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amazing number nine.on vedic maths

2009-11-05T15:56:39Z

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November 03, 2009
The Amazing Number 9 and the Mathematical Finger Print of God.

I am going to share what a Beautiful Number 9 is. This has been personified by the works of a mathematician Marko Rodin who calls his work Vortex Based Mathematics.Vortex-Based Mathematics (VBM) is completely different because it is a dynamic math that shows the relationships and thus the qualities of numbers rather than the quantities.

From his website :-

Marko studied all the world's great religions. He decided to take The Most Great Name of Bahaullah (prophet of the Bahai Faith) which is Abha and convert it into numbers. He did this in an effort to discover the true precise mystical intonation of The Most Great Name of God. Since the Bahai sacred scripture was originally written in Persian and Arabic, Marko used the Abjad numerical notation system for this letter to number translation. This was a sacred system of allocating a unique numerical value to each letter of the 27 letters of the alphabet so that secret quantum mechanic physics could be encoded into words. What Marko discovered was that (A=1, b=2, h=5, a=1) = 9. The fact that The Most Great Name of God equaled 9 seemed very important to him as everything he had read in both the Bahai scriptures and other religious text spoke of nine being the omni-potent number. So next he drew out a circle with nine on top and 1 through 8 going around the circle clockwise. Then he discovered a very intriguing number system within this circle. Marko knew he had stumbled upon something very profound. This circle with its hidden number sequence was the "Symbol of Enlightenment." This is the MATHEMATICAL FINGER PRINT OF GOD.

Follow along as the amazing properties of this symbol unveil themselves to you. Put your pencil on number 1 and without picking up your pencil, move your pencil in a straight line to number 2, then 4, then across the center to 8. Notice that you are doubling. So next should be 16 and it is, but 1+6=7. So move your pencil to 7. Then 16 doubled is 32, but 3+2=5. So move your pencil to 5. Then 32 doubled is 64 and 6+4=10 and 1+0=1. And you're back to 1. So move the pencil across the center and back up to 1. The significance of the Mayan calender is that 64 is one complete cycle of infinity. Then it begins again with 64 doubled is 128 and 1+2+8=11, then 1+1=2. And so on. You will never get off this track as you keep doubling. Notice the infinity symbol has formed underneath your pencil, creating an ever-repeating pattern of 1, 2, 4, 8, 7, 5. Amazingly, this number sequence stays intact as you half numbers as well. Start again at the 1 but this time go backwards on the infinity symbol. Half of 1 is .5, so move your pencil to the 5. Then half of .5 is .25, and 2+5=7. So move your pencil to the 7. And half of .25 is .125 and 1+2+5=8. So move to the 8. Next half of .125 is .0625 and 0+6+2+5=13 and 1+3=4. So go across to the 4. And half of .0625 is .03125 and 0+3+1+2+5=11 and 1+1=2. So move to the 2. Forever staying on the route of 1,2,4,8,7,5 even backwards.

At this point some of you might be thinking, "What in the world do these number patterns have to do with real world applications?" These number groupings piece together into a jig-saw-like puzzle pattern that perfectly demonstrates the way energy flows. Our base-ten decimal system is not man made, rather it is created by this flow of energy. Amazingly, after twenty years of working with this symbol and collaborating with engineers and scientists, Marko discovered that the 1,2,4,8,7,5 was a doubling circuit for a very efficient electrical coil. There was still one more very important number pattern to be realized. On the MATHEMATICAL FINGER PRINT OF GOD notice how the 3, 9, and 6 is in red and does not connect at the base. That is because it is a vector. The 1,2,4,8,7,5 is the third dimension while the oscillation between the 3 and 6 demonstrates the fourth dimension, which is the higher dimensional magnetic field of an electrical coil. The 3, 9, and 6 always occur together with the 9 as the control. In fact, the Yin/Yang is not a duality but rather a trinary. This is because the 3 and 6 represent each side of the Yin/Yang and the 9 is the "S" curve between them. Everything is based on thirds. We think that the universe is based on dualities because we see the effects not the cause.

Clearly Marko has used the principle of Digit Sums and the Vedic Square to create something beautiful, which later looks like this.

More on Marko's Website http://rodin.freelancepartnership.com/content/view/7/26/

After watching Bizza's One Eye and Now Marko's Vortex I think 9 has to be looked into seriously to bring out all the facets of it.

Another Contributor , Piyush Dadriwala from Noida, India sends us this on the number 9's amazing properties.

AMAZING NUMBER NINE

It is very interesting to note that when you take any number of digits like 25 AND 32,
NOW YOU CAN WRITE THEM IN FOUR WAYS LIKE THAT,
25*32=800
25*23=575
52*23=1196
52*32=1664

NOW VERY AMAZING,SUBTRACT BIGGER ONE TO ANY LOWER,ONE BY ONE

1664-1196=468=4+6+8=�18=1+8=9
1664-575=1089=1+0+8+�9=18=1+8=9
1164-800=864=8+6+4=1�8=1+8=9
1196-575=621=6+2+1=9��
1196-800=396=3+6+9=1�8=1+8=9
800-575=225=2+2+5=9

Always nine,it is amazing.
for any no of digits.it is always true!

Thanking you
Gaurav Tekriwal
www.vedicmathsinda.org

SQUARE THROUGH SQUARES

2009-11-03T13:33:50Z

MIRROR IMAGE MAN IN JAGRAN NEWS

2009-05-18T08:37:10Z

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JagranCityPlus / In Focus

Piyush Goel: 'Mirror Image Man' with multiple talents

Young Piyash Goel has a rare feat to his credit. He has written the world's first Shrimad Bhagwad Geeta in mirror image. Piyush says, "It is the first Bhagwad granth in the world written in mirror image. I wrote the epic in my own hand writing in two languages,� Hindi and English. One can read all the 18 chapters and 700 verses in front of a mirror."

The feat certainly shows the will power of a man who put everything readable in front of a mirror. He says, "Since my childhood I had a strong desire to copy everything in front of a mirror. Though I was not sure to achieve this uncommon art, yet I did it." He recalled how an accident had changed his life. I met with a serious accident in year 2000 and remained in bed for a long time. At that time I had developed this art, he reveals. A resident of Kaushambi, Piyush is now known as 'Mirror Image Man' and recently he was honoured with Holder Republic Award for this novel achievement.

