Thursday, August 22, 2013

Approximate value of Pi and its History

Param Poojya Bapu has been delivering his last 3 pravachans on the mathematical constant of Pi or Π. He has very lucidly and scientifically explained the relation, symbolism and commonality between this mathematical value and the Tripurari Trivikram Chinnah

Pi or Π is the value that is obtained whenever the circumference of the circle is divided by its diameter. This equation or ratio remains constant universally, regardless of the size, measurements of the circle. Another universal constant is the fact that the measure of any angle inscribed in a circle is a constant i.e. a right angle. 

Pi traces its roots and history to India. Indians were the first to observe that the perimeter (circumference) of a circle increases in proportion to its diameter. Indians established the relation: perimeter / diameter = constant. Indians knew about Pi since the Indus Valley civilization that dates back to 3000 BC. The value of Pi also finds a mention in Rigveda back to 1700-1100 BC. The Vedangas and Sulabasutras too mention the value of Π. The oldest of them, the Baudhayayana Sulabasutra claims that the perimeter of a pit is 3 times its diameter, therefore approximating the value of Π at 3. 

Yadnyavalkya
The legendary sage of Vedic India, Yajnavalkya writes astronomical calculations in the Shatapatha Brahmana (9th century BC) that led to a fractional approximation of Π ≈ 339/108 (which equals 3.13888, which is correct to two decimal places when rounded, or 0.09 percent below the exact value). 

Many other texts, including the Mahabharata (Bhishmaparva, XII: 44) and many Puranas approximate Π at the value of 3. Later, many other Sulabasutras mention the value of Π to be 18 * (3 – 2 √2) = 3.088. The Manava Sulabasutra approximates the value of Π to be 28/5= 3.125.

The ancient Jain school of mathematics preferred the approximation, Π = √10. This value of Π has been used not only by Jains, but also by the great Indian mathematicians like Varahamihira, Brahmagupta and Sridhara

Aryabhatta
With Aryabhatta (476 AD), a new era of mathematics dawned in India. Aryabhatta approximated value of Pi in the book Aryabhatiyam (chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAm AyutadvayaviShkambhasyAsanno vr^ttapariNahaH, Gaṇitapada 10) as Π = 62832/20000 = 3.1416. This was astonishingly correct to 4 decimal places. Aryabhata used the word asanna meaning approaching, to imply that not only is this an approximation but that the value is incommensurable or irrational. 
The Katapayadi system from 683 AD which is used to encode and encrypt numbers in many shlokas has a hymn (गोपीभाग्यमधुव्रात-श्रुग्ङिशोदधिसन्धिग ॥ खलजीवितखाताव लहालारसंधर ॥) dedicated to Lord Krishna which gives the value of Pi upto 31 decimal places. What is more astonishing is that they needed Pi upto 31 places! 

Many years later, another great mathematician of the Aryabhatta School of mathematics, named Madhava (1340 AD), gave the value of Π to be 2827,4333,8823,3 / 9*1011. This approximation yields correct value of Π to 11 decimal places. This value of Π is still in use in modern mathematics. Madhava's work on the value of Π is cited in the Mahajyanayana prakara ("Methods for the great sines"). This text gives the infinite series expansion of Π, now known as the Madhava-Leibniz series

As mentioned earlier, during the last three pravachans, Bapu explained how Pi is an integral part of the nature, may it be in human body or the Universe. He also mentioned, the Pi from the scientific world given by the Greeks is same as Shree Tripurari Trivikram Chinnah given by Indians, at the metaphysical or the spiritual level. 

Π finds its usage in geometry, trigonometry, complex numbers and analysis, number theory, probability, calculus, statistical physics, engineering, geology, computers, etc. In fact the list is endless. Mathematicians have so far been successful in devising the value of Pi till 5 million digits, post the decimal point. 

Bapu has also made it clear that; Pi or Π is an irrational number, which means that it cannot be expressed exactly as a ratio of any two integers. Fractions such as 22/7 are commonly used as an approximation of Π; no fraction can be its exact value.

Tuesday, August 6, 2013

MANY HAPPY RETURNS OF THE DAY AAI

Most wonderful and love filled eyes of my MOTHER.. my beloved NANDAI.. Teri Aankho ke siwah Duniya mein rakkha kya hai.. MANY HAPPY RETURNS OF THE DAY AAI