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Karatsuba during a lecture

Mathematicians find an amazing trick to multiply bigger numbers

The tables which were an amazing aid, pioneered first some 4000 years ago by the Babylonians is the most handy tool when it comes to multiplications. Well, its good for multiplication of small numbers but for bigger numbers , it becomes a bit tedious without the calculators and computers since we switch to the old school math of multiplying basically two numbers together. This method , though very accurate is slow since, for every single digit in a number , there needs to be a separate multiplication operation, before we add up the products. For every single digit in each number in the problem, you need to perform a separate multiplication operation, before adding all the products up.

Long multiplication is basically an algorithm, but not an efficient one, since the process is unavoidably pain-stricken. The problem is that as the numbers get bigger, the amount of work increases, represented by n raised to the power 2.

Well, the long multiplication algorithm was one of the most advanced multiplication algorithm we ever had until the 1960s, when a Russian mathematician, named Anatoly Karatsuba found that n raised to the power 1.58 was possible.

Soon after a decade, a pair of German mathematicians generated another shockwave with the most advanced breakthrough: the Schönhage–Strassen algorithm which in words of Harvey is- “They predicted that there should exist an algorithm that multiplies n-digit numbers using essentially n * log(n) basic operations,” as he posted his research paper which is yet to be peer- reviewed. However, It is in standard practice in mathematics to disseminate the results before undergoing any sort of peer review.

Using the Schönhage-Strassen algorithm and with new theoretical proof, it would take less than 30 minutes to solve the multiplication theoretically – and might be the fastest multiplication algorithm that is mathematically possible. Although the researchers have no idea about how big a number can they use to solve any multiplication problem using this method but the one they gave on the paper equates to 10214857091104455251940635045059417341952, which is really a very big number.

Fürer from Penn State University told the Science News that around a decade ago, he himself tried to make amendments the Schönhage-Strassen algorithm but eventually discontinued since it seemed quite hopeless to him. Now , that the mathematicians can verify it, his hopelessness has diminished.

In the meantime, Harvey and Joris van der Hoeven from École Polytechnique in France, say that their algorithm needs optimisation as they feel anxious if the results go wrong!

Srinivasa Ramanujan

Ramanujan: The man who knew Infinity

Srinivasa Ramanujan was one among India’s greatest mathematical geniuses. He created substantial contributions to the analytical theory of numbers and worked on elliptic functions, continuing fractions, and infinite series.

22 December 1897-26 April 1920, Ramanujan was a man of science who lived throughout the British decree in India. Although he had nearly no formal coaching in math, he created substantial contributions to mathematical analysis, range theory, infinite series, and continued fractions, as well as solutions to mathematical issues thought of to be insoluble.

Early life

When he was nearly 5 years young, Ramanujan entered the first faculty in Kumbakonam though he would attend many completely different primary faculties before getting into the city high school in Kumbakonam in Jan 1898. At the city high school, Ramanujan was doing well altogether in his faculty subjects and showed himself an in a position well-rounded scholar. In 1900 he began to figure on his own on arithmetic summing geometric and arithmetic series.

Ramanujan was shown a way to solve cube-shaped equations in 1902 and he went on to seek out his own methodology to unravel the fourth power. The subsequent year, not knowing that the quintic couldn’t be solved by radicals, he tried (and in fact failed) to unravel the quintic.

It was within the city high school that Ramanujan stumbled on an arithmetic book by G S Carr known as a summary of elementary leads to maths. This book, with its terribly cryptic vogue, allowed Ramanujan to show himself how to solve arithmetic problems, however, the design of the book was to possess a rather unfortunate impact on the method Ramanujan was later to jot down arithmetic since it provided the sole model that he had of written mathematical arguments. The book contained theorems, formulae, and short proofs. It conjointly contained an index of papers on maths that had been revealed within the European Journals of Learned Societies throughout the primary 1/2 the nineteenth century. The book, revealed in 1856, was, in fact, a feed of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake a deep analysis. He investigated the series ∑ (1/n) and calculated Euler’s constant to fifteen decimal places. He began to review the Bernoulli numbers, though this was entirely his own freelance discovery.

Ramanujan, on the strength of his smart faculty work, was given a scholarship to the govt. school in Kumbakonam that he entered in 1904. But the subsequent year his scholarship wasn’t revived. As a result of Ramanujan devoted a lot of his time to arithmetic and neglected his different subjects. While not having much cash, he was knee-deep in difficulties and, while not telling his oldsters, he ran away to the city of Vizagapatnam regarding 650 km far north of Madras. He continued his mathematical work, however, and then he worked on hypergeometric series and investigated relations between integrals and series. He was to find later that he had been learning elliptic functions.

Adulthood

In 1906 Ramanujan visited Madras wherever he entered Pachaiyappa’s school. His aim was to pass the primary Arts examination which might permit him to be admitted to the University of Madras. He attended lectures at Pachaiyappa’s school. However, he became unwell when 3 months of study. He took the primary Arts examination when having left the course. He passed in arithmetic however he was unsuccessful in all his alternative subjects and ultimately in the examination. This meant that he couldn’t enter the University of Madras. Within the following years, he worked on arithmetic problems, developing his own concepts with no one facilitating him and with no real plan of the then current analysis topics apart from that provided by Carr’s book.

g-h-hardy-ramanujan-mentor

Godfrey Harold Hardy, Ramanujan’s Mentor. Image Source: Wikipedia

Proceeding with his work on mathematical problems, Ramanujan studied continued fractions and divergent series in 1908. At this stage, he became seriously unwell once more and also underwent an operation in April 1909 when he took him some substantial time to recover. He married on 14th July 1909 once his mother organized for him to marry a ten-year-old woman S Janakiammal. Ramanujan did not accompany his partner, however, till she was twelve years of age.

