It was on one of these sunny days that I had a guest. He was a generous young man. Soon after he came nearby, he served his blood to the mosquitos. Considering all the things humans pick from nature, their blood is the fruit that nature can pick in return, right? Anyway, the reason that I called this guest generous is that he did not kill the mosquitos while they were sucking his blood. He sat near me, and started pondering over the nice valley below. I didn’t know at first if I should talk to him. The snail creeping on me preferred to keep silent too. So I decided not to speak to him.

Soon after, a friend of this young man arrived. They started talking about finding water here. I thought “How do you think water can be found at this height of the hill?” But they started digging with the special equipments they had. I had heard about these things when a group of students had come on a field trip. They talked about the different kinds of rocks and the minerals in them, their story of formation through the depths of the earth or under deep water, and how one could find invaluable fluids by digging through them. Seeing these two young men, I better understood what they meant.

Anyway, after a few hours of extreme noise, I felt the wetness that touched me from below. The water was indeed rising in the well the two young men had made, and soon they were screaming from joy. As they indulged in their celebration, I was engulfed by that uncomfortable feeling I had when they’d arrived. After all, this was my place, and they had invaded it.

As if this invasion wasn’t enough, the man who had arrived first stepped on me and spread a cloth. I was enraged at the thought of this man celebrating his victory over me. But my rage stumbled upon what I saw as I watched him. He started offering thanks and praying. It was then that a sense of warmth seeped between us, and I began talking to him everytime he came. During his visits, he used to drink some fresh water, sit on me looking towards the valley, and delve into reflections and prayers. I thanked God for giving me this flat surface so that someone could pray on me. Our relationship became so special that when he was not around, I used to pray for his return.

Years passed by. Things didn’t change much with me, but the man changed a lot. His once smooth face now looked like the wavy ocean surface. His body was bent like trees on steep hills. Things were changing incredibly fast for humans. I felt pity for this man who was troubled by his daily affairs. I prayed that I could be closer to him to help relieve his pains. My prayers were accepted and this respectable man spent more and more time on me. Sometimes lying, sometimes sitting, but always contemplating or praying.

At times, he brought his grandchildren. These kids did not enjoy being here as much as their grandfather. They constantly kept looking at the glasses in their hands that entertained them. But sometimes, they had funny talks with their grandfather. For example, this one time they talked about putting the stones in water or burrying them in the soil so that they can grow like plants. When I heard those words, I thought about my genesis through the ocean beds and the inferno compartments in the earth. My life story was so naively and elegantly put into those few words of the children. It is curious how the little ones can formulize complex things so easily.

Those times of amusement used to make me think about myself. I was a mature rock, and I wanted to have youngsters around me. I prayed to have little ones that were eager to “rock ‘n’ roll.” But perhaps those little ones would not show up as soon as they did for the humans, and they would not be as rambunctious as their human peers. Anyway, with my innate desire, I kept praying to have them around.

Strange enough, my prayers were answered sooner than I thought, and I started seeing these little pebbles parting from me. In my maturity, I had the blessing of giving birth to children all around me. Now, whenever the old man came with his grandchildren, they played with mine, and that gave me immense pleasure.

At last one day, I saw a group of people coming near by with a wooden box on their shoulders. Yes, that was him, brought to meet his Lord… He was burried in the cemetery at the peak of the hill.

…

You know, it’s not easy to lose someone you love. It’s feels like burying a part of your heart before you actually die. I cried a lot, and I still do, silently, in my own way. I don’t want my sorrow to assume the form of a complaint against the decree of our Lord. Yet, I cry nonetheless. Maybe this grief is what has made me so weak nowadays. I am feeling that my end is near, too. The increasing number of little ones all around is making this message impossible to ignore. But at the same time, I am developing a hope of continuity; because the water that is springing under me penetrates into my pores, and water means life.

However, the peace that comes with this hope is always followed by the grief of losing my friend, and I am crying all my water out. To quench my sorrow, God sent me lots of people. They are all around, now that my tears are making a waterfall. But I’d rather have my friend pray on me than these people laugh with negligence at my sorrow. “Oh my Lord, my friend in the heavens… Oh my Lord, my friend in the heavens…”

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.

References

• J. J. O’Connor and E. F. Robertson, www.gap.dcs.stand.ac.uk/history/HistTopics /Pi_through_the_ages.html
• D. Hazeghi, www.myownlittleworld.com/pi /history.html
• D. H. Bailey, J. M. Borwein and P. B. Borwein, www.cecm.sfu.ca/organics/papers/borwein /paper/html/paper.html
• Lazarus Mudehwe, www.geocities.com/CapeCanaveral /Lab/3550/pi.htm
• www gap.dcs.stand.ac.uk/history/HistTopics/Pi_chronology.html