𝗔𝗜𝗘𝗙 𝗚𝗹𝗼𝗯𝗮𝗹 𝗔𝗺𝗯𝗮𝘀𝘀𝗮𝗱𝗼𝗿

Happy to share that I am now officially 𝗔𝗜𝗘𝗙 𝗚𝗹𝗼𝗯𝗮𝗹 𝗔𝗺𝗯𝗮𝘀𝘀𝗮𝗱𝗼𝗿 for empowering educators ,and promoting Sustainable Development Goals, especially SDG-4.

Hey everyone,

I just wanted to share some exciting news with you all - I have officially received my honorary doctorate in Education from IIU University! 

Thanks to almighty God; my family, friends, and colleagues for their support and encouragement throughout this journey.

#DrAmitBajaj #HonoraryDoctorate #IIUUniversity

Happy Birthday to Srinivasa Ramanujan, the great Indian mathematician

Today is the birthday of Srinivasa Ramanujan, the great Indian mathematician who studied number theory, mastered modular and partition functions, and designed summation formulas. Ramanujan was born on December 22, 1887 in Erode, a city along the banks of the Cauvery River in the southern state of Tamil Nadu. He enrolled in a local high at the age of 10, but learned more about mathematics from the college students who boarded in parents' home. According to Robert Kanigel, Ramanujan's biographer and author of The Man Who Knew Infinity, the young mathematician was deeply influenced by two borrowed books: S.L. Loney's Plane Trigonometry and George Shoobridge Carr's Synopsis of Elementary Results in Pure Mathematics. Carr's work, a list of 5000 mathematical formulas, inspired Ramanujan to develop his own proofs for these theorems. By the age of 17, Ramanujan had calculated Euler's constant to 15 decimal places and proposed a new class of numbers. Although his peers "stood in respectful awe of him", said one contemporary, "we, including his teachers, rarely understood him".

Like Albert Einstein, Srinivasa Ramanujan struggled with school and even failed his high school exams because of difficulties concentrating. In 1909, the 22-year old college dropout moved from Erode to Madras and found work as a clerk in the Accountant General's Office. Ramachandra Rao, an Indian mathematician who helped Ramanujan obtain the clerkship, encouraged the young man to publish papers and seek broader support for his work. In 1911, Ramanujan's 17-page paper about Bernoulli numbers appeared in the Journal of the Indian Mathematical Society. Two years later, the young mathematician wrote a 10-page letter with over 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory. The letter's recipient, a Cambridge mathematician named G.H. Hardy, had ignored previous communications from Ramanujan, but shared this latest letter with J.E. Littlewood, a university colleague. According to Hardy, the English mathematicians concluded that Ramanujan's results "must be true because, if they were not true, no one would have the imagination to invent them."

With Hardy's help, Ramanujan was named a research scholar at the University of Madras, a position that doubled his clerk's salary and required only the submission of quarterly reports about his work. In March 1914, Ramanujan boarded a steamship for England and, upon his arrival at Cambridge University, began a five-year collaboration with G.H. Hardy. Together, the scholars identified the properties of highly composite numbers and studied the partition function and its asymptotics. They also identified the Hardy-Ramanujan number (1729), the smallest number expressible as the sum of two positive cubes in two different ways. Individually, Ramanujan made major breakthroughs with gamma functions, modular forms, divergent series, hypergeometric series, and mock theta functions. He also developed closed-form expressions for non-simple, continued fractions (Ramanujan's continued fractions) and defined a mathematical concept known as the Ramanujan prime. "I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people," Hardy later wrote, "'Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.'"

Srinivasa Ramanujan received an honorary bachelor's degree from Cambridge University in 1916, and was later appointed a Fellow of Trinity and a Fellow of the Royal Society. Despite his professional accomplishments, Ramanujan suffered from poor health and was eventually diagnosed with tuberculosis and amoebiasis, a parasitic infection of the liver. A vegetarian, he also suffered from a severe vitamin deficiency that may have been due to the shortage of fresh fruits and vegetables in wartime England. Srinivasa Ramanujan died on April 26, 1920 at the age of 33. Today, his home state of Tamil Nadu celebrates his birthday, December 22, to memorialize both the man and his achievements.

G. H. Hardy liked to rank mathematicians on a scale of 1 to 100, and he gave himself 25, Littlewood 30, David Hilbert 80, and Ramanujan 100, which shows just how great Ramanujan was.

 

Ancient grammatical puzzle solved after 2,500 years

 

A grammatical problem that has defeated Sanskrit scholars since the 5th century BC has finally been solved by an Indian Ph.D. student at the University of Cambridge. Rishi Rajpopat made the breakthrough by decoding a rule taught by "the father of linguistics," Pāṇini.

The discovery makes it possible to "derive" any Sanskrit word—to construct millions of grammatically correct words including "mantra" and "guru"—using Pāṇini's revered "language machine," which is widely considered to be one of the great intellectual achievements in history.

 

Leading Sanskrit experts have described Rajpopat's discovery as "revolutionary" and it could now mean that Pāṇini's grammar can be taught to computers for the first time.

 

While researching his Ph.D. thesis, published today, Dr. Rajpopat decoded a 2,500 year old algorithm that makes it possible, for the first time, to accurately use Pāṇini's "language machine."

 

Pāṇini's system—4,000 rules detailed in his greatest work, the Aṣṭādhyāyī, which is thought to have been written around 500 BC—is meant to work like a machine: Feed in the base and suffix of a word and it should turn them into grammatically correct words and sentences through a step-by-step process.

 

Until now, however, there has been a big problem. Often, two or more of Pāṇini's rules are simultaneously applicable at the same step, leaving scholars to agonize over which one to choose.

 

Solving so-called "rule conflicts," which affect millions of Sanskrit words including certain forms of "mantra" and "guru," requires an algorithm. Pāṇini taught a metarule to help us decide which rule should be applied in the event of "rule conflict," but for the last 2,500 years, scholars have misinterpreted this metarule, meaning that they often ended up with a grammatically incorrect result.

 

In an attempt to fix this issue, many scholars laboriously developed hundreds of other metarules, but Dr. Rajpopat shows that these are not just incapable of solving the problem at hand—they all produced too many exceptions—but also completely unnecessary. Rajpopat shows that Pāṇini's "language machine" is self-sufficient.

 

Rajpopat said, "Pāṇini had an extraordinary mind and he built a machine unrivaled in human history. He didn't expect us to add new ideas to his rules. The more we fiddle with Pāṇini's grammar, the more it eludes us."

Traditionally, scholars have interpreted Pāṇini's metarule as meaning that in the event of a conflict between two rules of equal strength, the rule that comes later in the grammar's serial order wins.

 

Rajpopat rejects this, arguing instead that Pāṇini meant that between rules applicable to the left and right sides of a word respectively, Pāṇini wanted us to choose the rule applicable to the right side. Employing this interpretation, Rajpopat found Pāṇini's language machine produced grammatically correct words with almost no exceptions.

 

Take "mantra" and "guru" as examples. In the sentence "Devāḥ prasannāḥ mantraiḥ" ("The Gods [devāḥ] are pleased [prasannāḥ] by the mantras [mantraiḥ]") we encounter "rule conflict" when deriving mantraiḥ "by the mantras." The derivation starts with "mantra + bhis." One rule is applicable to left part, "mantra'," and the other to right part, "bhis." We must pick the rule applicable to the right part, "bhis," which gives us the correct form, "mantraiḥ."

 

In the the sentence "Jñānaṁ dīyate guruṇā" ("Knowledge [jñānaṁ] is given [dīyate] by the guru [guruṇā]") we encounter rule conflict when deriving guruṇā "by the guru." The derivation starts with "guru + ā." One rule is applicable to left part, "guru" and the other to right part. "ā". We must pick the rule applicable to the right part, "ā," which gives us the correct form, "guruṇā."

 

Eureka moment

 

Six months before Rajpopat made his discovery, his supervisor at Cambridge, Vincenzo Vergiani, Professor of Sanskrit, gave him some prescient advice: "If the solution is complicated, you are probably wrong."

 

Rajpopat said, "I had a eureka moment in Cambridge. After 9 months trying to crack this problem, I was almost ready to quit, I was getting nowhere. So I closed the books for a month and just enjoyed the summer, swimming, cycling, cooking, praying and meditating. Then, begrudgingly I went back to work, and within minutes, as I turned the pages, these patterns starting emerging, and it all started to make sense. There was a lot more work to do but I'd found the biggest part of the puzzle."