ABOUT PIYUSH

He is a mechanical engineer working in a private firm of Greater Noida, Dadri. Collecting unusual things is also his passion. He says, "I came in contact with a bank employee in the year 1982. He was a stamp collector. I was very fascinated by this habit and I� started collecting various stamps and currencies of different countries." Later I started collecting match boxes, cigarette packets, pens, coins, currencies and autographs of celebrities, he adds. He has rare collection of autographs of great people like Indira Gandhi, Rajiv Gandhi, Sachin Tendulkar, Amitabh Bachan including several national and international personalities. About this particular habit he says, "I love to meet celebrities� and collect their signatures.� Though it is time consuming, for me it is like getting inspiration.

Presently, he has a rich collection of� various items. "Initially my family members used to get� irritated by my habit� since it is difficult to keep everything in a house. After seeing my craze and social recognition now my kids also� help in preserving� my collections".

BODY OF WORK

Apart from Shrimad Bhagwad Geeta he has written Shree Durga Sapt Satti in Sanskrit language, Sunderkand, Arti Sangrah and Shree Sai Sachcharitra (all 51 chapters, 308 pages, more than one lakh words).

He has� written a book on Mathematics, which is a juggle for most of mathematicians. He informs, "I am very� fond of Mathematics, I have done a lot of work on Mathematics, like Points Design of Pyramid and Equations, work on Pascal Triangle, A new triangle 'AP Right Angled Triangle' in which I have introduced a new strange Table and formula for two digits and Number Nine." It is very interesting way to understand the complications of Mathematics. The book is going to be published in the future," he adds.

FUTURE PLANS

Since his hand-written Bhagwad Geeta is to be adopted by Krishna museum of Kurukshetra University, he is feeling proud of the achievement. He accepts, "It is a fact that no one is going to read this holy book in front of a mirror but I have great satisfaction by writing an image of those great holy words and compiling them into a complete granth. I will continue with this writing and in the future write more holy books".

"People often ask me what would I do with these strange collections. I simply prove my point by organizing several exhibitions in various schools of Ghaziabad and Noida. My works and collections are informative for students and I have received so many invitations from schools and museums. So far as awards are concerned I never do anything� for the sake of any awards or remuneration. Though I have various recognitions and awards I don't like to mention them since I have a noble mission to preserve things for the future generation," he concludes.

�Manoj Sinha

� ��

�copyright Jagran CityPlus

�

Fermat's last theorem

2008-06-03T17:00:26Z

Fermat's last theorem

Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book.

There is a statue of Fermat and his muse in his home town of Toulouse:

(Click it to see a larger version)

Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus's Arithmetica.

Fermat's Last Theorem states that

xⁿ + yⁿ = zⁿ

has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote

I have discovered a truly remarkable proof which this margin is too small to contain.

Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus's Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.

In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with

x² + y² = z² such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat's theorem.

It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only. For if there were integers x, y, z with xⁿ + yⁿ = zⁿ then if n = pq,

(x^q)^p + (y^q)^p = (z^q)^p.

Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of Euler so perhaps it is not too unreasonable to attribute the n = 3 case to Euler.

Euler's mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form

p² + 3q²

and Euler shows that, for any a, b if we put

p = a³ - 9ab², q = 3(a²b - b³) then

p² + 3q² = (a² + 3b²)³.

This is true but he then tries to show that, if p² + 3q² is a cube then an a and b exist such that p and q are as above. His method is imaginative, calculating with numbers of the form a + b√-3. However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate.

The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xⁿ + yⁿ = zⁿ implies that one of x, y, z is divisible by n. Hence Fermat's Last Theorem splits into two cases.

Case 1: None of x, y, z is divisible by n.

Case 2: One and only one of x, y, z is divisible by n.

Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.

Case 2(i) was proved by Dirichlet and presented to the Paris Acad�mie des Sciences in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 was published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).

In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lam� in 1839. It showed why Dirichlet had so much difficulty, for although Dirichlet's n = 14 proof used similar (but computationally much harder) arguments to the earlier cases, Lam� had to introduce some completely new methods. Lam�'s proof is exceedingly hard and makes it look as though progress with Fermat's Last Theorem to larger n would be almost impossible without some radically new thinking.

The year 1847 is of major significance in the study of Fermat's Last Theorem. On 1 March of that year Lam� announced to the Paris Acad�mie that he had proved Fermat's Last Theorem. He sketched a proof which involved factorizing xⁿ + yⁿ = zⁿ into linear factors over the complex numbers. Lam� acknowledged that the idea was suggested to him by Liouville. However Liouville addressed the meeting after Lam� and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true. Cauchy supported Lam� but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Acad�mie an idea which he believed might prove Fermat's Last Theorem.

Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument

It is true for n = 2, n = 3 and n = 4 and one easily sees that the same argument applies for n > 4

was somewhat hopeful.

[Wantzel is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).]

On 24 May Liouville read a letter to the Acad�mie which settled the arguments. The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846. Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes. Kummer also said in his letter that he believed 37 failed his conditions.

By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B₂ , B₄ , ..., B_p-3 . The Bernoulli number B_i is defined by

x/(e^x - 1) = B_i xⁱ /i!

Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B₃₂ .

The only primes less than 100 which are not regular are 37, 59 and 67. More powerful techniques were used to prove Fermat's Last Theorem for these numbers. This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtw�ngler, Vandiver and others. Although it was expected that the number of regular primes would be infinite even this defied proof. In 1915 Jensen proved that the number of irregular primes is infinite.

Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.

In 1983 a major contribution was made by Gerd Faltings who proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xⁿ + yⁿ = zⁿ. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Faltings' arguments.

The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form y² = x³ + ax + b for constants a and b. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbr�cken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.

Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states

The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

In March 1994 Faltings, writing in Scientific American, said

If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.

Weil, also in Scientific American, wrote

I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.

In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.

In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.

On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem

Prime numbers

2008-06-03T16:58:10Z

Prime numbers

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.

The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.

A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.

A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.

You can see more about these numbers in the History topics article Perfect numbers.

By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

Euclid also showed that if the number 2ⁿ - 1 is prime then the number 2^n-1(2ⁿ - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.

In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.

There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.

The next important developments were made by Fermat at the beginning of the 17^th Century. He proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares.

He devised a new method of factorising large numbers which he demonstrated by factorising the number 2027651281 = 44021 46061.

He proved what has come to be known as Fermat's Little Theorem (to distinguish it from his so-called Last Theorem).

This states that if p is a prime then for any integer a we have a^p = a modulo p.

This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2ⁿ - 2 is divisible by n. The other half of this is false, since, for example, 2³⁴¹ - 2 is divisible by 341 even though 341 = 31 11 is composite. Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.

Fermat corresponded with other mathematicians of his day and in particular with the monk Marin Mersenne. In one of his letters to Mersenne he conjectured that the numbers 2ⁿ + 1 were always prime if n is a power of 2. He had verified this for n = 1, 2, 4, 8 and 16 and he knew that if n were not a power of 2, the result failed. Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the next case 2³² + 1 = 4294967297 is divisible by 641 and so is not prime.

