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Showing posts with label Srinivasa Ramanujan. Show all posts
Showing posts with label Srinivasa Ramanujan. Show all posts

## Saturday, October 22, 2016

### Ramanujan's Math still unsolved

Ramanujan’s second construction for the approximate squaring of a circle
To experience the greatness of great men one has to relive or redo some acts of theirs to the best of ones ability. In ones youth such enactments might inspire one to make a bid for greatness. Whether this happens or not is mostly up to your genetics. Nevertheless, through the enactments one can at least savor the experience of what it takes to get there. If there was one man in our midst who could have lived up to be a Gauss or an Euler it was Srinivasa Ramanujan.
By redoing some of his acts that are within the grasp of our limited intellect we experienced the monument that he was. He gave two constructions for the approximate squaring of a circle using a compass and a straight-edge. We had earlier described the first and more widely known of those. The second appears in his paper titled “Modular equations and approximations to \pi“. In this paper in addition to remarkable approximations for the perimeter of the ellipse, which we had also alluded to before, he gives several series for \pi=3.141592653589793.... One of these series with just the first term leads to the below approximation:
\pi \approx \dfrac{99^2}{2*\sqrt{2}*1103} = 3.141592730013305 which is correct down to 6 decimal places. It is this kind of accuracy he captures in his first construction for the quadrature of the circle. In the midst of the dizzying series he conjures in a very Hindu style of mathematics, he says that he came up with an empirical approximation which leads to the below construction for the approximate squaring of the circle:
1) Draw circle to be squared with center O.
2) Draw its diameter \overline{AB}.
3) Trisect its radius \overline{AO} to get a third of it as \overline{AF}.
4) Bisect the semicircle AB to get point C.
5) Draw \overline{BC}.
6) On \overline{BC} mark \overline{CG}= \overline{GH}=\overline{AF}.
7) Join point A to point H to get \overline{AH} and to point G to get \overline{AG}
8) With radius as \overline{AG} cut \overline{AH} to get point I.
9) Draw a line parallel to \overline{GH} through point I to cut \overline{AG} at point J.
10) Join points O and J to get \overline{OJ}.
11) Draw a line parallel to \overline{OJ} through point F to cut \overline{AG} at point K.
12) Draw the tangent to the circle at point A and cut it with radius as \overline{AK} to get point L.
13) Draw \overrightarrow{OL} to cut circle at point M.
14) Draw semicircle LM and perpendicular from point O to cut this semicircle at point N.
15) Triplicate \overline{ON} to get \overline{OQ}=3 \times \overline{ON}.
16) Produce \overline{OQ} in the opposite direction to cut circle at point R.
17) Draw semicircle RQ and a perpendicular from point O to cut it at point S.
18) Thus, we have \overline{OS} as the side of the square OSTU which has approximately the same area the starting circle.
Ramanujan tells us that his earlier construction gave an “ordinary” value \pi=\dfrac{355}{113}=3.141592920353982, which is correct to six decimal places. This one, however, gives us the value:
\pi=\left (9^2+\dfrac{19^2}{22}\right)^{\frac{1}{4}}= 3.141592652582646
This is correct to a whopping eight decimal places keeping with the Hindu love for big numbers.
source: Manasataramgini

## Wednesday, April 8, 2015

### Srinivasa Ramanujan

Deathbed theory dreamt by an Indian maths genius is finally proved correct - almost 100 years after he died
Theory came to Srinivasa Ramanujan in a dream on his deathbed in 1920 - but has never been proved
Discovery could now be used to explain the behaviour of parts of a black hole
Srinivasa Ramanujan, described as a 'natural genius', has finally had the mathematical functions he came up with on his deathbed proved correct
Researchers have finally solved the cryptic deathbed puzzle renowned Indian mathematician Srinivasa Ramanujan claimed came to him in dreams.
While on his death-bed in 1920, Ramanujan wrote a letter to his mentor, English mathematician G. H. Hardy, outlining several new mathematical functions never before heard of, along with a hunch about how they worked,
Decades years later, researchers say they've proved he was right - and that the formula could explain the behaviour of black holes.
'We've solved the problems from his last mysterious letters,' Emory University mathematician Ken Ono said.
'For people who work in this area of math, the problem has been open for 90 years,'
Ramanujan, a self-taught mathematician born in a rural village in South India, spent so much time thinking about math that he flunked out of college in India twice, Ono said.
Ramanujan's letter described several new functions that behaved differently from known theta functions, or modular forms, and yet closely mimicked them.
Functions are equations that can be drawn as graphs on an axis, like a sine wave, and produce an output when computed for any chosen input or value.
Ramanujan conjectured that his mock modular forms corresponded to the ordinary modular forms earlier identified by Carl Jacobi, and that both would wind up with similar outputs for roots of 1.
Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.
However, no one at the time understood what Ramanujan was talking about.
'It wasn’t until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,' Ono said.
Ono and his colleagues drew on modern mathematical tools that had not been developed before Ramanujan’s death to prove this theory was correct.
'We proved that Ramanujan was right,' Ono says.
'We found the formula explaining one of the visions that he believed came from his goddess.'
The team were also stunned to find the function could be used today.
'No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,' Ono says.
'Ramanujan's legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died,' Ono said.
The findings were presented last month at the Ramanujan 125 conference at the University of Florida, ahead of the 125th anniversary of the mathematician's birth on Dec. 22nd.
December 22, 2012 marks the 125th birth anniversary of Srinivasa Ramanujan, the self taught mathematician born into a modest and conservative family in Kumbakonam, a relatively small town in Tamilnad.
Ramanujan was self-taught and worked in almost complete isolation from the mathematical community of his time.
Described as a raw genius, he independently rediscovered many existing results, as well as making his own unique contributions, believing his inspiration came from the Hindu goddess Namagiri.
He spent so much time thinking about math that he flunked out of college in India twice.
He overcame several hurdles to find a place among the celebrated intellectuals of Cambridge.
Ramanujan passed away at the young age of 32 of tuberculosis, but he left behind formulations in mathematics that have paved the path for many scholars who came after him.