Category Archives: Mathematics

Symmetry and the Law: A Hidden Connection

If you ask someone to name a woman scientist, they will probably name Marie Curie. If you ask them to name another, most will not know what to say. And if you ask someone to name a woman mathematician, most people will not be able to name a single one. Why is that? Have there not been any woman mathematicians? Or have people not bothered to find about woman mathematicians. I think it is important for our children to know that there have been great woman mathematicians and scientists and that science, and more specifically mathematics is not just a province of men. In today’s VERITAS we will talk about a great woman mathematician who made huge contributions to physics and mathematics. Her work led to a new insight into the fundamental laws of Physics and resulted in major developments in modern physics. The name of this mathematician was Emmy Noether and this is what Einstein said of her: “Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

Continue reading Symmetry and the Law: A Hidden Connection

The search for the largest prime numbers and GIMPS

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Friends,

A few weeks back I started teaching my 11 year old son Python programming language. It is turning out to be an amazing experience as I see the delight on his face when his program works and when I see the worry when he is debugging a program that he thought he wrote “perfectly”. One of the programs that he recently wrote was to find the first n primes where n is a user specified value. When his programmed worked( after some debug), he asked me if we can run it long enough to be able to find the biggest prime ever discovered. A few years ago I had told him that there are prizes for finding the largest known prime number. He thought that he could win the prize by running his program overnight! That was an excellent starting point to discuss algorithms and why some algorithms are better than others. I also told him about the limits of a personal computer. We then searched on the internet for the prime number world records and how the largest prime numbers are found. And that is the subject of this VERITAS.

Continue reading The search for the largest prime numbers and GIMPS

Three icecream cones and a line

Infinitely Curious friends,

Every once in a while I meet a mathematical theorem that puts a smile on my face because of its simplicity and “cuteness”. In today’s VERITAS I will tell you about one such mathematical theorem. I read it about a week back and still continues to make me feel happy. So I decided to share this theorem with you.

MongesTheorem_1000

Here is what it is: Take a sheet of paper and make three circles of different sizes. The circles should be separate from each other. Let’s call these three circles A, B and C. Now draw two tangents to A and B. In other words draw one line that touches A and B but does not go inside it – this is a tangent to A and B. Now draw another tangent to A and B. So we have two common tangents to A and B. These tangents will cross( intersect) somewhere. Call this point P1. At this time it will look like an ice cream cone holding two scoops of ice cream of different sizes. Now draw two tangents to A and C. These tangents will cross at a point- call it P2. Similarly draw two tangents to B and C- these will intersect at P3. The cute mathematical fact is that P1, P2 and P3 lie on a straight line. Isn’t that so simple and yet so profound. See the attached image for a visual representation. It will look like three ice cream cones holding scoops A, B and C in pairs. The tips of the cones lie on a straight line.

This theorem is called Monge’s theorem. This theorem is valid even if the circles intersect.

Now a little bit about the mathematician who discovered this interesting mathematical theorem. Gaspard Monge was a french mathematician. Not many of us know his name but all the engineers of this world have used his invention- descriptive geometry. The engineers among you would remember the first year classes on technical drawing( I remember them with horror ). In technical drawing we learnt to represent three dimensional objects on 2D paper using different views and projections- eg top view, front view etc. These techniques were invented by Monge and are known as “descriptive geometry” in mathematics. After the French revolution Monge became very influential under Napoleon’s rule. He was instrumental in the setup of the Ecole Polytechnique- which is one of the great institutions of mathematics and sciences. He also was a part of Napoleon’s expedition to Egypt. In an earlier VERITAS( written in the year 2000), I had written in detail about Napoleon’s interest in mathematics and his trip to Egypt in which he had taken a number of scientists and mathematicians. In that VERITAS I had also written that Napoleon himself was a mathematician and even has a theorem by the name of Napoleon’s theorem. For details read VERITAS: Napoleon’s academic conquest of Egypt, 25th June 2000 (http://unvarnished-veritas.posterous.com/napoleons-academic-conquest-of-egypt).

Friends, I feel that there are three attitudes among most people about mathematics. The first two are “bad” attitudes: some people fear mathematics. Some others are arrogant about their mathematical abilities. Both these attitudes are caused by our educational system which uses mathematics to judge the intelligence of people. Mathematics becomes a scary competition in which 95% people are declared “bad at math” and 5% are considered “good in math”. This causes the 95% of people to develop a fear and 5% to develop an arrogance of their mathematical abilities. Both these attitudes are a result of comparison and comparisons can never result in true knowledge. There is a third attitude which is the correct one. It is about amazement and love for the beauty of mathematics. This attitude does not compare. It is about enjoying the infinite relationships between numbers and shapes. This attitude promotes a lifelong interest in mathematics and an appreciation of its beauty. This is the attitude that we need to promote in our kids. For our kids mathematics should be a way of looking at this amazing world and not about how much his friend scored.

