The relevance of Pi

No number has captured the attention and imagination of people through the ages as much as the ratio of the circumference of a circle to its diameter.

pi  (sometimes written pi) is a mathematical constant that is the ratio of the circumference of a circle to its diameter. p is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve p, which makes it one of the most important mathematical constants.

It is interesting to see many facets of p and humankind’s fascination with it—from the ancient Egyptians and Archimedes to Leonardo da Vinci and the modern-day computers that have calculated the value of pi to eight billion digits.

Throughout the history of mathematics, there has been much effort to determine the value of p more accurately and to understand its nature.

Decimal representation of p never ends and never repeats. The decimal representation of p truncated to 50 decimal places is: 3.1415926535897932384626 4338327950288419716939937510.

The fact that ratio of circumference to the diameter of a circle is a constant was known for ages. However, the first instance of mentioning something similar to pi still remains a mystery. Most probably Egyptians mentioned about this constant in their writings in papyrus scroll as early as 1650 BC. There is good evidence that value as 256/81 was used as a value for this constant. Babylonians around the same time used 25/8 as the value of this constant.

Archimedes provided the first theoretical calculation of pi around 200 BC. He said the constant takes the value between 223/71 and 22/7. The interesting thing is that he did not claim to know the exact value of p; rather he mentioned about the boundary of values between which pi exists. He used pure geometry, using the circle and regular polygons in deriving his expressions in term of fractions.

Indian mathematician Aryabhata made an approximation of pi using regular polygon of 384 sides and he gave the value as 62832/2000, which is equal to 3.1416 and was correct up to four decimal places.

During the year 1400, another Indian mathematical genius, Madhava, from Cochin, use series to calculate p. He used the following series:

p/14 = 1 – 1/3 + 1/5 …

And from this series, he calculated the approximate value of p as 3.14159265359, which was correct up to 11 decimal places. Historically, this was a great achievement since his European colleagues were still way behind this approximation during the same time.

The Indian mathematical prodigy Ramanujan discovered some new infinite series formula in 1910, but its importance was re-discovered around late 70s long after his death. With each addition of term in Ramanujan’s series could give approximately additional eight digits to p. During the year 1985, the value of pi up to 17 million digits was accurately computed using this formula.

The challenge of computing p has stimulated researches in many advanced areas of science and engineering which led to many new discoveries and many new algorithms in the field of mathematics.