Number of the form 2ⁿ - 1 also attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbers M_n because Mersenne studied them.

Not all numbers of the form 2ⁿ - 1 with n prime are prime. For example 2¹¹ - 1 = 2047 = 23 89 is composite, though this was first noted as late as 1536.

For many years numbers of this form provided the largest known primes. The number M₁₉ was proved to be prime by Cataldi in 1588 and this was the largest known prime for about 200 years until Euler proved that M₃₁ is prime. This established the record for another century and when Lucas showed that M₁₂₇ (which is a 39 digit number) is prime that took the record as far as the age of the electronic computer.

In 1952 the Mersenne numbers M₅₂₁, M₆₀₇, M₁₂₇₉, M₂₂₀₃ and M₂₂₈₁ were proved to be prime by Robinson using an early computer and the electronic age had begun.

By 2005 a total of 42 Mersenne primes have been found. The largest is M_25964951 which has 7816230 decimal digits.

Euler's work had a great impact on number theory in general and on primes in particular.

He extended Fermat's Little Theorem and introduced the Euler φ-function. As mentioned above he factorised the 5^th Fermat Number 2³² + 1, he found 60 pairs of the amicable numbers referred to above, and he stated (but was unable to prove) what became known as the Law of Quadratic Reciprocity.

He was the first to realise that number theory could be studied using the tools of analysis and in so-doing founded the subject of Analytic Number Theory. He was able to show that not only is the so-called Harmonic series ∑ (1/n) divergent, but the series

¹/₂ + ¹/₃ + ¹/₅ + ¹/₇ + ¹/₁₁ + ...

formed by summing the reciprocals of the prime numbers, is also divergent. The sum to n terms of the Harmonic series grows roughly like log(n), while the latter series diverges even more slowly like log[ log(n) ]. This means, for example, that summing the reciprocals of all the primes that have been listed, even by the most powerful computers, only gives a sum of about 4, but the series still diverges to ∞.

At first sight the primes seem to be distributed among the integers in rather a haphazard way. For example in the 100 numbers immediately before 10 000 000 there are 9 primes, while in the 100 numbers after there are only 2 primes. However, on a large scale, the way in which the primes are distributed is very regular. Legendre and Gauss both did extensive calculations of the density of primes. Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers). By the end of his life it is estimated that he had counted all the primes up to about 3 million. Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n). Legendre gave an estimate for π(n) the number of primes ≤ n of

π(n) = n/(log(n) - 1.08366)

while Gauss's estimate is in terms of the logarithmic integral

π(n) = ∫ (1/log(t) dt where the range of integration is 2 to n.

You can see the Legendre estimate and the Gauss estimate and can compare them.

The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem. Attempts to prove it continued throughout the 19^th Century with notable progress being made by Chebyshev and Riemann who was able to relate the problem to something called the Riemann Hypothesis: a still unproved result about the zeros in the Complex plane of something called the Riemann zeta-function. The result was eventually proved (using powerful methods in Complex analysis) by Hadamard and de la Vall�e Poussin in 1896.

There are still many open questions (some of them dating back hundreds of years) relating to prime numbers.

Some unsolved problems

The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.

Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.

Are there infinitely many primes of the form n² + 1 ?

(Dirichlet proved that every arithmetic progression : {a + bn | n N} with a, b coprime contains infinitely many primes.)

Is there always a prime between n² and (n + 1)² ?

(The fact that there is always a prime between n and 2n was called Bertrand's conjecture and was proved by Chebyshev.)

Are there infinitely many prime Fermat numbers? Indeed, are there any prime Fermat numbers after the fourth one?

Is there an arithmetic progression of consecutive primes for any given (finite) length? e.g. 251, 257, 263, 269 has length 4. The largest example known has length 10.

Are there infinitely many sets of 3 consecutive primes in arithmetic progression. (True if we omit the word consecutive.)

n² - n + 41 is prime for 0 ≤ n ≤ 40. Are there infinitely many primes of this form? The same question applies to n² - 79 n + 1601 which is prime for 0 ≤ n ≤ 79.

Are there infinitely many primes of the form n# + 1? (where n# is the product of all primes ≤ n.)

Are there infinitely many primes of the form n# - 1?

Are there infinitely many primes of the form n! + 1?

Are there infinitely many primes of the form n! - 1?

If p is a prime, is 2^p - 1 always square free? i.e. not divisible by the square of a prime.

Does the Fibonacci sequence contain an infinite number of primes?

Here are the latest prime records that we know.

The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in February 2005) is the 42^nd Mersenne prime: M_25964951 which has 7816230 decimal digits The largest known twin primes are 242206083 2³⁸⁸⁸⁰ 1. They have 11713 digits and were announced by Indlekofer and Ja'rai in November, 1995.

The largest known factorial prime (prime of the form n! 1) is 3610! - 1. It is a number of 11277 digits and was announced by Caldwell in 1993.

The largest known primorial prime (prime of the form n# 1 where n# is the product of all primes ≤ n) is 24029# + 1. It is a number of 10387 digits and was announced by Caldwell in 1993.

The development of group theory

2008-06-03T16:53:34Z

The development of group theory

The study of the development of a concept such as that of a group has certain difficulties. It would be wrong to say that since the non-zero rationals form a group under multiplication then the origin of the group concept must go back to the beginnings of mathematics. Rather we must take the view that group theory is the abstraction of ideas that were common to a number of major areas which were being studied essentially simultaneously.

The three main areas that were to give rise to group theory are:-

geometry at the beginning of the 19^th Century,

number theory at the end of the 18^th Century,

the theory of algebraic equations at the end of the 18^th Century leading to the study of permutations.

(1) Geometry has been studied for a very long time so it is reasonable to ask what happened to geometry at the beginning of the 19^th Century that was to contribute to the rise of the group concept. Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied. Also the movement to study geometry in n dimensions led to an abstraction in geometry itself. The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet. Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and J�nos Bolyai among others.

M�bius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.

(2) In 1761 Euler studied modular arithmetic. In particular he examined the remainders of powers of a number modulo n. Although Euler's work is, of course, not stated in group theoretic terms he does provide an example of the decomposition of an abelian group into cosets of a subgroup. He also proves a special case of the order of a subgroup being a divisor of the order of the group.

Gauss in 1801 was to take Euler's work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups. He examines orders of elements and proves (although not in this notation) that there is a subgroup for every number dividing the order of a cyclic group. Gauss also examined other abelian groups. He looked at binary quadratic forms

ax² + 2bxy + cy² where a, b, c are integers.

