The story starts in 2000 BC with attempts by the Egyptians and Babylonians to compute π. The Egyptians arrived at (4/3)4 3.1604, while the Babylonians found 25/8= 3.125. The Indians used A10 3.1622 for π 3.1415. These were very good approximations for their time, but they had an error that started from the second decimal place.
A major achievement in the computation of π was made around 250 BC. Archimedes of Syracuse (287-212 BC) had the brilliant idea to approximate π by using inscribed and circumscribed polygons that approached a circle (see the figure). The circumference of a circle of diameter 1 is π. He inscribed a polygon of n-side and computed its circumference.
This circumference is clearly less than π. Similarly, he circumscribed another polygon of n-side and computed its circumference. This time the circumference of the polygon is greater than the circumference of the circle which is π. So, he arrived at the inequality 3 < π
In the following centuries, some people used Archimedes’ technique, using polygons with more sides, and arrived at more accurate estimates for π. Ptolemy (c. 150 AD) found the value up to 4 places, Zu Chongzhi (c. 500) 6 places, al-Khwarizmi (c. 800) 4 places, al-Kashi (c. 1430) 14 places, Roomen (c. 1580) 17 places, while Van Ceulen (c. 1600) arrived at 35 places.
Unfortunately, Archimedes’ idea was the only mathematically significant approach for almost 2,000 years. The second major step in this direction came with the Renaissance. The general progress in theoretical mathematics gave a great push to the computation of π. With the aid of calculus and infinite series Gregory found the following formula: arctan(x) = 1- x/3 + x3/5 - x5/7 +... He then plugged in x=1, arriving at π/4 = 1- 1/3 + 1/5 – 1/7 +...
This formula looks very nice to begin with, but unfortunately it is not very useful for the computation of π. To get the first 4 places right, we need 10,000 terms of the series, making it very untidy. This formula was considerably improved later by others, and went on to become very useful for computation. By using trigonometry, Machin found the following formula for π: π/4 = 4arctan(1/5) - arctan(1/239).
With Gregory’s expression for arctangent, this formula became so powerful that one could arrive at 5 places of π in just 6 terms. Again, with such a formula, the only problem left is that the computation is quite tedious. In the 18th and 19th centuries, Machin, Rutherford, Shanks, and others improved the calculation of π to 700 digits.
On the other hand, amazing facts were discovered about the nature of π. In 1761, Lindemann was the first to show that π is irrational, which means that it cannot be written as the ratio of two integers, like 22/7, 353/113, etc. Then he proved that it is transcendental, or “very irrational,” i.e. it cannot be a root of any integer coefficient polynomial. In other words, it cannot be A10,A2 + A3, 3A29, etc. These facts imply that π is a very irregular number, that there is no pattern in its decimal places, and that one cannot express π in a simple algebraic way.
So, by the beginning of 20th century, we were able to compute only 700 digits of π. In the first quarter of the last century, the brilliant Indian mathematician Ramanujan accomplished the third great theoretical leap in or the computation of π, and came out with many impressive infinite series. He arrived at sound algebraic expressions which are very close to π.
In the second quarter of the last century, computers came onto the scene. With the arrival of computers, pen and paper calculations were rendered obsolete. After Ramanujan’s time, the people calculating π became programmers rather than mathematicians. In 1955, more than 3,000 digits were calculated at the Naval Ordnance Research Center in only 13 minutes, almost 500 times faster than the ENIAC only 6 years later. In 1959 an IBM 704 calculated more than 16,000 digits of π. In 1961 an IBM 7090 calculated over 100,000 digits of π in around 9 hours, in 1966 an IBM 7030 calculated 250,000 digits of π, while a year later a CDC 6600 calculated 500,000 digits and in 1973 a CDC 7600 calculated 1,000,000 digits of π in 23 hours.
However, the techniques for calculating π were still using arctangents which have a quadratic growth rate. In 1976 Eugene Salamin rediscovered a formula developed by Gauss. It was calculation intensive in Gauss’ time, but well suited for modern super computers the size of the Whitehouse and had a much lower growth rate than the arctangents formulas did. In 1982, a HITAC M-280H calculated 16 million digits of π in 30 hours, in 1988 a Hitachi S-820 calculated 201 million digits in 6 hours, while in 1989 both the 500 million and 1 billion π calculation records were broken. In 1995, 6 billion digits were calculated, in 1996, 8 billion, and finally in 1997 a Hitachi SR2201 calculated 51 billion digits of π in 29 hours. This Hitachi machine had 1,024 processors and 212 gigabytes of RAM. However, in September of 1999 a new record of 206,158,430,000 was announced. The calculation was set by Yasumasa Kanada and the University of Tokyo. The calculations took over 37 hours, with 43 hours more needed to verify them. The machine used contained 817GB of main memory and consisted of 128 Hitachi SR8000 processors.
So, we can see that this mysterious number has attracted the attention of many people for centuries; it seems set to continue to do so. It might not seem very interesting to compute billions of digits of one transcendental number, and it might even seem pointless. The real point here is not the billionth digit of the number, but the excitement and the beauty of the problem, and the challenge to human intelligence, like with so many other mathematical problems.