The 15 Most Famous Transcendental Numbers

In 1844, math genius Joseph Liouville (1809-1882) was the first to prove the existence of transcendental numbers. 

Hermite proved that the number e was transcendental in 1873.Lindeman proved that pi was transcendental in 1882. In 1882,  German mathematician F.

Lindemann proved that pi is transcendental. But are there other famous transcendental numbers. I made a list of the fifteen most famous transcendental numbers.

1.     pi = 3.1415 ...

2.     e = 2.718 ...

3.     Euler's constant, gamma = 0.577215 ... = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.)

4.     Catalan's constant, G = sum (-1)^k / (2k + 1 )^2 = 1 - 1/9 + 1/25 - 1/49 + ... (Not proven to be transcendental, but generally believed to be by mathematicians.)

5.     Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.

6.     Chaitin's "constant", the probability that a random algorithm halts. (Noam Elkies of Harvard notes that not only is this number transcendental but it is also incomputable.)

7.     Chapernowne's number, 0.12345678910111213141516171819202122232425... This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?)

8.     Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.)

9.     ln(2).

10. Hilbert's number, 2^{(sqrt 2 )}. (This is called Hilbert's number because the proof of whether or not it is transcendental was one of Hilbert's famous problems. In fact, according to the Gelfond-Schneider theorem, any number of the form a^b is transcendental where a and b are algebraic (a ne 0, a ne 1 ) and b is not a rational number. Many trigonometric or hyperbolic functions of non-zero algebraic numbers are transcendental.)

11. e^pi

12. pi^e (Not proven to be transcendental, but generally believed to be by mathematicians.)

13. Morse-Thue's number, 0.01101001 ...

14. iⁱ = 0.207879576... (Here i is the imaginary number sqrt(-1). Isn't this a real beauty? How many people have actually considered rasing i to the i power? If a is algebraic and b is algebraic but irrational then a^b is transcendental. Since i is algebraic but irrational, the theorem applies. Note also: iⁱis equal to e^{(- pi / 2 )} and several other values. Consider iⁱ = e^{(i log i )} = e^{( i times i pi / 2 )} . Since log is multivalued, there are other possible values for iⁱ. 
Here is how you can compute the value of iⁱ = 0.207879576...

1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2. 
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i. 
3. Therefore e^(iPi/2) = i. 
4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2). 
5. Therefore i^i = e^(-Pi/2) = .207879576...

15. Feigenbaum numbers, e.g. 4.669 ... . (These are related to properties of dynamical systems with period-doubling. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669 ... , and it has been discovered in many physical systems before they enter the chaotic regime. It has not been proven to be transcendental, but is generally believed to be.)

Math Software

Microsoft Mathematics is a free math software for your computer. With the help of this freeware math students can solve complex math problems easily. It basically helps math students to solve problems in algebra. Apart from that you can also draw 3D and 2D images with the help of this utility. The main objective of this freeware is to teach students the basic of math, physics and chemistry.

Math Editor is a free math software. With the help of this freeware you can create mathematical equations with various symbols such as Cyrillic, alpha, beta, Greek symbols, square root, integrate easily and quickly. It supports TIFF, GIF, BMP, JPEG and PNG formats. This freeware is very useful for creating mathematical equations on your PC. 

CompliCalc is a free mathematical software. With the help of this freeware you can solve complex mathematical problems easily and quickly. This freeware also includes various functional and algebraic calculators. CompliCalc lets you perform a wide variety of operations such as calculate square root, factorial, discount and distance. To use this freeware you need to specify the task you want to perform on the main interface of this freeware.

SpeQ Mathematics is a free math software for your computer. With the help of this freeware you can easily learn math and solve complex problems in mathematics. It has inbuilt support for a wide variety of variables, constants and mathematical functions. SpeQ Mathematics also lets you define functions and custom variables. You can also solve problems of trigonometry by using this freeware

Euler Math Toolbox is a free math software for your PC. With the help of this software you can carry out various calculations in mathematics such as subtraction, addition, calculus problems, algebra, matrices, functions and complex equations. This freeware is very useful for math students as they can solve nearly all types of mathematics problems by using this freeware.

Xfunc is a free math software for your computer. With the help of this freeware you can solve math problems easily. By using this software you can write various types of equations in mathematics solve them and also see their solutions. You can utilize various functions to make different mathematical expressions with the help of Xfunc. 

QMentat is a free software for practice and learning mental mathematics. Mental mathematics means performing mathematical calculations in your mind. There is no need to use calculators, pencil and pen in mental mathematics. With the help of this freeware you can quickly develop skills for mental mathematics. This freeware can perform various types of math operations such as roots, powers, divisions, multiplication, subtraction and addition.

TalkingMath is a free math software for your children. With the help of this freeware your kids can learn mathematics easily and quickly. This freeware also supports audio mode. TalkingMath works in 3 different modes such as timer, standard and interactive mode. TalkingMath lets you perform various types of mathematical calculations such as division, multiplication, subtraction and addition.

JXCirrus Maths is a free math software to learn mathematics. This freeware is specially created for small children to learn math in a exciting and fun way. With the help of this freeware you can easily create easy and basic exercises in math for your children. JXCirrus Maths teachs your children basic math operations such as division, multiplication, subtraction and addition. 

Geogebra is a free math software. With the help of this freeware math teachers can teach mathematics and math students can learn mathematics. This freeware provides a sound platform to math students to learn math and solve mathematical problems of various topics such as linear programming, complex numbers, vectors, probability, discrete mathematics, calculus, statistics, algebra, functions and graphs, geometry etc. 

GeoEnzo is a math software specially build for mathematics teachers. With the help of this freeware mathematics teacher can teach geometry to math students. This freeware lets you easily draw various types of geometrical shapes such as cone, triangle, circle, cube, line and many more. This freeware is very useful for teaching geometry to math students. 

Srinavasa Iyengar Ramanujan, Indian Mathematician          Srinavasa Iyengar Ramanujan was an Indian mathematical genius. Srinavasa Ramanujan Iyengar added a new dimension to the whole world of mathematics with his tremendous contributions

Srinavasa Iyengar Ramanujan, popularly known as S. Ramanujan, was amongst the most renowned and respected mathematicians in India. He is credited with making several numerous significant contributions to various fields of mathematics. This great mathematician was a self-taught learner and had no formal education in pure mathematics. He was born in a small town of Tamil Nadu. He struggled with poverty during his childhood days. However he was talented in mathematics. He had made astonishing contributions to number theory, continued fractions, mathematical analysis and infinite series. Ramanujan is generally considered as one of the soaring geniuses in mathematics. 

Early Life of S. Ramanujan 
Srinavasa Iyengar Ramanujan was born on 22nd of December, 1887 in Erode, Tamil Nadu. Ramanujan failed to complete his formal studies after Matriculation, due to his lack of attendance to the subjects other than mathematics. He failed twice in the Fine Arts degree exam. Once his formal education was over, he had to face a lot of difficulties in earning his livelihood. He actively started searching for jobs, after his marriage with Janaki. However, he had to wait for 5 long years before getting a job, as he did not have any formal educational certificate with him. 

Career of S. Ramanujan 
While pursuing for job, Srinavasa Iyengar Ramanujan travelled extensively to Chennaiand the neighbouring towns and during this course; he met several mathematicians who became his friends. All of them were actually the college teachers who taught mathematics and they all appreciated his mathematical skills. R. Ramchandra Rao, the Collector of Nellore soon granted S. Ramanujan a good scholarship of Rs 25 per month from his personal funds so that Ramanujan could pursue his passion for mathematics without even having a job. Apart from studying mathematics, Ramanujan also started contributing research papers in the newly started Journal of the Indian Mathematical Society. 

Ramchandra Rao also arranged a job for him and Ramanujan soon joined in a clerical post in the Madras Port Trust. He was a Class III-Grade-IV Clerk. 

