Ancient inventions: 3/14/15 at 9:26:53 - PI CELEBRATION DATE AND TIME π

Math and Science lovers prepare your self for the unique day in your lifetime. This year Annual celebration of Pi day will have longer sequence of Pi numbers 3.141592653 (3/14/15 at 9:26:53).
Just for fun you can start doing something on Pi date at Pi time for Pi length.

Pi, the ratio of the circumference to the diameter of a circle, has been calculated to over one trillion digits beyond its decimal point. Regardless of the circle's size, this ratio will always equal pi. As an irrational and transcendental number, it will continue infinitely without repetition or pattern and digits appear to be randomly distributed.

Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three. The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi.

Babylonian clay tablet dated 1900–1600 BC indicates a value of π as 25/8 = 3.1250. The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3.

The Egyptian Rhind Papyrus, dated around 1650 BC, copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as 4 × (8/9)2 = 3.1605.

India around 600 BC, the Shulba Sutras (Sanskrit texts rich in mathematical contents) treat π as (9785/5568)2 = 3.088. In 150 BC, or perhaps earlier, Indian sources treat π as

= 3.1622.

"Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of ten cubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular."

"The first calculation of pi was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71.
He obtained the approximation

223/71 < π < 22/7.

If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002. This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant". Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429). Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.

Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD).

In ancient China, values for π included 3.1547 (around 1 AD),

(100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556). Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416. Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.

Brilliant Chinese mathematician and astronomer Zu Chongzhi, (429–501 AD), calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of π available for the next 800 years. But because his book has been lost, little is known of his work.

In Fifth Century A.D. China, the mathematician Tsu Chung-Chi established that 3.1415926 <

< 3.1415927 an accuracy that was not attained in Europe until the 16th Century.

The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 1040 sides, which remains the most accurate approximation manually achieved using polygonal algorithms.

The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for π most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach was first discovered in India sometime between 1400 and 1500 AD. The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD. The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 – c. 1425. Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory–Leibniz series. Madhava used infinite series to estimate π to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm."