Arguably pi is one of the most intriguing numbers in mathematics. The concept is simple enough. If one divides the circumference of a circle by its diameter, then one gets this rather strange number 3.14159, and to make it even weirder, not only do the digits following the decimal point go on forever but there is no pattern in these numbers. But how did mathematicians arrive at this number in the first place?
One of the earliest references to pi can be found in the Bible.
And he made the molten sea; 10 cubits from the one brim to the other; it was round all about, and his height was five cubits: and a line of 30 cubits did compass it round about. (I Kings 7:23)
The Hebrews, it seems, had arrived at the conclusion that this ratio was close to three to 30 cubits divided by 10 cubits.
The ratio between the circumference and the diameter has fascinated mathematicians since ancient times. Indeed, the earliest-known record is an Egyptian papyrus written by an Egyptian scribe named Ahmes around 1650 BC, now known as the Rhind Papyrus. In problem 50, Ahmes writes:
Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as the circle.
What Ahmes is saying here is that the area of a circle with a diameter of nine units is equal to the area of a square whose side is eight units. Using the area πr2 of the circle (where r is the radius), this means that pi is 3.16049. However, as indicated above, this is not exact.
Subsequent attempts to arrive at the value of pi have adopted three approaches. The first, adopted by the great Archimedes, approximated the circumference of a circle by regular polygons circumscribing and inscribed in a circle. By calculating the perimeter of regular polygons having up to 96 sides, he arrived at values of pi that lie between 223/71 and 22/7. This is incredible when one considers that Archimedes had no calculator.
A giant step in the search for more accurate estimates was the discovery of differential calculus. Among his many achievements, the Scottish mathematician James Gregory discovered the arctangent series that still bears his name. He found that the area under the curve 1/(1 + x2) in the interval (0, x) is arctan x. Then by the simple process of long division and using a formula determined earlier by Cavalieri, he found that, for x at least -1 and at most 1,
arctan x = x - x3/3 + x5/5 - x7/7 + …
Putting x = 1, this series becomes
π/4 = 1 - 1/3 + 1/5 - 1/7 + …
The next chapter in the long story of pi begins with the invention of computers. With their arrival it was possible to produce values of pi which would have been unimaginable before. For example, in 1959, François Genuys computed pi to 16,167 decimal places, using an IBM 704, while in 1961, John Wrench and Daniel Shanks computed pi to 100,265 decimal places, using an IBM 7090. And if that is not mind boggling enough, in 1997, Yoshiaki Tamura and Yasumasa Kanada calculated over 52 billion decimal digits of pi on a Hitachi SR 2201 in just over 29 hours, setting a new world record.
Did you know!
• The symbol π has only been used regularly in its modern meaning for the past 250 years. Indeed it was used by William Jones in 1706, although it was Leonhard Euler who popularised it.
• Pi has made a number of appearances in movies. In an episode of the popular Star Trek series, Spock manages to foil the evil computer by asking it to compute the last digit of pi. In Darren Aronofsky’s movie Pi: Faith in Chaos, the main character attempts to find simple answers about pi, only to drive himself mad. In Alfred Hitchcock’s Torn Curtain, pi plays the role of a secret code.
• Pi Day is celebrated on March 14 because pi can be approximated by 3.14. Coincidentally, Albert Einstein was born on this day in 1879 in Ulm Wurttemberg, Germany.
• Pi has fascinated Egyptologists for the fact that the ratio between the vertical height of the Great Pyramid at Giza and the perimeter of its base seems to approximate pi.
For more trivia see: www.um.edu.mt/think
Sound bites
• Two number theorists who deserve a special mention are the brothers David and Gregory Chudnovsky. The brothers have held several world records for calculating the highest number of digits and they have also developed extremely sophisticated equations to describe pi. The Chudnovskys were born in the former Soviet Union and were reluctantly allowed to travel to Paris and then the US because Gregory suffered from an autoimmune disorder of the muscles. After their first record-breaking pi experience in 1989, the brothers built a super-computer in their apartment. The computer, called m-zero, took up much of the apartment and could raise the ambient room temperature to above 90 degrees Fahrenheit. David Chudnovsky once stated: “Exploring pi is like exploring the universe.”
• Jörg Arndt and Christoph Haenel argue that 39 digits are sufficient to perform most cosmological calculations as they are sufficient to calculate the circumference of the observable universe to the nearest atom. Great efforts towards computing pi to millions of digits still have been made and were partly motivated by the desire of breaking records, which often make headlines around the world. However, the resulting achievements have practical applications, such as testing supercomputers and testing numerical analysis algorithms. The data obtained is used in mathematics itself for evaluating the randomness of the digits of pi.
https://en.wikipedia.org/wiki/pi