True Life of Pi

                       Amazing world of Pi (π)

              (To some, more thrilling than celluloid Life of Pi)

 

Ratio of Circumference to Diameter for a circle is always constant. It means that if we choose any circle of any size, the ratio of the circumference to the diameter of that circle will remain the same and that its approximate value is 22/7 or 3.14 or a bit more accurately 355/113. Its exact value is denoted as π. This beautiful property of Circle has fascinated people for more than 4,000 years. In fact π is most researched number in the History of Mathematics. Indeed, no number can claim more fame than π.

Pi is an Irrational Number implying that it can not be written as the ratio of two Integers. In other words, the digits after the decimal do not terminate meaning there is not a finite or countable number of digits in π. The digits in π do not repeat and they do not occur in a repeating or recurring pattern. Modern Mathematicians even after using Super Computers to calculate trillions of digits in Pi have not found even a simple repeating pattern.

It is an Irrational and Transcendental Number not just occurring in Circles but in other branches of Mathematics also. (A Transcendental Number is a number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients.)

History of π and Life of Pi: While no one knows for sure who first calculated π, it is widely believed that Ancient Greek and Chinese mathematicians used this value in their calculations. Some mathematicians think that the Egyptians used π when they built the pyramids. The symbol π was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler.

Archimedes (225 BC) obtained the first rigorous approximation of π by constructing large polygons in a circle He found the value of π to be:

3+ (10/71) <π< 3+1/7

Since then, Mathematicians have been trying to find the more accurate value of π. The number π is thus one of the oldest subjects of research by mankind and possibly the one topic within mathematics which has been researched most and the longest. Humans have connected themselves with π for several thousand years by now. Mathematicians in the 1800s were able to calculate pi to about 1,000 digits. That was before they had calculators/ computers to assist them. They did their computations manually. In 1999, Dr. Yasumasa Kanada at the University of Tokyo calculated 206,158,430,000 decimal digits of pi. The current world record is more than 10trillion digits.

(Perhaps, calculating so many digits of π serves no apparent useful mathematical purpose - π goes on forever. Just 39 digits are enough to calculate the circumference of a circle the size of the universe with an error no larger than the radius of a hydrogen atom! Still, some π- Mathematicians believe that patterns in the digits can help in solving long pending problems of Mathematics. And the quest for searching more digits in π goes on and on…)

In nut-shell the history of Pi is quite fascinating and the Life of π is eternal…

Naif^* of Pi or should we say Rife is Pi! It has an uncanny ability to crop up in all sorts of unexpected places in maths besides Circles and Spheres. Only a few examples are listed below: 

(1)   In the probability that a pin dropped on a set of parallel lines intersects a line (Buffon’s Needle Problem). Buffon's Needle is one of the oldest problems in the field of geometrical probability. It was first stated in 1777. It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the page. The remarkable result is that the probability is directly related to the value of pi.

The Probability is equal to π/2= 0.6366197 approximately.

(2)   In the distribution of primes.                       

(3)   In the construction of numbers which are very close to integers, or almost integers. (The Ramanujan Constant)

e^π√163 =262 537412640768743.9999999999992

(4)    It occurs in the normalization of the normal distribution.

(5)   Pi appears to guide the lengths of meandering rivers. The ratio between the actual length of rivers from source to mouth and their direct length as the crow flies. Although the ratio varies from river to river, the average value is slightly greater than 3, that is to say that the actual length is roughly three times greater then the direct distance. In fact the ratio is approximately 3.14 which is close to the value of Pi.

(6)   Pi seems to crop up everywhere, even in places that have no ostensible connection to circles. For example, among a collection of random whole numbers, the probability that any two numbers have no common factor — that they are "relatively prime" — is equal to 6/π².

π – Series and Formulae:  There are scores of series and Formulae which define π. A few of the series are mentioned below for examples:

(1)The Liebniz-Gregory Series: 1/1- 1/3+ 1/5 -1/7+1/9-…= π/4

(2)Grampa's Series:(1/1x3) +(1/3x5)+(1/5x7)+ (1/7x9)+… = π/(2x4)

(3)The Euler Series: 1/1² + 1/2²   +1/3² +1/4² +… =  π²/6

(4)More Euler Series: 1/1² + 1/3²   +1/5² +1/7² +… =  π²/8
1/2² + 1/4²   +1/6² +1/8² +… =  π²/24
1/1² - 1/2²   +1/3² -1/4² +… =  π²/12

(5)  Viete’s Formula :  

                          

(6)Machin's formula :

π/4 = 4 cot^-1 5 - cot^-1 239

(7) Wallis Equation:

(π /2) =	(2*2) (1*3)	(4*4) (3*5)	(6*6) (5*7)	...

^*The meaning of the word Naif is Excess, Surplus in Arabic.