Before I end this VERITAS here is a little quote by G.H Hardy( the person who first recognized Ramanujan’s genius):

The mathematician’s patterns, like the painter’s or the poet’s must be
beautiful; the ideas, like the colors or the words must fit together in a
harmonious way. Beauty is the first test: there is no permanent place in
this world for ugly mathematics.

kanwar

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Go wondrous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run
correct old time, regulate the sun
======================================================================

abc Conjecture

 

Friends,

               I would like to thank Chandrakant Sakharwade for introducing me to this subject.

                This VERITAS is about a very deep and interesting relationship regarding prime numbers. In 2001 I had written a VERITAS article on prime numbers: http://unvarnished-veritas.posterous.com/the-prime-adventure

                Before I start Let me say that to read this VERITAS you may have to concentrate and maybe reread certain sections. I had to read several articles several times before I could understand the abc conjecture. I will try to make it as simple for you as possible by giving many examples and proceeding in a step by step fashion.

                Before I talk about the conjecture, let me give you an overview of its importance. It is generally considered one of the most important unsolved problems in mathematics. There are a lot of other problems in number theory which would be solved almost effortlessly if the abc conjecture was proven. One of them is Fermat’s last theorem. We all know that Fermat’s last theorem was proved by Andrew Wiles in 1995. That was a very complex proof- over a 100 pages long. Some people have suggested that if abc conjecture is proven then Fermat’s last theorem can be proven in less than 1 page! And there are many important unsolved problems which would be solved as soon as we have a proof of the abc conjecture.

                Now, let me explain what the conjecture is all about.

 Lets start with the simple equation: a + b = c.  a,b and c  are positive integers and should have no common factors except 1. So there should be no number except 1 which can divide the three numbers. To give you an example:  2 + 3 = 5 can be such an equation. 5,3 and 2 do not have any common factor. Another equation can be 1+ 2 = 3.  1,2 and 3 do not have any common factors. Such numbers which do not have any common factor are called co-prime numbers.

                Now lets take the numbers a, b and c and multiply them to form a number d. Now a very important theorem of mathematics( called The Fundamental theorem of Arithmetic) states that any number is either a prime or can be expressed as a product of primes. Take the number 17 – it is a prime. Take 18- it can be expressed as 2X 3X 3 ( and 2 and 3 are primes).

                So we take the number d ( which is the product of a, b and c) and find its prime factors- i.e express it as a product of primes. So for our equation 1+2 = 3 d is 1X2X3 = 6 and 6 can be expressed as 2 X 3. Similarly for our equation 2 + 3 = 5, d is 5*3*2 = 30 and 30 can be expressed as 2 * 5 * 3. Lets take another equation 4 + 5 = 9 ( you can check that 4, 5 and 9 do not have any common factors). For this equation d is 4 * 5 * 9 = 180. And 180 can be expressed as 2 * 2 * 3 * 3 * 5  and we know that 2,3 and 5 are all primes.

                Now we find the radical of d. What this means that in the prime factors of d remove any prime that occurs more than once and multiply the rest.

So in the equation 1+2 = 3, d is 6 and 6 is 2 X 3 – there is no duplicate primes so the radical of 6 is 6. We write rad(6) = 6. For the equation 2 + 3 = 5, d is 30 and 30 can be expressed as 2 * 3 * 5 – again there are no duplicates so rad(30) = 30.  Now lets take our equation 4 + 5 = 9. For this d is 4 * 5 * 9 = 180. And 180 can be expressed as 2 * 2 * 3 * 3 * 5. Here we have duplicates. Remove them – there are two 2s and two 3s. Remove the duplicates in the product so rad(180) is 2 * 3 * 5 = 30.

So we have three examples:

1+2 = 3  . d is 6 and rad(d) is 6

2+3 = 5  . d is 30 and rad(d) is 30

4 + 5 = 9. d is 180 and rad(d) is 30

In all the above cases rad(d) is greater than c.  For example in the last equation rad(d) is 30 and c is 9. So rad(d) > c.

Can it ever happen that rad(d) is less than c? Yes it can be. Lets take an example:

5 + 27 = 32

Now d is 5 * 27 * 32 = 4320. The unique primes in 4320 are 2,3 and 5 which makes rad(4320) = 30 which is less than c which is 32.

So there are cases in which rad(aXbXc)  is less than c. However these cases are rare. The funny thing is that they are infinite yet rare. For c < 10000 there are only 120 a,b,c triplets in which rad(d)  or rad( aXbXc) is less than c.

The abc theorem is about these rare, yet infinite abc triplets.

So c can be greater than rad(d). Can c be greater than d squared? Maybe – though we have not found any abc triplet that has this property. There are many examples for abc triplets in which c is greater than d to the power of something between 1 and 2.  For example for our example with 5 + 27 = 32 , c is 32 and rad d is 30. So c is greater than rad(d) to the power 1.018

Take another example:  1+ 8 = 9. d is 72 . 72 can be written as 2 * 2 * 2 * 3 * 3 . So rad (72) is 6. And 6 is less than c which is 9. So we have a abc triplet in which rad(d) is less than c.  So c is greater than rad(d) . c is even greater than  d ^ 1.2262 in this case.

