Thursday, July 07, 2005

It Began in Babylonia (currently sothern Iraq)

It Began in Babylonia (currently sothern Iraq)

In 2002, a Japanese professor of the University of Tokyo calculated pi (the ratio of the circumference of a circle to its diameter) to 1.24 trillion decimal points using a Japanese supercomputer, which is of course extremely remarkable.

In 2005, a Japanese psychiatric counselor has recited pi to 83,432 decimal points, speaking from his memory, which is also globally remarkable.

In the 17th century, when Isaac Newton calculated pi to 16 decimal points, a Japanese mathematician and samurai Takakazu Seki also calculated pi to 16 decimal points independently in Japan.

Seki’s disciple Katahiro Tatebe calculated pi to 41 decimal points in 1722, which was reported in his book where he also presented (arcsin x) * (arcsin x) in a series presentation which was regrettably recorded as and has been believed today to be Euler’s discovery in 1737 in Europe.

There is surely a reason why Seki and Tatebe could show such expertise in then most worldly advanced mathematical skills. Though Japan had closed its door to the outer world from the early 17th century to the middle of the 19th century, part of Western science was introduced to Japan by some Westerners, especially Jesuit priests who had risked their lives to enter Japan and preach the gospel. Their missions were mostly blocked and eradicated by a samurai regime but their knowledge of European science was secretly respected.

Therefore, long before Japan opened its door to the world in 1868, it had already reached a high level of modern mathematics, at least, in terms of practical calculation even in the field of differential and integral calculus. In addition, a handy manual calculator, an abacus, was a very common tool among ordinary people in Japan since 17th century.

After Japan's opening the country, many Western professors were hired by Japanese government doing its best to catch up with Western civilization. Some professors taught Japanese students the theory of differential and integral calculus. They found that the theory seemed to be somehow new to their Japanese students but the students could calculate equations and produce right answers based on their traditional mathematical skill.

So, it is no wonder that Japanese mathematicians had contributed much to the solution of Fermat's last theorem (X^n + Y^n = Z^n; X = Y = Z = 0 if n >2), which was achieved by the English man Andrew Wiles in 1995.
(Still other Japanese mathematicians made significant contributions to understanding one of the most important problems of the last 300 years: the proof of Fermat's last theorem. In this endeavor, Yutaka Taniyama and Goro Shimura helped lay the groundwork that led to Andrew Wiles's (1953- ) 1994 proof of the validity of Fermat's last theorem. The Shimura-Taniyama conjecture, which describes the relationship between certain families of curves, proved key to solving this theorem.
However, the issue here is Pythagoras. Whose famous theory has been traditionally a key to obtaining a series expression of arctan x, a key function to calculate pi (= 4 * (1 – 1/3 + 1/5 – 1/7..) ).

Pi is a transcendental. This number cannot be expressed in the form of M/N using two integers. Also, the number x that satisfies the equation x * x = 2 is a transcendental.

In the age of Buddha, Confucius, and ancient Jew’s Babylonian captivity, such number as the square root of two was hard to comprehend. Pythagoras who loved integers so much was very unhappy to know existence of the type of numbers; thus he made it a secret kept only to his sect.

It is said that one of Pythagoras’ disciples who made public the existence of the transcendental was put to death by the sect. It is also said that Pythagoras himself was later excused by opponents to his sect.

It is a story of 2500 years ago. But, even today, some powerful people who love super-integers so much might try to make secret the very existence of a super- transcendental in any fields. For example, a leader of one religious sect might find such a thing in other religions, but he might be very unhappy. He might rather prefer to make it secret, which might create dangerous prejudice even in the 21st century.

Ancient Babylonians were great in that they left the oldest record of a value of pi, though there is a theory that very ancient Romans had recorded it prior to Babylonians’. In BC2000 or so, Babylonians found that pi = 25/8 = 3.125, while Egyptians found that pi = 256/81 = 3.1605. Their offspring should be treated more respectfully if our civilization still relies on pi.

There is long history in anything valuable. And you cannot make it secret. You had better recite it from memory to show your love to the transcendent existence of the Creator of Universe.