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Showing posts with label Basic mathematical in Sanskrit. Show all posts
Showing posts with label Basic mathematical in Sanskrit. 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

## Sunday, April 5, 2015

### Ancient Indian Math- Madhava of Sangamagrama

Madhava of Sangamagrama (c. 1340 – c. 1425), was an Indian mathematician-astronomer from the town of Sangamagrama (present day Irinjalakuda) near Thrissur, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite PROCEDURES of ancient mathematics to treat their limit-passage to infinity". His discoveries opened the doors to what has today come to be known as Mathematical Analysis. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.

Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in ANALYSIS and calculus.

They are the infinite series of the functions, sine, cosine and the arctangent. These infinite series are called by his name, Madhava sine series and Madhava Cosine series.

While the power series expansion of ArcTan is called Madhava-Gregory series, the power series are collectively called the Madhava Taylor series. The Pi series is known as Madhava Gregory series.

Sankara Varier, one of the foremost disciples of Madhava, had translated his poetic verses in in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441

Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc.

Madhava's formula for Pi was discovered in the West by Gregory and Liebniz.

sin x = x - x^3/3! + x^5/5! - x^7/7!+......

The Europeans encountered difficulties in using these precise sine values for determining longitude, as in the Indo-Arabic navigational techniques or in the Laghu Bhâskarîya. This is because this technique of longitude determination also required an accurate estimate of the size of the earth. Columbus had underestimated the size of the earth to facilitate funding for his project of sailing to the West. His incorrect estimate was corrected in Europe only towards the end of the 17th century CE.

Even so, the Indo-Arabic navigational technique required calculations while the Europeans lacked the ability to calculate. This is because algorismus texts had only recently triumphed over abacus texts and the European tradition of mathematics was “spiritual” and “formal” rather than practical, as Clavius had acknowledged in the 16th century and as Swift (of ‘Gulliver’s Travels’ fame) had satirized in the 17th century. This led to the development of the chronometer, an appliance that could be mechanically used without any application of the mind.

# || संस्कृतम ||

|| संस्कृत अध्ययन कार्ये स्वागतम् अस्तु ||

Welcome to learning learning Sanskrit

सरल गणित-प्रक्रिया: -

१. योगे युति: स्यात् क्षययो: स्वयोर्वा धनर्णयोरन्तरमेवयोग:||

अन्वय - क्षययो: (राश्यो: -,-) योगे युति: स्यात्, (एवं) स्वयो: (+,+) (राश्यो: योगे युति:) वा| धन (+) ऋणयो: (-)

अन्तरं एव योग: स्यात्|

Meaning : Two positive or negative numbers will be added together but positive and negative number's difference is their addition.

Explanation : This algorithm explains very basic rule of addition of signed integers.

२. स्वयोरस्वयो स्‍वम् वध: स्वर्णघाते|

क्षयो भागहारे अपि चैवम् निरुक्तम्||

अन्वय - स्वयो: (+*+), अस्वयो: (-*-) वध: स्वम् (+) (भवति|) स्व-ऋण-घाते (+*-)

वध: क्षय: (-) (भवति)| भागहारे (/) अपि च एवम् निरुक्तम्|

Meaning : Multiplication of two positive or negative numbers is positive. Multiplication of positive and negative number is negative. Same in case of division.

Explanation : This algorithm explains rule for multiplication and division of signed integers.

३. योगान्तरं तेषु समानजात्योर्विभिन्नजात्योश्च पृथक् स्थितिश्च||

अन्वय - तेषु (अव्यक्तयोगान्तरेषु) समानजात्यो: योग: अन्तरं च (भवति) विभिन्नजात्यो: पृथक् स्थिति: च

(भवति)|

Meaning : (In case of variables' adition/subtraction) Their similar terms are added and subtracted, different terms remain separate.

Explanation : For example, in following addition of variables -

5a+3b+4c+7a+2b-c-2a+2c

adding/subtracting similar terms, it will give this result -

(5a+7a-2a)+(3b+2b)+(4c-c+2c) = 10a+5b+5c

४. अस्मिन् विकार: खहरे न राशावपि प्रविष्टेष्वपि निसृतेषु|

बहुष्वपिस्याल्लयसृष्टिकाले अनन्ते अच्युते भूतगणेषु यद्वद||

अन्वय - अस्मिन् खहरे राशौ बहुषु प्रविष्टेषु अपि (अथवा राशौ) निसृतेषु अपि विकार: न स्यात्|

यद्वत् सृष्टिलयकाले बहुषु अपि भूतगणेषु अनन्ते अच्युते (प्रविष्टेषु) विकार: न स्यात्|

Meaning : This infinite number does not change even after adding or subtracting any number from it. Like, the 'Brahmanda' (universe) is not altered when at the end of the world, many lives enter into it.

Explanation : This algorithm explains the concept of Infinity.