Gauss examined the behaviour of forms under transformations and substitutions. He partitions forms into classes and then defines a composition on the classes. Gauss proves that the order of composition of three forms is immaterial so, in modern language, the associative law holds. In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss's works, found a basis for this abelian group.

(3) Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations. Lagrange's main object was to find out why cubic and quartic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x', x'' and x'''. Then, taking 1, w, w² as the cube roots of unity, he examines the expression

R = x' + wx'' + w²x'''

and notes that it takes just two different values under the six permutations of the roots x', x'', x'''. Although the beginnings of permutation group theory can be seen in this work, Lagrange never composes his permutations so in some sense never discusses groups at all.

The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini's work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls permutazione and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely permutazione semplice which are cyclic groups in modern notation, and permutazione composta which are non-cyclic groups. The permutazione composta Ruffini divides into three types which in today's notation are intransitive groups, transitive imprimitive groups and transitive primitive groups.

Ruffini's proof of the insolubility of the quintic has some gaps and, disappointed with the lack of reaction to his paper Ruffini published further proofs. In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.

Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations. However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right. He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term syst�me des substitutions conjugu�es for a group. Cauchy calls two permutations similar if they have the same cycle structure and proves that this is the same as the permutations being conjugate.

Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.

Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a proper decomposition if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

Galois' work was not known until Liouville published Galois' papers in 1846. Liouville saw clearly the connection between Cauchy's theory of permutations and Galois' work. However Liouville failed to grasp that the importance of Galois' work lay in the group concept.

Betti began in 1851 publishing work relating permutation theory and the theory of equations. In fact Betti was the first to prove that Galois' group associated with an equation was in fact a group of permutations in the modern sense. Serret published an important work discussing Galois' work, still without seeing the significance of the group concept.

Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations. He defines isomorphism of permutation groups and proves the Jordan-H�lder theorem for permutation groups. H�lder was to prove it in the context of abstract groups in 1889.

Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry. Groups were certainly becoming centre stage in mathematics.

Perhaps the most remarkable development had come even before Betti's work. It was due to the English mathematician Cayley. As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

Cayley's papers of 1854 were so far ahead of their time that they had little impact. However when Cayley returned to the topic in 1878 with four papers on groups, one of them called The theory of groups, the time was right for the abstract group concept to move towards the centre of mathematical investigation. Cayley proved, among many other results, that every finite group can be represented as a group of permutations. Cayley's work prompted H�lder, in 1893, to investigate groups of order

p³, pq², pqr and p⁴.

Frobenius and Netto (a student of Kronecker) carried the theory of groups forward. As far as the abstract concept is concerned, the next major contributor was von Dyck. von Dyck, who had obtained his doctorate under Klein's supervision then became Klein's assistant. Von Dyck, with fundamental papers in 1882 and 1883, constructed free groups and the definition of abstract groups in terms of generators and relations.

Group theory really came of age with the book by Burnside Theory of groups of finite order published in 1897. The two volume algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra published in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to bring group theory into perhaps the most major theory of 20^th Century mathematics.

The fundamental theorem of algebra

2008-06-03T16:50:41Z

The fundamental theorem of algebra

The Fundamental Theorem of Algebra (FTA) states

Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.

In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors.

Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x³ = 15x + 4 gave an answer involving √-121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics.

Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.

Vi�te gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en alg�bre . However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C. In fact this was to become the whole problem of the FTA for many years since mathematicians accepted Albert Girard's assertion as self-evident. They believed that a polynomial equation of degree n must have n roots, the problem was, they believed, to show that these roots were of the form a + bi, a, b real.

Now Harriot knew that a polynomial which vanishes at t has a root x - t but this did not become well known until stated by Descartes in 1637 in La g�om�trie, so Albert Girard did not have much of the background to understand the problem properly.

A 'proof' that the FTA was false was given by Leibniz in 1702 when he asserted that x⁴ + t⁴ could never be written as a product of two real quadratic factors. His mistake came in not realising that √i could be written in the form a + bi, a, b real.

Euler, in a 1742 correspondence with Nicolaus(II) Bernoulli and Goldbach, showed that the Leibniz counterexample was false.

D'Alembert in 1746 made the first serious attempt at a proof of the FTA. For a polynomial f he takes a real b, c so that f(b) = c. Now he shows that there are complex numbers z₁ and w₁ so that

|z₁| < |c|, |w₁| < |c|.

He then iterates the process to converge on a zero of f. His proof has several weaknesses. Firstly, he uses a lemma without proof which was proved in 1851 by Puiseau, but whose proof uses the FTA! Secondly, he did not have the necessary knowledge to use a compactness argument to give the final convergence. Despite this, the ideas in this proof are important.

Euler was soon able to prove that every real polynomial of degree n, n ≤ 6 had exactly n complex roots. In 1749 he attempted a proof of the general case, so he tried to proof the FTA for Real Polynomials:

Every polynomial of the nth degree with real coefficients has precisely n zeros in C.

His proof in Recherches sur les racines imaginaires des �quations is based on decomposing a monic polynomial of degree 2ⁿ into the product of two monic polynomials of degree m = 2^n-1. Then since an arbitrary polynomial can be converted to a monic polynomial by multiplying by ax^k for some k the theorem would follow by iterating the decomposition. Now Euler knew a fact which went back to Cardan in Ars Magna, or earlier, that a transformation could be applied to remove the second largest degree term of a polynomial. Hence he assumed that

x^2m + Ax^2m-2 + Bx^2m-3 +. . . = (x^m + tx^m-1 + gx^m-2 + . . .)(x^m - tx^m-1 + hx^m-2 + . . .)

and then multiplied up and compared coefficients. This Euler claimed led to g, h, ... being rational functions of A, B, ..., t. All this was carried out in detail for n = 4, but the general case is only a sketch.

In 1772 Lagrange raised objections to Euler's proof. He objected that Euler's rational functions could lead to 0/0. Lagrange used his knowledge of permutations of roots to fill all the gaps in Euler's proof except that he was still assuming that the polynomial equation of degree n must have n roots of some kind so he could work with them and deduce properties, like eventually that they had the form a + bi, a, b real.

Laplace, in 1795, tried to prove the FTA using a completely different approach using the discriminant of a polynomial. His proof was very elegant and its only 'problem' was that again the existence of roots was assumed.

