What is Pi?
Pi is a mathematical constant defined as the ratio of a circle’s circumference to its diameter. Pi is also found in many relationships throughout mathematics, physics, and statistics. Being an irrational number, Pi cannot be expressed by a fraction, and its decimal representation is infinitely long. Therefore, the only way to depict Pi numerically is through approximations. Here are the first 100 decimals of Pi:
π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679.......
π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679.......
History of Finding Pi
Throughout history, better approximations of Pi have always been sought after. The Ancient Egyptians, Babylonians, and Chinese struggled to specify Pi thousands of years ago. Records show that mathematicians used 3 as the first approximation of Pi. In 250 BCE, Archimedes offered a range of possible values that were accurate to 2 decimal places. He did so by inscribing polygons into a circle. The next major milestone came over 700 years later when Chinese mathematicians correctly approximated 7 decimals of Pi. They took Archimedes' method one step further and created a “12,288-gon” for their approximation. It would not be until 1000 years later when 10 decimal places was reached by Indian mathematician Madhava. Throughout the 1700 and 1800’s, mathematicians including Newton, Euler, and Rutherford produced over 500 digits of Pi. By now, the process was carried out with infinite power series, most notably Leibniz's formula for Pi. With the advent of computers, it took until only 1973 to reach 1 million digits of Pi, 1 billion in 1989, and 1 trillion in 2002. Currently (2018), Peter Trueb holds the record for computing 22,459,157,718,361 digits of Pi. The process took 105 days on his home computer with commercially available parts. Learn more here.
Current Methods to Find Pi
To reach 22,459,157,718,361 digits of Pi (the current world record, 2018), one cannot simply inscribe a polygon in a circle. Even Leibniz's formula for Pi is unfeasible, as it converges very slowly (the sum of 5 billion terms of Leibniz's formula only produces 10 correct decimal places of Pi). Instead, more efficient methods to calculating Pi have been discovered. The current best method is the Chudnovsky Algorithm, which has been used in most of the recent world records. The power series converges at a very rapid rate: the first 358 terms of the algorithm produce 5000 digits of Pi, much faster than Leibniz's formula. Learn more here.
Leibniz's Formula
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Chudnovsky Algorithm
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How Many Digits of Pi are Necessary?
So what exactly is the point of knowing trillions of digits of Pi? Billions of digits? Millions of digits? Or even thousands? The short answer: there is no point. For everyday calculations on earth, Pi to 6 digits is sufficient. 3.14159 is only 0.000084 percent off from the actual value of Pi. If you were to build a fence in a circle of radius 100 meters, and used Pi to 6 digits to estimate how much fencing is needed (the circumference), the resulting calculation would be only half a millimeter short. NASA’s JPL uses only 15 digits of Pi for their calculations. With the approximation at 3.141592653589793, the circumference of the circle with Voyager 1 at the edge (radius of 12.5 billion miles away) can be calculated within a 1.5 inch error. Such a value of Pi is plenty for any current applicable use. However, how many digits of Pi are necessary to calculate the largest thing to within a margin of error of the smallest thing? With 39 digits of Pi, the circumference of the observable universe can be measured to within the width of a single hydrogen atom. Unless you are estimating the size of the universe to pointless accuracy, anywhere from 4 to 15 digits will work for very accurate calculations. Learn more here.
FindingPi.org
The basis of FindingPi.org is to document and test various approximations of Pi through "experimental" methods. Each method could have been completed thousands of years ago, if the relation to Pi was known at the time. There are no supercomputers involved, however, most methods are additionally automated in Excel. Each method uses random chance that is possible with or without a computer.
Experimental Methods
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The probability of two random numbers being co-prime is said to equal (π^2)/6. Do by hand and computer simulations confirm this proof? What are the limitations?
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Pi is used when calculating the area of a circle. Therefore, the area of a circle (determined by randomly placing dots over a circle inscribed in a square) can be used to determine Pi.
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The period of a pendulum involves the value of Pi. Instead of determining the period, we can time it, as well as account for air resistance and initial angle, in order to find Pi.
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Buffon's Needles is a problem that involves randomly throwing sticks onto specifically marked paper. The proportion that land on the lines can be used to approximate Pi.
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