One of the most important incidences of Ramanujan's life also occurred during those days, when he got the chance to go to England and pursue systematic studies in higher mathematics. His friends and well-wishers in Chennai gave a lot of efforts to send him to England and they sent some of Ramanujan's best works to the eminent British mathematician, the Cayley Lecturer in Mathematics and a Fellow of Trinity College, Cambridge, Mr. G. H. Hardy. Apart from that, Srinavasa Iyengar Ramanujan also got a fellowship in the University of Madras for pursuing research in mathematics, in spite of lacking formal education. 

G. H. Hardy was impressed by his works. He told Ramanujan to go to England, as a student in Trinity College, Cambridge, England. Though initially he was reluctant to go he eventually agreed and started for England on 17th of March, 1914. After reaching London, S. Ramanujan faced a lot of difficulties, due to the completely different lifestyle in England However, he got some good Indian friends there and the eminent Indian statistician, P.C. Mahalnobis, who was then studying at Cambridge, was the most notable amongst them. 

Achievements of S. Ramanujan 
Srinavasa Iyengar Ramanujan spent nearly 5 years in Cambridge and collaborated with the famous mathematicians of the world. A part of his findings was also published in England. He was awarded a B.A. degree in March 1916 for his job on highly composite numbers that was published as a paper in the Journal of the London Mathematical Society. He was elected to the London Mathematical Society on 6th December 1917. He was also elected a Fellow of the Royal Society in 1918 for his investigation in Elliptic Functions and the Theory of Numbers. He was the second Indian to become a Fellow and also one of the youngest Fellows in the entire history of the Royal Society. Ramanujan also became the first Indian to be elected a Fellow of Trinity College, Cambridge for his contribution in developing the Number Theory, on 13th of October, 1918. 

Ramanujan was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a hospital. After suffering from various diseases for some months, Srinavasa Iyengar Ramanujan eventually came back to India on 27th of March, 1919. He received a grand welcome. Unfortunately, he died on 26th of April, 1920, at the age of 32 years. 

The Government of India has issued a commemorative stamp to pay tribute and honour the 75th anniversary birth of Ramanujan in 1962. Tamil Nadu celebrates his birthday as the State IT Day. The Ramanujan Journal was launched as an international publication, which included his work in areas of mathematics. 

Shakuntala Devi

Shakuntala Devi is a famous and much venerated Indian mathematician who is considered as a genius and is popularly known as the Human Computer. In 1977, Shakuntala extracted the 23rd root of a 201-digit number mentally, 12 seconds faster than the Univac-1108. She even occupies a place on the 1995 Guinness Book of Records for her skills. It was her passionate interest in exploring and increasing the learning capacity of the human mind that led her to build up the concept of Mind Dynamics. 

Early Life of Shakuntala Devi 
Shakuntala Devi was born on 4th of November, 1939 in Bengaluru in a well-known Brahmin priest family. She did card tricks with her father when she was only three. Shakuntala Devi received her early lessons in mathematics from her grandfather. By the age of 5, Shakuntala Devi became an expert in complex mental arithmetic and was recognised as a child prodigy. She demonstrated her talents to a large assembly of students and professors at the University of Mysore a year later. When she was 8 years old, she demonstrated her talents at Annamalai University. 

Contributions of Shakuntala Devi 
In the year 1977, Shakuntala Devi mentally solved the 23rd root of a 201 digit number without any help from mechanical aid. In 1980, 18th June she solved the multiplication of 13 digit number 7,686,369,774,870 and 2,465,099,745,779 picked up by the computer science department of imperial college, London. Shakuntala solved the question in a flash and took 28 seconds to solve the entire problem, and her answer was 18,947,668,177,995,426,462,773,730. This amazing incident helped her get a place in the Guinness book of world record. She has been travelling around the globe performing for the student community, Prime Ministers, Presidents, Politicians and Educationalists. 

Shakuntala Devi as an Author 
Shakuntala Devi has authored a few books. She shares some of the methods of mental calculations in her world famous book, Figuring: The Joy of Numbers. Puzzles to puzzle You, More Puzzles to puzzle you, The Book of Numbers, Mathability: Awaken the Math Genius in Your Child, Astrology for you, Perfect Murder, In the Wonderland of Numbers are some of the popular books written by her. Her book, In the Wonderland of Numbers, talks about a girl Neha, and her fascination for numbers. Her other famous books include Astrology for You and The World of Homosexuals. 

