1 + 1/n)ⁿ as n approaches infinity.

It is believed that this equation was first solved by Jacob Bernoulli way back in 1683 when trying to solve a problem on the compound interest.

Solving this, he came across that magical number now known as the letter  or the Euler constant.

This Euler constant is one of the most famous constants that is both irrational (cannot be expresses as a ratio of 2 integers) and transcendental (not a root of any non-zero polynomial with rational coefficients).

So over the years, this remarkable branch of mathematics called analysis and along with it algebra became so powerful and useful that now they began to be used to define geometry.

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

November 29, 2016 Tuesday

Bedtime Story 

Cantor Defends His Ideas

After this 1874 paper, in a span of five years between 1879 and 1884 Cantor published 6 papers of which the 5^th paper was most significant.

This 5^th paper was titled “Foundations of a General theory of Aggregates”.

Already a lot of hostility and aversion was building up against Cantor’s ideas by the established professors at the helm of which was his own mentor Kronecker.

Kronecker found the concept of multiple infinites to be preposterous, as that would eventually lead to such insidious paradoxes that would destroy the foundations of mathematics.

Cantor would have none of it.

He had to pursue logic and reason and see where it took him; if it ended up in paradoxes then so be it.

In this fifth paper of the series that he published in 1883, he was not only laying down the foundations of set theory but was also defending himself from the belligerent attacks that were being hurled at him.

In this paper, he carefully began by defining the concept of well-ordered sets and ordinal numbers (order types of well ordered sets).

Then we went on to demonstrate how transfinite numbers are not something extraordinary, but merely an extension of natural numbers.

In this paper, Cantor also defined the rules of addition and multiplication for the cardinal and ordinal numbers.        

These were formidable papers in the sense that they were challenging the long standing beliefs of the existing establishment.

Yet it has the first 1874 paper that was published in the Crelle’s Journal that was most revolutionary.

It laid the background for his dealing with the notion of Euclidean space or real coordinate space of n dimensions that is denoted by the symbol Rⁿ.

  

Let me deal with the Euclidean space in the nights to come.

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

November 28, 2016 Monday

Bedtime Story 

Dedekind Provided the Proof for the First Theorem

The most common example that is cited is the division of the number line with the root of 2, the most famous irrational number that cannot be expressed as a ratio of two integers or fraction.

In this case, the number that lies exactly to the left hand side of √2 will be another new irrational number.

In words of Dedekind himself:

“Whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut…

From now on, therefore, to every definite cut there corresponds a definite rational or irrational number…”

Dedekind himself was dealing with number theory and these kinds of ideas related to set, so that he was possibly the first and the only mathematician (besides Cantor’s former teachers Karl Weierstrass and Leopold Kronecker) who could not only understand Cantor work but even contribute to it through correspondence.

His greatest contribution was to provide the proof to Cantor’s first theorem that defined the set of real algebraic numbers to be countable.

Dedekind did something which the likes of us can never imagine.

He saw a connection between algebraic numbers and polynomials.

He then defined the height of polynomials and then ordered the real roots of the same height by the numeric order.

This orderings put the real algebraic number into a sequence.

Quite understandably, this is a difficult idea for us non-mathematicians to grasp.

Cantor was very thankful to Dedekind for his help which he wrote to him in private, but left it out in the paper for which Dedekind never forgave him and even stopped his correspondence for years.

So in that way, even mathematicians are just like other human apes.

They too seek acclaim, acknowledgement and whatever glory mathematics is capable of giving from their own peer group.  

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

November 27, 2016 Sunday

Bedtime Story 


Cantor's Uncountability Theorem

Cantor makes the following remark after stating his second theorem:

“The reason why collections of real numbers forming a so-called continuum (such as all real numbers which are ≥0 and ≤1) cannot correspond one-to-one with the collection (v) [the collection of positive integers];

Thus I have found a clear difference between a so-called continuum and a collection like the totality of real algebraic numbers.”

This is Cantor’s uncountability theorem.

In the paper, Cantor did not actually prove it but merely stated it.

It should be emphasized that a benign statement such as:

“The set of real algebraic numbers can be written as an infinite sequence in which each number appears only once.”

needed an immense effort from many mathematicians to be proved.

Cantor himself could not prove it.

The proof would come from another German mathematician and perhaps the only mathematician in the early days to both appreciate and admire Cantor’s work.

It was Richard Dedekind who met Cantor accidentally on a holiday at the beautiful town of Interlaken situated in the Bern canton of Switzerland in the year 1872.

Dedekind was the first mathematician to give a precise definition of an infinite set and real numbers.

With what goes now by the name of Dedekind cut, he gave a whole new perspective of looking and constructing real numbers.

It is a beautiful idea and I would like to share it with you.

He considered the famous number line that contains all the rational and irrational numbers.