Gauss is usually credited with the first proof of the FTA. In his doctoral thesis of 1799 he presented his first proof and also his objections to the other proofs. He is undoubtedly the first to spot the fundamental flaw in the earlier proofs, to which we have referred many times above, namely the fact that they were assuming the existence of roots and then trying to deduce properties of them. Of Euler's proof Gauss says

... if one carries out operations with these impossible roots, as though they really existed, and says for example, the sum of all roots of the equation x^m+ax^m-1 + bx^m-2 + . . . = 0 is equal to -a even though some of them may be impossible (which really means: even if some are non-existent and therefore missing), then I can only say that I thoroughly disapprove of this type of argument.

Gauss himself does not claim to give the first proper proof. He merely calls his proof new but says, for example of d'Alembert's proof, that despite his objections

a rigorous proof could be constructed on the same basis.

Gauss's proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our present day standards required for a rigorous proof.

In 1814 the Swiss accountant Jean Robert Argand published a proof of the FTA which may be the simplest of all the proofs. His proof is based on d'Alembert's 1746 idea. Argand had already sketched the idea in a paper published two years earlier Essai sur une mani�re de repr�senter les quantiti�s imaginaires dans les constructions g�ometriques. In this paper he interpreted i as a rotation of the plane through 90� so giving rise to the Argand plane or Argand diagram as a geometrical representation of complex numbers. Now in the later paper R�flexions sur la nouvelle th�orie d'analyse Argand simplifies d'Alembert's idea using a general theorem on the existence of a minimum of a continuous function.

In 1820 Cauchy was to devote a whole chapter of Cours d'analyse to Argand's proof (although it will come as no surprise to anyone who has studied Cauchy's work to learn that he fails to mention Argand !) This proof only fails to be rigorous because the general concept of a lower bound had not been developed at that time. The Argand proof was to attain fame when it was given by Chrystal in his Algebra textbook in 1886. Chrystal's book was very influential.

Two years after Argand's proof appeared Gauss published in 1816 a second proof of the FTA. Gauss uses Euler's approach but instead of operating with roots which may not exist, Gauss operates with indeterminates. This proof is complete and correct.

A third proof by Gauss also in 1816 is, like the first, topological in nature. Gauss introduced in 1831 the term 'complex number'. The term 'conjugate' had been introduced by Cauchy in 1821.

Gauss's criticisms of the Lagrange-Laplace proofs did not seem to find immediate favour in France. Lagrange's 1808 2^nd Edition of his treatise on equations makes no mention of Gauss's new proof or criticisms. Even the 1828 Edition, edited by Poinsot, still expresses complete satisfaction with the Lagrange-Laplace proofs and no mention of the Gauss criticisms.

In 1849 (on the 50th anniversary of his first proof!) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots. The proof is similar to the first proof given by Gauss. However it is adds little since it is straightforward to deduce the result for complex coefficients from the result about polynomials with real coefficients.

It is worth noting that despite Gauss's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a whole hierarchy of imaginary quantities of which complex numbers were the simplest. Gauss called them a shadow of shadows.

It was in searching for such generalisations of the complex numbers that Hamilton discovered the quaternions around 1843, but of course the quaternions are not a commutative system. The first proof that the only commutative algebraic field containing R was given by Weierstrass in his lectures of 1863. It was published in Hankel's book Theorie der complexen Zahlensysteme.

Of course the proofs described above all become valid once one has the modern result that there is a splitting field for every polynomial. Frobenius, at the celebrations in Basle for the bicentenary of Euler's birth said:-

Euler gave the most algebraic of the proofs of the existence of the roots of an equation, the one which is based on the proposition that every real equation of odd degree has a real root. I regard it as unjust to ascribe this proof exclusively to Gauss, who merely added the finishing touches.

The Argand proof is only an existence proof and it does not in any way allow the roots to be constructed. Weierstrass noted in 1859 made a start towards a constructive proof but it was not until 1940 that a constructive variant of the Argand proof was given by Hellmuth Kneser. This proof was further simplified in 1981 by Martin Kneser, Hellmuth Kneser's son.

References (8 books/articles)

algebra

2008-06-03T16:47:43Z

Quadratic, cubic and quartic equations

It is often claimed that the Babylonians (about 400 BC) were the first to solve quadratic equations. This is an over simplification, for the Babylonians had no notion of 'equation'. What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation. The method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.

In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities.

Hindu mathematicians took the Babylonian methods further so that Brahmagupta (598-665 AD) gives an, almost modern, method which admits negative quantities. He also used abbreviations for the unknown, usually the initial letter of a colour was used, and sometimes several different unknowns occur in a single problem.

The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. However al-Khwarizmi (c 800) gave a classification of different types of quadratics (although only numerical examples of each). The different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e. x, x² and numbers.

Squares equal to roots.

Squares equal to numbers.

Roots equal to numbers.

Squares and roots equal to numbers, e.g. x² + 10x = 39.

Squares and numbers equal to roots, e.g. x² + 21 = 10x.

Roots and numbers equal to squares, e.g. 3x + 4 = x².

Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.

Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.

A new phase of mathematics began in Italy around 1500. In 1494 the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita, now known as the Suma, appeared. It was written by Luca Pacioli although it is quite hard to find the author's name on the book, Fra Luca appearing in small print but not on the title page. In many ways the book is more a summary of knowledge at the time and makes no major advances. The notation and setting out of calculations is almost modern in style:


	                    6.p.R.10
	18.m.R.90
	____________________________
	108.m.R.3240.p.R.3240.m.R.90

hoc est 78.

In our notation

(6 + √10)

(18 - √90) =

(108-√3240 + √3240 - √900)

which is 78.

The last term in the answer 90 is an early misprint and should be 900 but the margin was too narrow so the printer missed out the final 0!

Pacioli does not discuss cubic equations but does discuss quartics. He says that, in our notation, x⁴ = a + bx² can be solved by quadratic methods but x⁴ + ax² = b and x⁴ + a = bx² are impossible at the present state of science.

Scipione dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in 1501-2. dal Ferro is credited with solving cubic equations algebraically but the picture is somewhat more complicated. The problem was to find the roots by adding, subtracting, multiplying, dividing and taking roots of expressions in the coefficients. We believe that dal Ferro could only solve cubic equation of the form x³ + mx = n. In fact this is all that is required.

For, given the general cubic y³ - by² + cy - d = 0, put y = x + b/3 to get

x³ + mx = n where m = c - b²/3, n = d - bc/3 + 2b³/27.

However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations. Remarkably, dal Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.

Fior was a mediocre mathematician and far less good at keeping secrets than dal Ferro. Soon rumours started to circulate in Bologna that the cubic equation had been solved. Nicolo of Brescia, known as Tartaglia meaning 'the stammerer', prompted by the rumours managed to solve equations of the form x³ + mx² = n and made no secret of his discovery.