 

"Over the next few weeks I was so excited, I couldn't sleep and would spend hours in the library, including in the middle of the night to check what I'd found and solve related problems. That work took another two and half years."

 

Significance

 

Professor Vincenzo Vergiani said, "My student Rishi has cracked it—he has found an extraordinarily elegant solution to a problem which has perplexed scholars for centuries. This discovery will revolutionize the study of Sanskrit at a time when interest in the language is on the rise."

 

Sanskrit is an ancient and classical Indo-European language from South Asia. It is the sacred language of Hinduism, but also the medium through which much of India's greatest science, philosophy, poetry and other secular literature have been communicated for centuries. While only spoken in India by an estimated 25,000 people today, Sanskrit has growing political significance in India, and has influenced many other languages and cultures around the world.

 

Rajpopat said, "Some of the most ancient wisdom of India has been produced in Sanskrit and we still don't fully understand what our ancestors achieved. We've often been led to believe that we're not important, that we haven't brought enough to the table. I hope this discovery will infuse students in India with confidence, pride, and hope that they too can achieve great things."

 

A major implication of Dr. Rajpopat's discovery is that now that we have the algorithm that runs Pāṇini's grammar, we could potentially teach this grammar to computers.

 

Rajpopat said, "Computer scientists working on natural language processing gave up on rule-based approaches over 50 years ago... So teaching computers how to combine the speaker's intention with Pāṇini's rule-based grammar to produce human speech would be a major milestone in the history of human interaction with machines, as well as in India's intellectual history."

 

The research is published in the journal Apollo—University of Cambridge Repository.

Source: https://phys.org/news/2022-12-ancient-grammatical-puzzle-years.html

Check out My classroom teaching notes & Video Lectures

Dear Mathematics Educators & Students,

You may like to visit my Mathematics website which contains a lot of Mathematical resources (useful for students studying in classes VI-XII) such as worksheets, assignments, classroom teaching notes and video lectures, quizzes, self-assessment tests, and many more...

                                                  https://www.amitbajajmaths.com/

Amit Bajaj

https://sites.google.com/view/amitbajaj

HAPPY PI APPROXIMATION DAY ( 22-07-2022)

 Dear fellow educators,

The mathematical constant π (pi) is special for a number of reasons. One of them is that there are at least two holidays dedicated to pi: Pi Day celebrated on March 14 and Pi Approximation Day observed on July 22.

The number pi is the ratio of the circle's circumference to its diameter. It is an irrational number, which means it can't be expressed as a common fraction. However, fractions and other rational numbers are commonly used to approximate it in order to facilitate calculations.

The fraction 22/7 is one of the most widely used approximations of pi. It dates from Archimedes. 22/7 is accurate to two decimal places (3,14). Pi Approximation Day is celebrated on July 22 since this date is written 22/7 in the day/month date format, which is viewed as a reference to the fraction 22/7.

Pi Approximation Day was first celebrated at the Chalmers University of Technology, Gothenburg, Sweden. Both Pi Day and Pi Approximation Day are marked with cooking and eating pie, as the words “pi” and “pie” are homophones in the English language.

Happy Pi Approximation Day…

Amit Bajaj

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Amit Bajaj

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Dear All,

Pi Day is held to celebrate the mathematical constant π (pi). Pi Day is observed on March 14 (3/14) , due to π being approximately equal to 3.14.

Pi Minute is also sometimes celebrated on March 14 at 1:59 p.m. If π is truncated to seven decimal places, it becomes 3.1415926, making March 14 at 1:59:26 p.m., Pi Second (or sometimes March 14, 1592 at 6:53:58 a.m.).

The Pi Day celebration includes public marching, consuming fruit pies and playing pi games... The founder of Pi Day was Larry Shaw, a now retired physicist at the Exploratorium who still helps out with the celebrations.

      

Pi has been calculated to be over one trillion digits beyond its decimal point. As an irrational and transcendental number, it will continue infinitely without repetition or pattern. While only a handful of digits are needed for typical calculations, Pi’s infinite nature makes it a fun challenge to memorize and to computationally calculate more and more digits.

HAPPY PI DAY :)