Shakuntala Devi is an excellent astrologer as well and provides remedies based on date and time of birth and place. Her clients include celebrities and well known personalities from various fields.

Game theory experts to analyse space debris removal

October 22, 2015

      Researchers from the University of Liverpool are using strategic game theory to analyse the accumulation of space debris, and to assess removal efforts and mitigation measures to limit its growth.

      Space debris is non-manoeuvrable, human-made objects which orbit the Earth which poses a significant collision risk for operational spacecraft, especially in low-Earth orbit.

       A collision with a piece of space debris of about ten centimetre in diameter would cause a spacecraft to disintegrate resulting in more space debris fragments, which in turn increases collision risk.

       The build-up of space debris could eventually result in a catastrophic cascade of collisions, called the Kessler syndrome. Currently, there are more than 23,000 objects larger than 5-10cm in Earth's orbit.

        Supported by the European Space Agency (ESA), University Computer Scientists will develop theoretical models to study the accrual and removal of space debris as well as assess the different policies and actions that could be used to limit its growth.

Game theory

       Game theory is an economical theory that models interactions between rational agents as games of two or more players that can choose from a set of strategies and the corresponding preferences. It is the mathematical study of interactive decision making in the sense that the agents involved in the decisions take into account their own choices and those of others.

        The Liverpool team (Dr Daan Bloembergen, Professor Karl Tuyls , Dr. Rahul Savani, Richard Klima) at the ESA in the Netherlands.

        Choices are determined by stable preferences concerning the outcomes of their possible decisions and agents act strategically, in other words, they take into account the relation between their own choices and the decisions of other agents. Different economical situations lead to different rational strategies for the players involved.

Dynamic game

       

        Professor Karl Tuyls who is leading the project said: "We will be using game theory techniques to model debris accumulation and active removal efforts from space agencies and governments as a dynamic game, to determine optimal behaviour assuming both cooperative and self-interested parties, and hopefully to propose a mechanism to steer the behaviour of these involved parties to a desirable outcome.

    

       "Space debris accumulation has become a critical problem that can potentially affect all of us, not just governments, space agencies and industry."

      

          Professor Tuyls added: "It is expected that this study will provide a deeper understanding of the space debris problem and its potential economic ramifications, and will provide an insight into the use of game theory solution strategies to address complex real-life issues."

       

          The ESA is investing in mission concepts to clean up and deorbit space debris. Removing active debris represents a number of technical complex challenges as well as being a very costly undertaking.

Game theory: Army of agents to tackle corrupt officials, tax evaders, terrorists

October 20, 2016

Credit: University of Warwick

    Game theory has long been used to apply mathematical models of conflict and cooperation between intelligent rational decision-makers.

     However, our world has evolved from great power conflicts to one where many of our major problems are spawned not from monolithic blocks of self-interest, but from a vast array of single entities making highly individual choices: from lone wolf terrorists to corrupt officials, tax evaders, isolated hackers or even armies of botnets and packages of malware.

     Game theory needs to catch up and new research by mathematicians, led by Professor Vassili Kolokoltsov at the University of Warwick, has just found the way to do that by giving game theory calculations an enormous army of "agents".

     In a paper, entitled 'The evolutionary games of pressure (or interference), resistance and collaboration', Professor Kolokoltsov, from Warwick's Department of Statistics, has been able to take Game Theory far beyond some of its early applications of two opposing sides in zero sum games, and equipped it with the ability to model the impact of a vast array of individual actors - an "infinite state-space of small players".

      The paper says the new tool can be "applied to the analysis of the processes of inspection, corruption, cyber-security, counter-terrorism, banks and firms merging [...] and many others"

To take just one application: tax fraud costs the UK government £16bn a year, according to the National Audit. HM Revenue & Customs (HMRC) has faced questions about both how it decides to deal with individual large companies and how it balances its efforts between pursuing large corporations and individual tax payers.