So once again: There are infinitely many abc triplets. Yet abc triplets are rare!

Now after this background( phew!) we come to the abc conjecture:

For any number t greater than 1  there are a finite number of abc triplets which satisfy : c > [rad(d)] ^ t

So in plain English, for any number t  which is greater than 1, you can find limited or finite number of abc triplets so that c is greater than rad( aXbXc) raised to power t. So you can take any number t greater than 1: say you take 1.2234. There would be a finite number of abc triplets in which c is greater than rad(abc) ^ 1.2234. For t >2 there may be no such triplets.

Note the most interesting part: I said that abc triplets are infinite yet rare. So there are an infinite number of triplets which satisfy c > [ rad(d)] ^ 1 but only a finite number of triplets which satisfy c > [ rad(d)] ^ 1.01 or c > [ rad(d)] ^ 1.001 or c > [ rad(d)] ^ 1.0000001 or rad(d) raised to the power of 1 + the tiniest possible thing that you can imagine. So the infiniteness suddenly becomes finite even for a tiny variation from 1 to 1.01 or 1.00001 or 1.0000000001! That is the AMAZING part! So before you my dear friends lies the wonderful boundary between the finite and the infinite!

Well it might seem that mathematicians are getting crazy about details. But this is indeed a very important theorem. This conjecture was proposed by David Masser and Joseph Oesterle in 1985. And since that day there has been a huge excitement and work by mathematicians around the world to prove this. And the exciting news is that last week Shinichi Mochizuki of Kyoto University  has claimed to have solved it. It is a 500 page paper – incredibly interesting and suspenseful, I am sure. 🙂

However note that in science we should not go by the first reports. There would be verifications and discussions before this paper is accepted. In science there is always a process of testing by different teams before a fact is considered to be reasonably accepted as true. However the press seems to jump to conclusions when the first paper on a subject is presented. And that is why people think that scientists keep claiming different things every month. On some days you will see a newspaper report that coffee is good for you and then a few days later, it is the worst thing in the world. The problem is not science or its method – the problem is the newspaper reporters who want “exciting” science news to fill their paper and so will publish something even before it is verified through hundreds of experiments by independent teams.

Finally before I end, here is an interesting poem on prime numbers. Before you read the poem let me tell you what the stanzas refer to. The first stanza is about the fact that nothing can divide them but they can form all other numbers( see fundamental theorem of arithmetric described above). The second stanza tells us that in the sequence of natural numbers, prime numbers appear unexpectedly. The third paragraph refers to the fact that in the sequence of natural numbers, first there are many primes but as the numbers get larger, the primes get more and more rare and less dense. The last stanza refers to the fact that mathematicians have tried hard to find a formula for prime numbers but have all failed. You can find all these facts in my 2001 VERITAS article on prime numbers: link at the top of this article.

 

Let Us Now Praise Prime Numbers      

by Helen Spalding

 

Let us now praise prime numbers 

With our fathers who begat us: 

The power, the peculiar glory of prime numbers 

Is that nothing begat them, 

No ancestors, no factors, 

Adams among the multiplied generations. 

 

None can foretell their coming. 

Among the ordinal numbers 

They do not reserve their seats, arrive unexpected. 

Along the lines of cardinals 

They rise like surprising pontiffs, 

Each absolute, inscrutable, self-elected. 

 

In the beginning where chaos 

Ends and zero resolves, 

They crowd the foreground prodigal as forest, 

But middle distance thins them, 

Far distance to infinity 

Yields them rare as unreturning comets. 

 

O prime improbable numbers, 

Long may formula-hunters 

Steam in abstraction, waste to skeleton patience: 

Stay non-conformist, nuisance, 

Phenomena irreducible 

To system, sequence, pattern or explanation.

 

Kanwar

 

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

======================================================================

The House of Wisdom

Dear lovers of knowledge,

                A few days back when I was researching for my next book( “Pilgrims of Episteme”, expected in 2013) I came across some very interesting information. I will share that with you in this VERITAS.

                As you are aware, the Greeks had a great tradition of science, mathematics and philosophy. People like Socrates, Aristotle, Euclid, Plato etc – there is a long list of Greek greats in these fields. In the second century BC after the Punic and Macedonian wars, the Greek city states declined and the Romans became the dominant power in the Mediterranean.  The Romans were very powerful in terms of military and had a great political system. However they were weak in Science, Arts, Philosophy etc. So they admired the Greeks for these things. On the other hand the Greeks considered themselves much superior in terms of culture and knowledge but they feared the military might of the Romans.

                After the Punic wars Greek culture became very popular amongst the Romans. They learnt the Greek language, copied their architecture, even identified their gods with the Greek gods. The good thing that the romans did was to preserve the Greek culture, language, philosophy and science. The Romans did not add anything to Greek thought but did their best to preserve it. So Greek science and philosophy continued to progress under Roman rule.

