~William L. Schaaf, Nature and History of Pi

Pi is a name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. Pi isoften written using the symbol  and is pronounced "pie", just like the dessert.

                                                             William Jones

The Man Who Invented Pi. In 1706 a little-known mathematics teacher named William Jones first used a symbol to represent the platonic concept of pi, an ideal that in numerical terms can be approached, but never reached.

A Brief History of Pi (π)

Pi (π) has been known for almost 4000 years—but even if we calculated the number of seconds in those 4000 years and calculated π to that number of places, we would still only be approximating its actual value. Here’s a brief history of finding π.

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. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for π, which is a closer approximation.

The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π.

The first calculation of π 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 π but only an approximation within those limits. In this way, Archimedes showed that π is between 3 1/7 and 3 10/71.

A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for π, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.

An eighteenth-century French mathematician named Georges Buffon devised a way to calculate π based on probability. You can try it yourself at the Exploratorium's Pi Toss exhibit.

Digits of Pi

First 100 decimal places

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...

First 1000 decimal places
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

Where does pi occur?

Pi occurs in many areas of mathematics, far too many to list here.

The study of pi begins around middle school, when students learn about the circumference and area of circles. 

CIRCUMFERENCE and AREA of CIRCLES

The definition of pi gives us a way to calculate circumference. The circumference of a circle is the distance around a circle. If π = Cd, then C = πd. You can also calculate the circumference of a circle with C = 2πr.  

The area of a circle is A = πr2.

The Archimedian method

Archimedes of Syracuse was the first to attempt calculating pi. Through observation, he hypothesised that perimeters of polygons drawn inside and outside of a circle would be close to the circumference of the circle. Subsequently, he used Pythagoras Theorem to find the areas of a polygon inscribed in a circle and a polygon within which the circle was circumscribed. Archimedes could successfully find the limits of the circle’s area as it is bigger than the area of the polygon inside the circle, but smaller than the area of the polygon outside the circle. He started this rigorous calculation by drawing hexagons. He first drew a circle with a diameter of 1 and used the diameter and the fact that an angle of a hexagon is 60 degrees to find the measurements for each of the sides of the inner and outer hexagons. He figured out that the perimeter of the inscribed hexagon is 3, and the perimeter of the circumscribed hexagon is about 3.46. He concluded that pi would be bigger than 3 but smaller than 3.46. Archimedes used polygons with bigger numbers of sides so that he could get a better approximation for pi. He ultimately used polygons with 96 sides and utilised these limits driven from calculation to support that pi is between 3.1408 and 3.14285.

Modern approaches

William Jones in 1706 first used the modern symbol for pi, $\pi$. The reason why $\pi$ was chosen instead of other Greek letters was that, in Greek, is pronounced like the alphabet ‘p,’ which stands for the perimeter. The symbol was later popularised in 1737 by Leonhard Euler. Since then, pi has been widely used in diverse fields by engineers, physicists, architects, and designers… anywhere that mathematics is involved! Some people suggest that the Great Pyramid at Giza was built based on pi because the ratio of the perimeter to the height of the pyramid is strikingly close to pi. In fact, the Pyramid has an elevation of 280 cubits and a perimeter of 1760 cubits, and the ratio of the perimeter to the height is 6.285714, which is approximately two times pi.

Since pi is a non-repeating decimal, attempts to calculate pi up to as many digits as possible also have some history. The most accurate calculation of pi before computers was one done in 1945, by D.F. Ferguson. He calculated pi up to 620 digits. This was the most accurate and lengthy calculation since, before Ferguson, William Shanks was known to have calculated pi up to 707 digits–in which only 527 digits were correct. Later on, pi was calculated up to trillions of digits with the help of digital programs. To list a couple of people who made world records by carrying out the most extended calculation of pi, Takahashi Kanada calculated pi up to 206,158,430,000 digits with a Hitachi SR8000 in 1999, and Shigeru Kondo calculated up to 10 trillion digits with Alexander Yee’s y-cruncher program in 2011.

In the past, pi was only used by specialists such as mathematicians, scientists, and architects. However, nowadays, pi is commonly used by non-specialists and is even used in literature, movies, and museums. For instance, Pi is the nickname of the main character in the award-winning novel Life of Pi. An episode of StarTrek: The Original Series included a scene in which Spock orders a computer to calculate pi up to the last digit. What’s more, ‘3.14159’ is included in cheers of Georgia Institute of Technology and MIT.

Today, perhaps the most popular cultural phenomenon regarding pi is Pi day. It is celebrated on 14 March all around the world. On Pi day, people who are enthusiastic about mathematics write and talk about pi, eat pie, and make bad jokes. Now that you have learned more about pi, maybe you can tell people around you about all the exciting things about pi on Pi Day!

Various Formulas for Computing Pi

• Wallis
p/2=(2.2.4.4.6.6.8.8. ...)/(1.3.3.5.5.7.7.9. ...)
• Machin
p/4=4 arctan(1/5)-arctan(1/239)
• Ferguson
p/4= 3 arctan(1/4)+arctan(1/20)+arctan(1985)
• Euler
p/4= 5 arctan(1/7)+2 arctan(3/79)
• Euler
p²/6=1/2²+1/3²+ 1/4²+1/5²+ ...
• Euler
e^ip+1=0
• Borwein and Borwein
1/p=12S [(-1)ⁿ(6n)!/(n!)³(3n)!] [(A+nB)/C^n+1/2];
whereA=212175710912Ö(61) +1657145277365;
B=13773980892672Ö(61) +107578229802750;
C=[5280(236674+30303Ö(61)]³
• Borwein, Bailey, and Plouffe
p=S [4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6)](16)^-n
This formula enables one to calculate the nth digit of pi, in hexadecimal notation, without being forced to calculate the preceding n-1 digits.


CONCLUSION:In the light of the above i discuss about the "Pie".I got information from Wikipedia,piday.org,Mathy.com,exploratorium,pcworld.com etc.Watch video.Writtn By Raktim Bar