      

      This evolution of Game theory could greatly assist it in simultaneously model the best approach to manage the great number of participants in the process and create efficient disincentives for both individual and corporate tax evasion.

The modelling tools this evolution of Game Theory will provide can also deal with a tax system's budget inputs and the potential for corruption within any tax system.

Professor Vassili Kolokoltsov comments:

"Our method has a potential to be used in a variety of situations where one big player, referred to as the principal agent, confronts the behaviour of a large pool of individuals with different agendas."

"Of course, as usual for the applications of mathematical tools to socio-economic systems, any concrete applications of the method would require a serious additional input of concrete experimental data to feed various key parameters the model relies upon," he continues.

Professor Kolokoltsov is now working with colleagues to apply the new Game Theory technique to specific types of problem such as internet Denial of Service attacks by botnets. 

5 Greatest Mathematical Inventions In History

Math is all about loving numbers and understanding how life revolves around it. Some of us still revel in the math fun games that we participated in. To hunker down a subjective list of the greatest mathematical discoveries of all time may be difficult, but here are a list of 5 greatest mathematical inventions in history:

  

  

1. The Euler’s identity is a stunning formula that is both useful and deceiving in its essence. The Euler’s number is the base of the natural algorithm and is equal to 2.718. Euler is often regarded as the father of mathematics and the greatest physicist of all time Richerd Feynman regarded the identity as a remarkable formula ever. Euler’s equation helps answer the most difficult of questions in arithmetic.

Richard Phillips Feynman, was an American theoretical physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics.

2. The fast Fourier Transforms are the pillars of the modern computing age. The discrete Fourier Transform was first introduced by Fourier in the early part of the 19^th Century and can break the signals of sound waves and wireless notifications into composite frequencies. There are many applications of the fast and discrete Fourier transform. It remains the single biggest algorithm ever discovered in mathematics.

Jean-Baptiste Joseph Fourier was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

  

3. Godel’s theorems are the next big axiomatic system, which is an imperfect analogy found in the liar paradox. In this paradox, if you begin with a machine, you can feed it any statement and output possible with an unfailing accuracy. The results of Godel’s theorems are in use even today and the computational systems still use this systematically to discover newer theorems.

Known for Godel’s incompleteness theorems, Godel’s completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel metric, Godel’s ontological proof.

 4.Pierre de Fermat’s analysis of numbers and his examining of the Diophantine equations remains the cornerstone for work done in later mathematical research in the 20^th and 21^st century, hundreds of years after his death.

A French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

5. If you are a math lover, you will never forget the mathematical achievements of Greek antiquity. The most seminal and influential of all Greek mathematicians is Euclid. Euclid covered almost all areas of mathematics – such as algebra and plane geometry in his book- Elements. This book remain a staple in all graduate level mathematics classes and even after 2000 years of its creation, has been the centerpiece of geometry and its laws. Written in the year 300 BC, Euclid introduced a set of axioms that went around to demonstrate the mathematical exactitude of the theorems that follow naturally. Along with Pythagoras, Euclid remains the father of geometry in mathematics. From Dostoevsky to Albert Einstein, Euclid’s Elements remains a path breaking work in mathematics.

our students are participated in Auxilium college, Inter collegiate paper presentation on 7th october, 2016

our department students are participated in Gonzaga college of Arts & Science, National level seminar on 27th september 2016

our department conducted National Level Seminar on 30th september 2016

our department students are participated in Islamiah women's college for women and won prizes and championship on 21st september

our department students are participated in Islamiah women's college for women and won prize and over all championship on 21st september 2016

M.Phil students submitted thesis in Thiruvalluvar University on 30.8.2016 & 31.8.2016

The Abel Prize Laureate 2016

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to

Sir Andrew J. Wiles

The Abel Prize was established on 1 January 2002. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.

Discussing about the state level seminar

Our department staff Ms.L.shalini participated at state level seminar in K.M.G college of arts & science, Gudiyattam on 26.8.2016

Our department students participated and presented a paper at K.M.G college of Arts and Science, Gudiyattam on 26.8.2016

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