Fior challenged Tartaglia to a public contest: the rules being that each gave the other 30 problems with 40 or 50 days in which to solve them, the winner being the one to solve most but a small prize was also offered for each problem. Tartaglia solved all Fior's problems in the space of 2 hours, for all the problems Fior had set were of the form x³ + mx = n as he believed Tartaglia would be unable to solve this type. However only 8 days before the problems were to be collected, Tartaglia had found the general method for all types of cubics.

News of Tartaglia's victory reached Girolamo Cardan in Milan where he was preparing to publish Practica Arithmeticae (1539). Cardan invited Tartaglia to visit him and, after much persuasion, made him divulge the secret of his solution of the cubic equation. This Tartaglia did, having made Cardan promise to keep it secret until Tartaglia had published it himself. Cardan did not keep his promise. In 1545 he published Ars Magna the first Latin treatise on algebra.

Here, in modern notation, is Cardan's solution of x³ + mx = n.

Notice that (a - b)³ + 3ab(a - b) = a³ - b³

so if a and b satisfy 3ab = m and a³ - b³ = n then a - b is a solution of x³ + mx = n.

But now b = m/3a so a³ - m³/27a³ = n,

i.e. a⁶ - na³ - m³/27 = 0.

This is a quadratic equation in a³, so solve for a³ using the usual formula for a quadratic.

Now a is found by taking cube roots and b can be found in a similar way (or using b=m/3a).

Then x = a - b is the solution to the cubic.

Cardan noticed something strange when he applied his formula to certain cubics. When solving x³ = 15x + 4 he obtained an expression involving √-121. Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation. He wrote to Tartaglia on 4 August 1539 in an attempt to clear up the difficulty. Tartaglia certainly did not understand. In Ars Magna Cardan gives a calculation with 'complex numbers' to solve a similar problem but he really did not understand his own calculation which he says is as subtle as it is useless.

After Tartaglia had shown Cardan how to solve cubics, Cardan encouraged his own student, Lodovico Ferrari, to examine quartic equations. Ferrari managed to solve the quartic with perhaps the most elegant of all the methods that were found to solve this type of problem. Cardan published all 20 cases of quartic equations in Ars Magna. Here, again in modern notation, is Ferrari's solution of the case: x⁴ + px² + qx + r = 0. First complete the square to obtain

x⁴ + 2px² + p² = px² - qx - r + p²

i.e.

(x² + p)² = px² - qx - r + p²

Now the clever bit. For any y we have

(x² + p + y)² = px² - qx - r + p² + 2y(x² + p) + y²

= (p + 2y)x² - qx + (p² - r + 2py + y²) (*)

Now the right hand side is a quadratic in x and we can choose y so that it is a perfect square. This is done by making the discriminant zero, in this case

(-q)² -4(p + 2y)(p² - r + 2py + y²) = 0.

Rewrite this last equation as

(q² - 4p³ + 4 pr) + (-16p² + 8r)y - 20 py² - 8y³ = 0

to see that it is a cubic in y.

Now we know how to solve cubics, so solve for y. With this value of y the right hand side of (*) is a perfect square so, taking the square root of both sides, we obtain a quadratic in x. Solve this quadratic and we have the required solution to the quartic equation.

The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra.

In the years after Cardan's Ars Magna many mathematicians contributed to the solution of cubic and quartic equations. Vi�te, Harriot, Tschirnhaus, Euler, Bezout and Descartes all devised methods. Tschirnhaus's methods were extended by the Swedish mathematician E S Bring near the end of the 18^th Century.

Thomas Harriot made several contributions. One of the most elementary to us, yet showing a marked improvement in understanding, was the observation that if x = b, x = c, x = d are solutions of a cubic then the cubic is

(x - b)(x - c)(x - d) = 0.

Harriot also had a nice method for solving cubics. Consider the cubic

x³ + 3b²x = 2c³

Put x = (e² - b²)/e. Then

e⁶ - 2c³e³ = b⁶

which is a quadratic in e³, and so can be solved for e³ to get

e³ = c³ +√(b⁶ + c⁶).

However

e³(e³ - 2c³) = b⁶ so that b⁶/e³ = -c³ +√(b⁶ + c⁶).

Now x = e - b²/e and both e and b²/e are cube roots of expressions given above.

Leibniz wrote a letter to Huygens in March 1673. In it he made many contributions to the understanding of cubic equations. Perhaps the most striking is a direct verification of the Cardan-Tartaglia formula. This Leibniz did by reconstructing the cubic from its three roots (as given by the formula) as Harriot claimed in general. Nobody before Leibniz seems to have thought of this direct method of verification. It was the first true algebraic proof of the formula, all previous proofs being geometrical in nature.

blogadda

2008-06-03T16:18:53Z

words

2008-06-01T11:41:22Z

1.ween (ween) verb tr., intr.

� To think, suppose, believe.

[From Old English wenan (to expect), from the Indo-European root wen-

(to desire or to strive for) that's also the source of wish, win,

venerate, venison, Venus, and banya. It's the same word that shows

up in "overweening".]

2.sweven (SWEV-uhn) noun

� Dream; vision.

[From Old English swefn (sleep, dream, vision).]

3.scrannel (SKRAN-l) adjective

� 1. Thin.

� 2. Unmelodious.

[Of unknown origin.]

4.point-device (point di-VYS) adverb

� Completely; perfectly.

adjective

� Perfect; precise; meticulous.

[From the phrase "at point devis" meaning "at a fixed point" or

"to perfection".]

5.Fashions come and go. One year it's bell-bottoms that are cool, another

time it might be torn jeans. What is hip for one age is pass� for another.

The same goes for words. Yesterday's street slang becomes respectable today,

suitable for office memos and academic theses. Words once in everyday use

may be labeled archaic a few hundred years later.

As I see it, there's no reason to despatch any word to the attic of time.

Each word on our verbal palette -- whether new or old -- helps us bring out

a nuance in conversation and in writing.

The words featured here this week are considered archaic but are still in

good shape. They're old but have not yet retired from the language. They

still faithfully report for duty, as shown by some of the examples from

newspapers.

garboil (GAHR-boil) noun

� Confusion; turmoil.

[Via French and Italian from Latin bullire (to boil).]

how to square,it is totally new

2008-05-29T16:52:15Z

http://www.youtube.com/watch?v=DpKTsLSDvic

it is my new creation

piyushdadriwala

know the truth history of tajmahal

2008-04-26T07:46:13Z

BBC says about Taj Mahal--- Hidden Truth - Never say it is a Tomb

Aerial view of the Taj Mahal�

BBC says about Taj Mahal--- Hidden Truth - Never say it is a Tomb

Aerial view of the Taj Mahal

The interior water well

Frontal view of the Taj Mahal and dome

Close up of the dome with pinnacle

Close up of the pinnacle

Inlaid pinnacle pattern in courtyard

Red lotus at apex of the entrance

Rear view of the Taj & 22 apartments

View of sealed doors & windows in back

Typical Vedic style corridors

The Music House--a contradiction

A locked room on upper floor

A marble apartment on ground floor

The OM in the flowers on the walls

Staircase that leads to the lower levels

300 foot long corridor inside apartments

One of the 22 rooms in the secret lower level

Interior of one of the 22 secret rooms

Interior of another of the locked rooms

Vedic design on ceiling of a locked room

Huge ventilator sealed shut with bricks

Secret walled door that leads to other rooms

Secret bricked door that hides more evidence

Palace in Barhanpur where Mumtaz died

Pavilion where Mumtaz is said to be buried

NOW READ THIS.......

No one has ever challenged it except Prof. P. N. Oak, who believes the

whole world has been duped. In his book Taj Mahal: The True Story, Oak says

the

Taj Mahal is not Queen Mumtaz's tomb but an ancient Hindu temple palace of

Lord Shiva (then known as Tejo Mahalaya ) .. In the course of his research O

ak discovered that the Shiva temple palace was usurped by Shah Jahan from

then Maharaja of Jaipur, Jai Singh. In his own court chronicle,

Badshahnama,

Shah Jahan admits that an exceptionally beautiful grand mansion in Agra

was taken from Jai SIngh for Mumtaz's burial . The ex-Maharaja of Jaipur

still

retains in his secret collection two orders from Shah Jahan for

surrendering the Taj building. Using captured temples and mansions, as a

burial place for

dead courtiers and royalty was a common practice among Muslim rulers.

For example, Humayun,Akbar, Etmud-ud-Daula and Safdarjung are all buried

in such mansions. Oak's inquiries began with the name of Taj Mahal. He says

the term " Mahal " has never been used for a building in any Muslim countries

from Afghanisthan to Algeria. "The unusual explanation that the term Taj

Mahal derives from Mumtaz Mahal was illogical in atleast two respects.

Firstly, her name was never Mumtaz Mahal but Mumtaz-ul-Zamani ," he writes.

Secondly, one cannot omit the first three letters 'Mum' from a woman's

name to derive the remainder as the name for the building."Taj Mahal, he

claims, is a corrupt version of Tejo Mahalaya, or Lord Shiva's Palace . Oak

also says the love story of Mumtaz and Shah Jahan is a fairy tale created

by

court sycophants, blundering historians and sloppy archaeologists . Not a

single royal chronicle of Shah Jahan's time corroborates the love story.

Furthermore, Oak cites several documents suggesting the Taj Mahal predates

Shah Jahan's era, and was a temple dedicated to Shiva, worshipped by

Rajputs of Agra city. For example, Prof. Marvin Miller of New York took a

few

samples from the riverside doorway of the Taj. Carbon dating tests revealed

that the door was 300 years older than Shah Jahan. European traveler Johan

Albert Mandelslo,who visited Agra in 1638 (only seven years after Mumtaz's

death), describes the life of the cit y in his memoirs. But he makes no

reference to the Taj Mahal being built. The writings of Peter Mundy, an

English visitor to Agra within a year of Mumtaz's death, also suggest the

Taj was a noteworthy building well before Shah Jahan's time.

Prof. Oak points out a number of design and architectural inconsistencies

that support the belief of the Taj Mahal being a typical Hindu temple

rather

than a mausoleum. Many rooms in the Taj ! Mahal have remained sealed

since Shah Jahan's time and are still inaccessible to the public . Oak

asserts they contain a headless statue of Lord Shiva and other objects

commonly used for worship rituals in Hindu temples .. Fearing political

backlash, Indira Gandhi's government tried to have Prof. Oak's book

withdrawn from the bookstores, and threatened the Indian publisher of the

first edition dire consequences . There is only one way to discredit or

validate Oak's research.

The current government should open the sealed rooms of the Taj Mahal under

U.N. supervision, and let international experts investigate.

Do circulate this to all you know and let them know about this reality..... � � �


=
The interior water well 
 
Frontal view of the Taj Mahal and dome 
 
Close up of the dome with pinnacle 
 
Close up of the pinnacle 
 
Inlaid pinnacle pattern in courtyard 
 
Red lotus at apex of the entrance 
 
Rear view of the Taj & 22 apartments 
 
View of sealed doors & windows in back 
 
Typical Vedic style corridors 
 
The Music House--a contradiction 
 
A locked room on upper floor 
 
A marble apartment on ground floor 
 
The OM in the flowers on the walls 
 
Staircase that leads to the lower levels 
 
300 foot long corridor inside apartments 
 
One of the 22 rooms in the secret lower level 
 
Interior of one of the 22 secret rooms 

Interior of another of the locked rooms 
 
Vedic design on ceiling of a locked room 
 
Huge ventilator sealed shut with bricks 
 
Secret walled door that leads to other rooms 
 
Secret bricked door that hides more evidence 

Palace in Barhanpur where Mumtaz died 

Pavilion where Mumtaz is said to be buried 
 
NOW READ THIS....... 
No one has ever challenged it except Prof. P. N. Oak, who believes the 
whole world has been duped. In his book Taj Mahal: The True Story, Oak says 
the
Taj Mahal is not Queen Mumtaz's tomb but an ancient Hindu temple palace of 
Lord Shiva (then known as Tejo Mahalaya ) .. In the course of his research O 
ak discovered that the Shiva temple palace was usurped by Shah Jahan from 
then Maharaja of Jaipur, Jai Singh. In his own court chronicle, 
Badshahnama,
Shah Jahan admits that an exceptionally beautiful grand mansion in Agra 
was taken from Jai SIngh for Mumtaz's burial . The ex-Maharaja of Jaipur 
still
retains in his secret collection two orders from Shah Jahan for 
surrendering the Taj building. Using captured temples and mansions, as a 
burial place for
dead courtiers and royalty was a common practice among Muslim rulers. 
For example, Humayun,Akbar, Etmud-ud-Daula and Safdarjung are all buried 
in such mansions. Oak's inquiries began with the name of Taj Mahal. He says 
the term " Mahal " has never been used for a building in any Muslim countries 
from Afghanisthan to Algeria. "The unusual explanation that the term Taj 
Mahal derives from Mumtaz Mahal was illogical in atleast two respects. 
Firstly, her name was never Mumtaz Mahal but Mumtaz-ul-Zamani ," he writes. 
Secondly, one cannot omit the first three letters 'Mum' from a woman's 
name to derive the remainder as the name for the building."Taj Mahal, he 
claims, is a corrupt version of Tejo Mahalaya, or Lord Shiva's Palace . Oak 
also says the love story of Mumtaz and Shah Jahan is a fairy tale created 
by
court sycophants, blundering historians and sloppy archaeologists . Not a 
single royal chronicle of Shah Jahan's time corroborates the love story. 
Furthermore, Oak cites several documents suggesting the Taj Mahal predates 
Shah Jahan's era, and was a temple dedicated to Shiva, worshipped by 
Rajputs of Agra city. For example, Prof. Marvin Miller of New York took a 
few
samples from the riverside doorway of the Taj. Carbon dating tests revealed 
that the door was 300 years older than Shah Jahan. European traveler Johan 
Albert Mandelslo,who visited Agra in 1638 (only seven years after Mumtaz's 
death), describes the life of the cit y in his memoirs. But he makes no 
reference to the Taj Mahal being built. The writings of Peter Mundy, an 
English visitor to Agra within a year of Mumtaz's death, also suggest the 
Taj was a noteworthy building well before Shah Jahan's time. 
Prof. Oak points out a number of design and architectural inconsistencies 
that support the belief of the Taj Mahal being a typical Hindu temple
rather
than a mausoleum. Many rooms in the Taj ! Mahal have remained sealed 
since Shah Jahan's time and are still inaccessible to the public . Oak 
asserts they contain a headless statue of Lord Shiva and other objects 
commonly used for worship rituals in Hindu temples .. Fearing political 
backlash, Indira Gandhi's government tried to have Prof. Oak's book 
withdrawn from the bookstores, and threatened the Indian publisher of the 
first edition dire consequences . There is only one way to discredit or
validate Oak's research. 
The current government should open the sealed rooms of the Taj Mahal under 
U.N. supervision, and let international experts investigate. 
Do circulate this to all you know and let them know about this reality..... � � �

palm reading

2008-04-26T07:40:34Z

�

Heart Line

Placement: Upper Palm

The heart line runs horizontally across the upper part of your palm.

�

Heart Line

Basic Heart Line Meanings:

�� Long Line: Idealistic, Dependent on partner

�� Short Line: Self-centered

�� Deep Line: Stressful

�� Faint Line: Sensitive Nature, Weak Heart

�� Straight Line: Intense Feelings

�� Curved Line: Intellectual Bent

�� Broken Line: Troubled relationships

�� Chained Line: Intertwined relationships, Karmic relationships

�� Forked Line: Heartbreak, Divorce

�� Absent Line: Ruthlessness, Logic rules the heart

Head Line

Placement: Middle of the Palm

The head line represents intellect and reasoning.

Head Line

Basic Head Line Meanings:

�� Long Line: Ambitious

�� Short Line: Intelligent, Intuitive

�� Deep Line: Excellent Memory

�� Faint Line: Poor Memory

�� Straight Line: Materialistic

�� Broken Line: Disappointment

�� Chained Line: Mental Confusion

�� Forked Line: Career Change

�� Double Line: Talented, Inspired by a Muse

�� Absent Line: Laziness, Mental Imbalance

Life Line

Placement: Mid to Lower Palm

The life line begins somewhere between your thumb and index finger and runs downward toward wrist. Life line is generally curved.

Basic Life Line Meanings:

�� Long Line: Good Health, Vitality

�� Short Line: It is a myth that a short life line means a short life. If the life line is short, look closer to other signs (broken, deep, faint, etc.)

�� Deep Line: Smooth Life

�� Faint Line: Low energy

�� Broken Line: Struggles, Losses

�� Chained Line: Multiple Walks (meaning that your life path is multifold)

�� Forked Line: Various meanings depending on fork placement on the hand. Generally forks indicate diversion or life change. Although they can also mean scattered or split energies.

� �Double Line: Partner with Soul Mate, or there is someone near (friend or family member) that serves as a guardian or caretaker.

�� Absent Line: Anxious, Nervous

Fate Line

Also called "Destiny"

Placement: Center of Palm, vertical or slanted line divides the palm in half

Fate Line

Basic Meaning of Fate Line

�� Absent Line: Preplanned Life

�� Deep Line: Inheritance

�� Faint Line: Failures, Disappointments

�� Forked Line: Conflict or Dual Destiny

�� Jagged Line: Struggle, Indecisiveness

�� Broken Line: Trauma, Difficult Circumstance

�� Chained Line: Highs and Lows

Fame Line

Success, Wealth, Talent

Placement: Parallels Fate Line

Fame Line

Fame line gives light to the a person's fate or destiny, indicating brilliance or artistic ability enhances life purpose. Note: This line is not always present.

Love Lines

Also called "Marriage Lines"

Love lines are short horizontal lines found on the side of the hand underneath the pinky.

�

Love Lines

Love lines indicate the number of significant relationships there are in a lifetime. Sometimes it is easier to see these lines if you bend your pinky slightly toward your palm to see the line creases.

Children Lines

Placement: Vertical lines between pinky fingers

Children lines commonly root out of marriage lines (Love Lines) indicating births that are a result of corresponding relationships.

Children Lines

Intuition Line

Placement: Parallel to Life Line (either side)

Intuition lines generally shadow the life line because intuition indicates keen insight into one's life.

Intuition Line

Basic Intuition Line Meaning:

The more prominent this line appears (deeper, longer) the stronger the indication that psychic ability is a dominant characteristic for the person. Intuition lines are not the easiest to detect, and may be absent entirely.

Health Line

Placement: Vertical line begins below ring finger

An absent health line usually indicates that health is not an issue. Degree of sickness is indicated by the strength or weakness of this line.

Health Line

Bracelets

Also called "Rascettes"

Placement: Bracelets are the lines at the bend of your inner wrist.

It is most common to have two or three bracelets. Although, some people have only one bracelet, and having four or more is possible. More bracelets indicate a longer life, broken bracelets indicate ill health or lowering of chi energies (It's the basic circulating energy of life).

Travel Lines

Placement: Mid to Lower Palm Underneath Pinky Finger

Travel lines indicate travel, but can also merely indicate a desire to travel.

Travel Lines

Girdle of Venus

Placement: Semi-circle between index and pinky fingers

The shape of the Girdle of Venus is similar to a crescent moon hanging over the heart line. This palm line configuration intensifies the emotions.

�

Girdle of Venus appears on the hands of individuals who tend to be ultra-sensitive. Symbolically it can indicate a need for shielding or creating emotional boundaries.

whoo....1

2008-04-19T14:35:25Z

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WHOOO....

2008-04-19T14:33:08Z