                In the 4th and 5th century AD the Roman empire declined. Also there was a lot of conflict between the new Christian religion and the older pagan religion. This resulted in the destruction of the Great Library at Alexandria in 391 AD( See VERITAS: Loss of a Great Library, 27 July 2000, http://unvarnished-veritas.posterous.com/the-loss-of-a-great-library). Rome was attacked and looted several times by Gothic tribes between 409 and 476 AD. One of the tribes, the Vandals were extremely ruthless and destroyed just about everything. The word vandalism is derived from this tribe. In 476 AD the emperor of Rome abdicated and the Roman empire finished. Europe descended into the dark ages. The knowledge of Greek was lost, the Pope acquired great powers and the tradition of science and questioning came to an end.

                The eastern part of the Roman empire became the Byzantine empire and it is here that the Greek language and culture continued to flourish to some degree. However the Byzantines could not add much to the great works of the Greek.

                So the Romans and The Byzantines cannot be considered the real inheritors of Greek science and mathematics.  To find the real succesors of the great Greeks we will have to look elsewhere- a different time and a different place.

                Lets now talk about Baghdad in the 8th century AD.  At that time the Abbasid Caliphate was in power. The Abbasid Caliphs had come to power in 750 AD and in 762 AD shifted their capital to a new city: Baghdad. The Abbasids were influenced by the Hadith( sayings by Prophet Mohammed) such as “The Ink of the Scholar is more precious than the blood of the martyr” and “Seek knowledge from the cradle to the grave” and they started making Baghdad a centre of learning and knowledge. The third Caliph of the Abbasid Calipate, Harun al-Rashid formed Bayt Ul-Hikma or “The House of Wisdom”.  Harun al-Rashid was a great patron of science, music and art. We all know him from the book “A Thousand  and one Nights” also known as “Arabian Nights”.  Many stories in this book involve Harun al-Rashid and his vizier, Jafar.

  The House of Wisdom became the centre of what is known as the Translation movement. The Abbasids had formed a library at the House of Wisdom and they were obsessed with getting every book in the world translated into Arabic and kept in the library. First the Persian texts were translated. Then the focus shifted to Greek texts. Harun al-Rashid’s son, Al-Mamun who ruled from 813 to 833 took the House of Wisdom to great heights. Al-Mamun was hugely influenced by Greek thought. It is said that when he was a young man he once dreamt of Aristole. In the dream Aristotle came to him and spoke about philosophy and politics. Under Al-Mamun’s rule, the works of Plato, Aristotle, Galen, Eulid, Archimedes – every great Greek philosopher, mathematician, physician were translated into Arabic.

                Now the translation of works of Philosophy, Science, Mathematics and medicine is not an easy task. The translator has to be knowledgeable about these subjects to  be able to translate them well. Al-Mamun was not just a collector of books, he was also a collector of the best minds. So everyday, translators, scribes, scientists, mathematicians, physicians met at the House of Wisdom to translate works by the great Greeks. The House of Wisdom became so famous that scholars from around the world came to join it.

                Now what happens when scholars get together under one roof? New ideas, new thoughts emerge. So what started as a Translation Movement soon became a thriving place where original work was carried out in science, mathematics and other subjects.  The scholars at the House of Wisdom extended the works of the great Greeks and took science and mathematics forward.

                Another thing that happened at the House of Wisdom was that texts from India and China were also translated. The scholars were able to combine the works from the east and the west to make great leaps in knowledge. Let me take an example: the Greeks and Romans could not multiply the way we can. Yes, Aristotle, Plato, Euclid, Archimedes etc could not multiply as easily as a school kid can in today’s age. They used to add repeatedly. Why? Because they did not have a decimal system: Imagine trying to multiply XCVIII with DCCVII! Now lets convert it to another notation: 98 X 707. Now it becomes so simple. Why? Because the decimal system gives each place a value- it is a positional system. A positional system requires the number 0. The Arab scholars got this system from India and were quick to realize how superior it was over the Greek-Roman system. Al-Khwarizimi wrote “On the calculation with Hindu numerals” in 825 AD. Al-Kindi wrote a 4 volume subject on the same subject in 830 AD.  

                Now that was one example. The House of Wisdom was the centre of world science and mathematics at that time. Let me give you a few more examples: The word Algorithm comes from Al-Khwarizimi who as we saw earlier wrote the book on Indian number system. Al-Khwarizimi described a step by step way of solving algebraic equations- we call it Algorithm these days. Talking about algebra- the word algebra comes from Al-Khwarizimi ‘s book Kitab Al-jebr. The words Alchemy, Chemistry, Almanac, Amber, Alcohol, Alkali, Azimuth come to us from Arabic and are all associated with some major contributions towards science or mathematics by the scholars at the House of Wisdom.