Ramanujan continued to develop his mathematical concepts and started to cause issues and solve issues within the Journal of the Indian Mathematical Society. He developed relations between elliptic standard equations in 1910. When he published a superb analysis paper on Bernoulli numbers in 1911 within the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of university education, he was turning into being acknowledged within the Madras space as a mathematical genius.

The pursuit of a career in mathematics

In 1911 Ramanujan approached the founding father of the Indian Mathematical Society for a recommendation on employment. After this, he was appointed to his first job, a short-lived post within the Comptroller General’s workplace in Madras. It was then absolutely urged upon him that he approach Ramachandra Rao, who was a Collector at Nellore. Ramachandra Rao was a founder of the Indian Mathematical Society. The agency which had helped him begin the arithmetic library.

Ramachandra Rao told him to come to Madras and he tried, unsuccessfully, to rearrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk within the accounts section of the Madras Port Trust.

Despite the very fact that he had no university education, Ramanujan was clearly documented to the university mathematicians in Madras for, along with his letter of application, Ramanujan enclosed a reference from E W Middlemast who was the faculty member of arithmetic at The Presidency faculty in Madras. Middlemast, a graduate of St John’s faculty, Cambridge.

On the strength of the advice, Ramanujan was appointed to the post of clerk and commenced his duties on 1st March 1912. Ramanujan was quite lucky to possess a variety of individuals operating around him with coaching in arithmetic. In fact, the Chief comptroller for the Madras Port Trust, S N Aiyar, was trained as a scientist and printed a paper on the distribution of primes in 1913 on Ramanujan’s work. The faculty member of engineering at the Madras Engineering faculty C L T Griffith was conjointly fascinated by Ramanujan’s talents and, having been educated at University faculty London, who knew the faculty member of arithmetic there, particularly M J M Hill. He wrote to Hill on 12th November 1912 referring him a number of Ramanujan’s work and a duplicate of his 1911 paper on Bernoulli numbers.

Hill replied in an exceedingly fairly encouraging method however he showed that he had did not perceive Ramanujan’s results on divergent series. The advice to Ramanujan that he browsed Bromwich’s Theory of infinite series failed to please Ramanujan a lot of. Ramanujan wrote to E W Hobson and H F Baker attempting to interest them in his results however neither replied. In Jan 1913 Ramanujan wrote to G H Hardy having seen a duplicate of his 1910 book Orders of time.

Mathematical achievements

On 18th of February, 1918 Ramanujan was appointed as a fellow of the Cambridge Philosophical Society then 3 days later, the best honor that he would receive, his name appeared on the list for election as a fellow of the Academy of London. He had been planned by a powerful list of mathematicians, specifically Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a fellow of the academy was confirmed on 2nd May 1918, and then on 10th Oct 1918, he was appointed as a Fellow of Trinity faculty Cambridge, the fellowship to last for six years.

The honors that were presented on Ramanujan perceived to facilitate his health improvement and he revived his efforts at manufacturing arithmetic. By late November 1918, Ramanujan’s health had greatly improved.

Ramanujan sailed to India on 27th February 1919 and he reached on thirteen March. But his health was terribly poor and, despite medical treatment, he died there the subsequent year.

Posthumous recognition

The letters Ramanujan wrote to Hardy in 1913 had contained many desirable results. Ramanujan figured out the Riemann series, the elliptic integrals, hypergeometric series and therefore the practical equations of the zeta function. On the opposite hand, he had solely an imprecise plan of what constitutes a proof. Despite several sensible results, a number of his theorems on prime numbers were utterly wrong.

Riemann Zeta Function

This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin. (Source: wikimedia.org)

Ramanujan discovered the results of Gauss, Kummer, et al on the hypergeometric series on his own. Ramanujan’s own work on partial sums and results of hypergeometric series have shed major light and led to a significant development within the topic. Maybe his most notable work was on the amount p(n) of partitions of an integer number n into summands. MacMahon had made tables of the worth of p(n) for tiny numbers n, and Ramanujan used this numerical information to conjecture some outstanding properties a number of that he tried exploitation elliptic functions. Various other results were proved posthumously.

Ramanujan gave an asymptotic formula for p(n) after collaborating on a paper with Hardy. It had the outstanding property that it seemed to offer the right worth of p(n), and this was later verified by Rademacher.

Random Quiz

What is Hardy-Ramanujan number?

Correct! Wrong!

1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy's words: “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."” The two different ways are these: 1729 = 1^3 + 12^3 = 9^3 + 10^3


Wrapping up!

Ramanujan left a variety of unpublished notebooks full of theorems that mathematicians have continued to check. G N Watson, Mason academic of maths at Birmingham from 1918 to 1951 revealed fourteen papers underneath the final title Theorems declared by Ramanujan and altogether he revealed nearly thirty papers that were impressed by Ramanujan’s work. Hardy passed on to Watson an outsized range of manuscripts of Ramanujan that he had, each written before 1914 and a few written in Ramanujan’s last year in India before his death. Ramanujan left behind 3 notebooks and a bundle of pages (also referred to as the “lost notebook”) containing several unpublished results that mathematicians continuing to verify long since his death.

Read More:

  1. Ramanujan’s lost notebook
  2. Ramanujan surprises again
  3. Mathematical proof reveals magic of Ramanujan’s genius