If you cut the line anywhere with an irrational number, you end up creating 2 non-empty sets A and B of rational numbers such that all the elements of A are less than all the elements of B.

That seems very apparently obvious.

What is less obvious is this next statement:

Moreover, the set A will contain no greatest element of rational number.

This statement baffles me till this date.

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

On the left hand side of the root 2, there is no greatest element. For there will always be yet another irrational number left of root 2. 

November 26, 2016 Saturday

Bedtime Story 


The 1874 Paper of Georg Cantor Contained the First Theorems of Transfinite Set Theory 

Remember, Cantor was more interested in numerical sets that were infinite and until him, no one even knew how to tackle them.

Anybody who had handled them earlier had landed up in paradoxes and left things at that.

Mathematicians consider the beginning of modern set theory with the publication of Cantor’s seminal paper in 1874 titled “On a Property of the Collection of All Algebraic Real Numbers”.

This paper is also known as Georg Cantor’s first set theory article and is rather short for its greatness.

It spans a mere 4 full pages and 1/3^rd of another.

At the very onset, Cantor in his paper states his first theorem.

The set of real algebraic numbers can be written as an infinite sequence in which each number appears only once.

It does not seem to be very extraordinary theorem at first glance, not even at the second well-pondered glance.

What Cantor was implying in his first theorem is that the entire set of real algebraic numbers can have one-to-one correspondence with the set of positive integers.

Following this, Cantor goes on to state the second theorem which pertains to numbers in a closed interval of a and b where a and b are certain integers with a ≠ b:

Given any sequence of real numbers x1, x2, x3…and any interval [a, b], there is a number in the interval [a, b] that is not contained in the given sequence. Hence, there are infinitely many such numbers.

That is not very difficult to imagine.

Let us consider say a set of positive integers {4, 5} only.

The let us consider a set of real numbers in the interval [4, 5].

The real numbers between 4 and 5 can be {9/2, 19/4, 23/5, 24/5…) and many more.

It is evident that there does not exist a one-to-one correspondence between the set of positive integers {4, 5} and the set of real numbers lying in the interval [4, 5].

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

November 25, 2016 Friday

Bedtime Story 


Basic Notations Devised by Cantor

Consider the infinite set of natural numbers:

N = {1, 2, 3, 4, 5…} has the least element as 1.

So it is a well ordered set with a total order (because 1, 2, 3 and so on obey the conditions of antisymmetry, transitivity and totality) and has the least element 1.

Every non-zero natural number has a unique number that precedes it (For example, 99 will always and only precede 100).

Another ordering in it would be that all the even numbers are less than the odd numbers (2 < 3, 4<5, 6<7 and so on).

Now consider the infinite set of integers:

Z = {…-3, -2, -1, 0, 1, 2, 3, 4…}

Now Z has all the conditions fulfilling total order but since it has no least element, it is not a well ordered set.

The ordinal number deals only with well-ordered sets.

Another idea that Cantor developed was that of the order type of the set.

The order type gives the length of any well-ordered infinite set, the smallest infinite ordinal being represented by the Greek small letter omega ω.

The natural numbers as you will intuitively understand is the smallest totally ordered set with no upper limit or no upper bound.

The first infinite ordinal is ω and that comes after all the natural numbers.

With this, I have tried to define the basic technical terms that were coined by Cantor to develop his astounding ideas.

They are:

1. One-to-one correspondence or bijection

2. Transfinite numbers (numbers that are more than finite but less than uncountable infinite)

3. Cardinal numbers or cardinality of infinite sets which is denoted by the symbol aleph and which can be ₀, ₁, _2…

The natural numbers are infinite but countable and have been assigned the lowest cardinality 0.

The cardinality of the real numbers  also known as the cardinality of the continuum is denoted by the letter  which is the lower case letter of Fraktur script.

4. Ordinal numbers represented by the symbol ω and the idea of total-order and well-ordered sets.

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

The first is the Hebrew letter aleph and the second is the lower case of fraktur script "c" which is used to show the size of the set of Real numbers

November 24, 2016 Thursday

Bedtime Story 

Defining Total Order in a set and Well Ordered Set

I will keep coming back to Cantor’s work in parallel with defining the basic notions that arose out of his work.

While the cardinal number has something to do with the size of the set, the concept of ordinal number arose when Cantor needed to deal with the positions of the elements of various infinite sets.

Cantor decided to represent the cardinality of infinite sets with the Hebrew letter aleph.

So for example, take the natural numbers.

They start from 1 and go on endlessly.

Cantor took the cardinality of natural numbers to be aleph-zero or _0. 

This was an audacious step though it may not seem so to us ordinary mortals.

So two sets can have the same cardinality or the same cardinal numbers but due to their difference in the linear sequence, can have different ordinal numbers.

Ordinal numbers deal with the ordering of elements in a set.

The difference between the cardinality and ordinality is subtle and not obviously apparent in finite sets.

Here arises the concept of a well-ordered set and total order.

A set say A which has elements a, b and c is said to be totally ordered if it satisfies the following conditions:

If a ≤ b and b ≤ a then a = b; possessing antisymmetry.

If a ≤ b and b ≤ c then a ≤ c; possessing transitivity.

a ≤ b or b ≤ a; possessing totality.

Thus a total ordered set is antisymmetric, transitive and is total.

A well-order on a set A implies that the set A has total order and if non-empty, has a least element in its ordering.

It is understandably a bit confusing, so allow me clear it with an example which I shall do in the following bedtime story.

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

November 23, 2016 Wednesday

Bedtime Story 

Cardinality of a Set

As Cantor worked on the open problem of representing a function by trigonometric series he came upon the idea of transfinite ordinals.

Transfinite numbers are such that that they are certainly larger than anything that can be defined as finite yet are not absolutely infinite.

Just think about it for a moment and see how bizarre and counter intuitive it sounds.

This was truly a novel concept back in 1870 something which so many mathematical geniuses had thought and were distressed about but could never come to this conclusion.

In a way, it was almost Einsteinian moment of mathematics, similar to suggesting that time could contract or dilate.

The only difference was that while Einstein was made a hero for his ideas, Cantor was cursed and condemned.

How does one understand the ordinal numbers?

In a set, the natural numbers can be used do define set in at least 2 possible ways.

A natural number can show how many elements a set contains and then it would become a cardinal number.

Consider these two sets:

P = {2, 3, 5, 7, 11, 13}

Z = {-2, -1, 0, 1, 2, 3}

So |P| = |Z| = 6

Both these 2 sets have the same cardinality namely 6 as these two sets satisfies the definition of one-to-one correspondence between their respective elements.      

Another word that is used to show this 1-to-1 correspondence is bijection or existence of bijection function.

Cantor’s greatness lay in using this bijection to infinite sets say for example the set of natural numbers.

The natural numbers 1, 2, 3, 4… are infinite.

So also are its multiples of 10 namely 10, 20, 30, 40… are infinite.

If I were to ask you which would be more, the natural numbers or its multiples of 10, one would always go for the former.

“The natural numbers has to be more since the tens of it are a subset of the natural numbers” would be your argument.

Cantor using the idea of bijection showed both these infinite series could be paired using the 1-to-1 correspondence and hence their cardinality must be the same.

Cantor went further ahead to show that similar bijections with natural numbers could be shown for other numbers that were apparently subsets of the natural numbers.

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

November 22, 2016 Tuesday

Bedtime Story 


Galileo's Paradox was published in his last book: The Discources and Mathematical Demonstrations to Two New Sciences (1638)

We all have a vague intuitive sense of the words finite and infinite.

Yet we are all unaware the emphasis on rigor and precision of definitions that pure mathematicians demand of their subject.

Finite and infinite have troubled nearly all the mathematicians of the past including the pre-Socratic Zeno of Elea (somewhere around 450 B.C.) and Galileo.

A famous mathematical paradox on infinity has been named after Galileo that describes the essence of trouble one get into while dealing with infinity.

Galileo had published this paradox in the form of a dialogue between 3 fictional men who go by the name of Simplicio, Salviati and Sagredo.

The book “The Discourses and Mathematical Demonstrations Relating to Two New Sciences” was Galileo’s final treatise published in 1638 that was gleefully banned by the Roman Inquisition.

The Christianity of Medieval ages was like the Islam of today, far different and ominous from what it portrays itself at present.

In the dialogue, Simplicio first states that some numbers are squares (such as 4 and 9 are of 2 and 3) whereas other are not (such as 5 or 13).

Salviati then states that it must follow that the sum of all the squares and non-squares ought to me more than just the squares.

Salviati further states that it is also true that very square has to have a root and hence every square will have a corresponding root.

Salviati then poses a question.

What if I were to ask how many roots there are?

The roots must be same as there are squares.

Also the number of squares must be same as there are number since every number can be squared.

But this goes against our first statement which said that there are more numbers than there are squares as not all numbers are squares.

Sagredo (playing the role of an idiot like me) then asks what can be inferred from this paradox.

Salviati (playing the role of wise man) says we can conclude the following:

First, that the numbers are infinite, their squares are infinite and so are their roots.

Secondly, the terms such as “equal”, “greater” and “lesser” are not applicable to infinite quantities but only to finite.

Cantor disagreed with Galileo and like him, nearly 400 years later, he was brutally disparaged, maligned, vilified and castigated so much so that he was thrown into bouts of depression, madness and eventually suicide.  

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

                              

Good night and my fellow cousin ape.

         

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, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg

The book has 4 parts in the form of 4 days and in each day a different aspect of physics is discussed