                The works of many greek philosophers and scientists would have been unknown to us if they were not carefully translated and preserved by the scholars at the House of Wisdom. Lets take the case of Galen, the great Greek man of medicine. The 3 million words that he wrote are available to us only because the Arabs translated them and kept them so carefully. Mohammed al-Razi was known as the “Galen of the Arabs”. He used Galen’s works as a basis and then  wrote a 23 volume text on medicine. He formed a fuction between Persian, Greek and Indian systems of medicine.

  The Islamic Golden Age of Science lasted from 750 Ad to 1258 AD- 5 centuries of scientific progress by the greatest minds of the day based on the great works of Greeks, Indians and Chinese. But then, as it typically happens, lust for power destroys love for knowledge. Baghdad was conquered in the year 1258 by the Mongols led by Hulgau Khan who was the grandson of Genghiz Khan. The House of Wisdom was destroyed. The Mongols took the books from the House of Wisdom and threw them into the Tigris to form a bridge on the river so that they could cross it!

                Friends, from 476 AD to the 14th century, Europe was in dark ages. It was a place of orthodox religion, superstition and ignorance. The centre of science, philosophy, culture and mathamatics was in Baghdad, in the House of Wisdom. When Renaissance began in Italy in the 14th century, it was based on what the Arabs had preserved. The House of Wisdom prepared Europe for Renaissance. Without the House of Wisdom, Renaissance would not have happened, or would have been delayed by several centuries.

                In the end I will leave you with a thought by Al-Kindi, a mathematician and a scholar from the House of Wisdom: “We ought not to be embarrassed of appreciating the truth and of obtaining it wherever it comes from, even if it comes from races distant and nations different from us. Nothing should be dearer to the seeker of truth than the truth itself.”

 

 Like you, a lover of Knowledge

Kanwar

 

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

======================================================================

 

Read my latest book, “Shadows of Lost Time”. http://shadows-of-lost-time.posterous.com/

Fibonacci series in nature

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Today I will tell you about a concept of mathematics and then apply it to biology, music, history, poetry, literature.
In fact, I will show you the world with this one concept.

Lets take a series of numbers:

1, 1 , 2 , 3, 5, 8, 13, 21, 34, 55, 89…….

Each number is just the sum of previous 2 numbers. So we start with 1 and 1. Add to make 2. Then we add
the 2 to the one before it and get 3. 5 is got by adding 3 and 2. And it goes on and on forever. You can check
that 89 is formed by adding 55 and 34. Of course we can add 89 and 55 to find the next number in the sequence.
So every number is the sum of its immediately 2 previous numbers.

This series is known as the Fibonacci series after a person named Leonardo of Pisa, also known as
Fibonacci, who studied it in detail in 1202 AD. The numbers in this series are called Fibonacci numbers. So 89
is a Fibonacci number but 85 is not.

Fibonacci numbers had always been known to Indian mathematicians. Pingala in 200 BC had used this number
sequence in describing the metrical structure of Sanskrit poetry. All of Sanskrit poetry is based on Fibonacci numbers!

Virgil used the Fibonacci sequence to structure his epic poem Aeneid.

There are studies that state that Beethoven and Mozart used this series of numbers while composing their music.

Now, lets examine the beauty of different kinds of flowers using the Fibonacci number sequence.
A sunflower is beautiful isn’t it? Now lets take a look at its mathematical beauty. If you look into the seeds at the
centre of the flower, you will see them arranged in a tightly packet spiral. The number of spirals is a Fibonacci
number. So you can have 34 spirals inside the sunflower but not 35 or 33. 34 is a Fibonacci number as we have
seen above!

How many petals does a buttercup have : 5 ( it is a Fibonacci number).
How many does a marigold have? 13( Fibonacci number).
How many does an aster have? 21 ( Fibonacci no!).
How many does a daisy have ? 34, 55 or 89 ( ALL FIBONACCI NUMBERS!)

The petals of all flowers follow the Fibonacci number sequence! The number of leaves on a plant in each turn as
you go from bottom to top follow the Fibonacci sequence! Pine cones have a Fibonacci number of spirals and “petals”
at every set of “winding”. Same is true for pineapples!

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Leonardo Da Vinci knew about this. He suggested that plants and flowers set themselves up in such a way so as
to preserve moisture. Leonardo Da Vinci used the Fibonacci number sequence in his painting: Mona Lisa and in
the study of the proportions of man, ” The Vetruvian man”.

A golden rectangle is the one whose sides are the ratios of any two consecutive Fibonacci numbers. Divide any
Fibonacci number by its previous and you get about 1.6. So a golden rectangle is one whose sides are of the
ratio 1.6. So a rectangle with the sides 34 and 21 is a golden rectangle because the sides are Fibonacci numbers.
A golden rectangle is most pleasing to the eye. So since time immemorial, artists have employed it in their art and
architects have employed it in their buildings.

