## Over the years, Pi has been calculated in many different ways by many different people. Examine some of the more famous methods of calculating Pi.

People have known the ratio of the circumference of a circle to its diameter is a constant value since before recorded history. Even from the earliest records of Babylonian, Mesopotamian and Egyptian societies agree to this fact. The Bible references this several times. But what is that value, the value we call Pi?

Archimedes, a well-known Greek mathematician and physicist from the 2nd Century B.C.E., is credited with the first theoretical calculations of Pi. Before that point, all values were derived from measurements. He inscribed and circumscribed regular polygons around a circle. He knew the area of the inner polygon was less than that of the circle and the area of the outer polygon was greater. Using a 96 sided polygon, Archimedes determined that 223/71 < pi="">< 22/7.="" this="" was="" a="" very="" good="" approximation="" of="" pi.="" this="" method="" of="" calculating="" pi="" is="" called="" "squaring="" the="" circle."="" many="" others="" including="" ptolemy="" continued="" using="" archimedes'="" method="" of="" calculating="" pi="" to="" further="" refine="" its="" known="" value.="" ludolph="" von="" ceulen="" used="" this="" method="" with="" a="" 2^68-sided="" polygon="" to="" calculate="" pi="" to="" 35="" decimal="" places.="" for="" many="" years,="" pi="" was="" called="" the="" ludolphin,="" named="" after="" von="" ceulen.="">

Until the Renaissance hit Europe, Archimedes' method seems to be the sole method used to calculate Pi. With the advent of higher mathematics including calculus in the 17th Century, several mathematical formulas for Pi were derived. The most famous of these, Pi/4 = 1 - 1/3 + 1/5 -... , is generally credited to Gottfried Liebniz, but in actuality was first discovered by James Gregory. This calculation provides fairly accurate results but is very tedious...you have to calculate many terms before it starts approaching the value of Pi.

Many other famous mathematicians derived other formulas for Pi:

John Wallis derived Pi/2 = 2*2*4*4*6*6*.../1*1*3*3*5*5*...

Isaac Newton derived Pi/6=1/2+1/2*(1/3*2^3)+(1*3)/(2*4)*(1/5*2^5)+...

Leonard Euler derived Pi^2/2=1/1^2 + 1/2^2 + 1/3^2 + ...

In 1761, Johann Lambert proved that Pi is an irrational number. That means that Pi cannot be written as the ratio of two integers. He also conjectured that Pi was transcendental, or a real number which is not the root of a polynomial equation with integer coefficients. However, he could not prove it. Carl von Lindemann finally proved that Pi is transcendental in 1882.

The advent of computers in the mid-20th Century led to faster and more accurate calculations of Pi to more decimal places. ENIAC was used to calculate Pi to 2037 decimal places in 1949 - a calculation that took 70 hours. By 1955 the Naval Ordinance Research Center (NORC) had calculated Pi to 3089 digits in only 13 minutes. Today, Pi is calculated by computers to many more decimal places than were ever though possible. In 1997, Y. Kanada and D. Takahashi calculated Pi to 51,539,600,000 digits in just over 29 hours.

The value of Pi has been a fascination for centuries. Many different mathematicians from many different cultures and times have tried to calculate it using many different methods. Even the politicians have gotten involved. In 1897, the Indiana State Legislature passed a bill decreeing that Pi = 16/5. Luckily, that law is off the books and the people of Indiana are free to calculate their own values of Pi.