Nature is a painter, a musician, a thinker, a philosopher and a great mathematician. Mathematics is Nature’s
attempts to create beauty in numbers. Nature is beautiful; Mathematics is beautiful. There is no other way!

Now let’s learn a very interesting thing about honeybees. I have two parents. You have two parents. All animals
have two parents. But a honeybee is different! It can have one or two parents! A queen bee lays down many
eggs. If an egg is fertilized by a male drone, it hatches to be a female. But if an egg remains unfertilized then
it hatches to be a male drone. So, a male bee only has a mummy! A female bee has a mummy and a daddy!
Females usually end up as worker bees but some( very few) are fed a special kind of jelly which makes them
queens- they fly away to form their own colonies.

Now lets take a male bee. How many parents does it have? 1: the queen female.
How many grand parents does it have : 2 ( the queen female has to have a father and a mother)
How many great grand parents does it have : 3 ( The father will have only a mother but the mother will have two parents)
How many great-great grand parents does it have : 5 ( two for the each of the great grand mothers. One for the great grand father)
How many great great great grand parents: 8 ( you can calculate this….. it is simple)
How many great-4 grand parents: 13
How many great-5 grand parents: 21
How many great-6 grand parents : 34
How many great -7 grand parents : 55
How many great-8 grand parents: 89

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This is the Fibonacci series! So the number of ancestors of honey bees at each generation follows the Fibonacci series!
Isn’t this WOW! This is true for the female bee also!

Isn’t the mathematical structure of Nature beautiful?

Now let me tell you about limerick. A Limerick is a 5 line poem with the rhyming sequence: AABBA.
It is always a funny poem and has the following structure: The 1,2 and 5th lines are longer and the 3rd and 4 th
lines are shorter. They are shorter by a metrical foot. The 1,2 and 5 lines should have 9 syllables and the 3rd and 4th
lines have 6 syllables. That is the structure. Here is an example:

This is Dixon Merrit’s famous limerick on pelicans. It has become so popular that it has been quoted several times
in scholarly books of Ornithology. Here it is:

        A wonderful bird is the pelican,
        His bill will hold more than his belican,
        He can take in his beak
        Enough food for a week
        But I’m damned if I see how the helican!

Inspired by this and in reference to our above discussion about Fibonacci numbers and bees, I have written a
Limerick:

The fibonacci number sequence,
is for the bees of consequence,
because the female bee,
has a daddy you see,
but the drone is due to his absence.

Kanwar

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Go wonderous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run
correct old time, regulate the sun
======================================================================

Creative Commons License
Veritas by Kanwarpreet Grewal is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

Cantor

 

 A very interesting article on the mathematician Georg Cantor posted for VERITAS by Soumyaroop Roy. See also VERITAS: History of Set Theory, 8 May 2002) 

 

Given the old maxim about an infinite number of monkeys and typewriters, one can assume that said simian digits will type up the following line from Hamlet an infinite number of times.

“I could confine myself to a nutshell and declare myself king of infinity”.

This quote could almost be an epithet for the mathematician Georg Cantor, one of the fathers of modern mathematics. Born in 1845, Cantor obtained his doctorate from Berlin University at the precocious age of 22. His subsequent appointment to the University of Halle in 1867 led him to the evolution of Set Theory and his involvement with the until-then taboo subject of infinity.

Within Set Theory he defined infinity as the size of the never-ending list of counting numbers (1, 2, 3, 4….). Within this he proved that sub-sets of numbers that should be intuitively smaller (such as even numbers, cubes, primes etc) had as many members as the counting numbers and as such were of the same infinite size. By pairing off counting and even numbers together, we see that the number of counting and even numbers must be the same:

 

 

1 -> 2

2 -> 4

3 -> 6

4 -> 8

5 -> 10

6 -> 12

 

He then went on to demonstrate the impossibility of pairing off all the real numbers (those including irrational decimals like Pi) with the counting numbers, concluding that one was larger than the other. The result, confusing though it may seem, is that some infinities are bigger than others!

 

Cantor’s work represented a threat to the entrenched complacency of the old school mathematicians. Up until then infinity, to quote mathematician Carl Friedrich Gauss, had been treated as a “way of speaking and not as a mathematical value”. This stonewalling inevitably brought Cantor into conflict with his less enlightened peers. His most vocal critic was Leopold Kronecker (ironically one of Cantor’s past professor) who undertook a personal crusade to discredit his lapsed protg. Using his position at the University of Berlin he dedicated himself to rubbishing Cantor’s ideas and ruining him personally.

His coup de grace was blocking Cantor’s lifelong ambition of gaining an appointment at the University of Berlin.

 

In 1884, consigned to a backwater University and under constant attack from Kronecker, Cantor had his first nervous breakdown. He spent the rest of his life in and out of mental institutions, his serious work at an end.

Cantor’s later years may have been defined by tragedy but his contribution to modern mathematics is colossal. His one-time collaborator David Hilbert once said of him in tribute “No one will drive us from the paradise that Cantor has created.”

The Copernicus of Geometry

 

     A few weeks back we had discussed the method of mathematical proof called “Reductio Ad Absurdum”. We had also learnt that Euclid used this method in several proofs. In that VERITAS I had mentioned that Euclid made a wrong proof of the fifth postulate or the parallel postulate.

        In today’s VERITAS we will discuss the parallel postulate and also read the story of the person who dared to challenge this postulate.

My inspiration for this VERITAS is E.T Bell’s book “Men of Mathematics”.

I can safely bet that anyone who reads this book cannot escape from falling in love with mathematics and a desire to become one of them!

      The parallel postulate can be stated in various ways. The simplest definition is this: ” if there is a straight line and a point X not on that line then there is only one line that can pass through X and never intersect the given line( even if they are both extended to infinity).

This line is parallel to the given line”. This looks pretty obvious if you draw lines on a piece of paper. Euclid tried to prove this but was never really satisfied. So he used it as an axiom. That was about 300 BC. No one challenged this for about 2200 years. And then came Lobatchewsky, the Copernicus of geometry ….

       Nicolas Lobatchewsky was the son of a minor government official. He was born on November 2, 1793 in Russia. He did his schooling at Kazan. He entered the Kazan University in 1807. In 1811 he obtained his degree from the university. In 1813 he was appointed to a teaching post in that University.

      The Kazan University was a new and a small one. And it was in a small town. Lobatchewsky professorship in the university was not restricted to teaching mathematics. There was little order in the university and in the town. So Lobatchewsky did nearly everything : he taught Mathematics, Physics, Astronomy. He was inchange of the laboratories, he was inchange of the library, he had to supervise the elementary schools in the area, he would even dust the books and catalog the books in the library himself. So he did just about everything.

In 1827 Lobatchewsky was appointed the rector of the university.

That was one of the top posts there. But that did not deter Lobatchewsky from doing everything. He would still take a mop and clean the lab.

When the government decided to modernize the university by making a few new buildings, Lobatchewsky was incharge of overseeing the project. But Lobatchewsky could not just oversee! He had to be in it! So he learnt architecture and designed the buildings himself! The university was his life!

         But mathematics was his first love. In 1823 he produced a work that would challenge the Great Euclid himself. He created a new geometry in which the parallel postulate is not valid. And this new geometry was fully self consistent( or complete). This geometry later came to be known as non-Euclidian geometry. Euclid’s geometry was the geometry of flat surfaces, Lobatchewsky’s geometry was the geometry of curved surfaces.

      To get an idea of geometries other than the euclidian lets take an example of our own curved earth. When we travel between 2 distant places the shortest path is not a straight line but a curve. ( Remember the last time you flew from Jalandhar to Southall the TV in the aeroplane showed you a curved path). The shortest distance between two points on the earth is called a geodesic. How is this defined: Imagine a flat plane that passes through the 2 points and cuts the earth and passes through the centre of the earth. The line that this plane makes on the surface of the earth is called a geodesic.

      The geometry on a sphere is different from euclidian geometry. On a plane surface ( euclidian geometry) two lines will either be parallel, or cross each other at exactly one point. On a sphere 2 geodesics will either be parallel or cross each other at TWO points. There are several other differences.

      On a sphere the sum of the angles of a triangle is always more than 180 degrees.

      And there are several other geometries: the hyperbolic geometry and the geometry on a pseudo-sphere. On a pseudo sphere there are exactly 2 lines which pass through a point and are parallel to a line. Then there is Riemann’s geometry in which 2 angles of a rectange are obtuse. Riemann’s geometry was used by Einstein in his General theory of Relativity in which he showed that gravity curves space-time.

      So Lobatchewsky’s genuis was in challenging a 2200 year old axiom. He opened the gates for several types of geometries. He showed that Euclid’s geometry was only a special case.

        But his genuis as a mathematician was never recognized in his lifetime. In 1846 he was charged with mismanagement in the university and was asked to leave. He loved the university, it was HIS university.

This broke him. His health suffered and he became blind. In 1856 he died a sad and broken man.

      Physicists are now realizing that all of Physics and all the laws that govern the Universe may be manifestations of the Geometry of space time.

      A few poetic words on geometry by Emily Dickenson:

 

      Best Witchcraft is Geometry

      —————————

 

      Best Witchcraft is Geometry

        To the magician’s mind —

        His ordinary acts are feats

        To thinking of mankind.

 

Kanwar

 

|======================================================|

  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                           

  Correct old time and regulate the Sun;                             

|======================================================|

Reductio ad absurdum

 

                  Friends, in today’s VERITAS we will try to understand a technique of mathematical proof. This is a very powerful technique and helps prove some very complicated mathematical theorems. The technique is called “Reductio ad absurdum” which means reduction to an absurdity. This technique is also called proof by contradiction.

        This technique is very old. It has been there since the time of Euclid(330 – 275 BC). Euclid used this very effectively in the proof of several theorems. This is how it works: suppose we want to prove a theorem which can be stated by a statement S. We assume that S is not true i.e we assume that our theorem is false. Then we show that the consequences of such an assumption is something absurd which can never be true. So this means that our assumption that S is false is not valid. Thus S is a true statement.

 Theorem proved.

  The mathematician Hardy( Ramanujan’s friend) wrote in his book, A Mathematician’s apology: “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game”

        Notice that our technique makes use of the law of the “excluded middle”. According to this law a statement can either be true or false.

And if a statement is not true then it MUST be false. Not every school of mathematical thought accepts this technique as a valid method of mathematical proof. The school of intuitionism does not take the law of the excluded middle to be universally valid and thus does not agree that Reductio ad  absurdum should be a valid method of mathematical proof.

        Lets look at an example of a proof by Reductio ad absurdum. This is Euler’s proof that the number of prime numbers are infinite i.e there is no largest prime number. But before we start our proof we should remember that every non prime  natural number can be written as a product of primes in essentially one way. This is called the fundamental theorem of arithmetic. We will use this theorem in our proof that the number of prime numbers is infinite:

proof: Lets assume that our theorem is false i.e there are a finite number of prime numbers. That means that there is a largest prime. Now lets multiple all the primes together and add one to the product. This new number leaves a remainder of 1 when divided by ANY prime. Therefore it cannot be  divided by any prime ie it is not divisible by ANY prime.

So it HAS to be another prime. But that is absurd because we had already taken all the primes and multiplied them . And now we have found another prime! SO this contradicts the statement with which we started. So our statenment that  the number of primes is finite is wrong. So we have proved that the number of primes is infinite!!

        See what a powerful technique this is! But it is also a very dangerous one. Euclid made a mistake in his proof of the fifth postulate(the parallel postulate). The problem is caused when a statement looks false but is not really PROVEN false.

      A question to you: do you think that God listens and acts upon prayers?? Should be apply Reductio ad absurdum here? Ivan Turgenev once said ” Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'” !!!!!

How very absurd ! 🙂

Kanwar

 

|======================================================|

  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                          

  Correct old time and regulate the Sun;                             

|======================================================|

 

Reductio Ad Absurdum

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Friends, in today’s VERITAS we will try to understand a technique of mathematical proof. This is a very powerful technique and helps prove some very complicated mathematical theorems. The technique is called “Reductio ad absurdum” which means reduction to an absurdity. This technique is also called proof by contradiction.

       This technique is very old. It has been there since the time of Euclid(330 – 275 BC). Euclid used this very effectively in the proof of several theorems. This is how it works: suppose we want to prove a theorem which can be stated by a statement S. We assume that S is not true i.e we
assume that our theorem is false. Then we show that the consequences of such an assumption is something absurd which can never be true. So this means that our assumption that S is false is not valid. Thus S is a true statement.
 Theorem proved.

 The mathematician Hardy( Ramanujan’s friend) wrote in his book, A Mathematician’s apology: “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or
even a piece, but a mathematician offers the game”

       Notice that our technique makes use of the law of the “excluded middle”. According to this law a statement can either be true or false. And if a statement is not true then it MUST be false. Not every school of mathematical thought accepts this technique as a valid method of mathematical proof. The school of intuitionism does not take the law of the excluded middle to be universally valid and thus does not agree that Reductio ad  absurdum should be a valid method of mathematical proof.

       Lets look at an example of a proof by Reductio ad absurdum. This is Euler’s proof that the number of prime numbers are infinite i.e there is no largest prime number. But before we start our proof we should remember that every non prime  natural number can be written as a product of primes in essentially one way. This is called the fundamental theorem of arithmetic. We will use this theorem in our proof that the number of prime numbers is infinite:

proof: Lets assume that our theorem is false i.e there are a finite number of prime numbers. That means that there is a largest prime. Now lets multiple all the primes together and add one to the product. This new number leaves a remainder of 1 when divided by ANY prime. Therefore it cannot be  divided by any prime ie it is not divisible by ANY prime. So it HAS to be another prime. But that is absurd because we had already taken all the primes and multiplied them . And now we have found another prime! SO this contradicts the statement with which we started. So our statenment that  the number of primes is finite is wrong. So we have proved that the number of primes is infinite!!

       See what a powerful technique this is! But it is also a very dangerous one. Euclid made a mistake in his proof of the fifth postulate(the parallel postulate). The problem is caused when a statement looks false but is not really PROVEN false.

       A question to you: do you think that God listens and acts upon prayers?? Should be apply Reductio ad absurdum here? Ivan Turgenev once said ” Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'”
!!!!!
How very absurd ! 🙂

Kanwar

|=============================

=========================|
 Go, wondrous creature! mount where Science guides:
 Go, measure earth, weigh air, and state the tides:
 Instruct the planets in what orbs to run,
 Correct old time and regulate the Sun;
|======================================================|

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Veritas by Kanwarpreet Grewal is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.