Chudnovsky Formula To Compute 960 example essay topic

π is the mysterious number that most people think is merely 3.14. π is the ratio of a circle's circumference to its diameter. π has been calculated to 206,158,430,000 digits, which was accomplished by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo in September of 1999. On the other hand, there were the Babylonians and Hebrews who lived and died, believing that π was simply 3. Evidence from the Rhin d papyrus shows that the Egyptians knew 3.16, but implied in their Great Pyramid is the even better value 3.14. Much like the Babylonians, the early Hindus and Chinese accepted 3.

Archimedes led the Greeks to believe that the limits of π were 3.141 π 3.143. The later Greeks, with the knowledge of Ptolemy, knew 3.1416. In the 5th century A.D. Tsu Ch " ung-chih, a Chinese mechanician, gave π correct to six decimals. This feat was not paralleled in the western world for 1000 years.

Another common belief was that Π was the square root of ten, which was somehow equal to 3 1/7. A little known verse from the Old Testament leads us to believe that the Hebrews used 3 as their value of Pi". And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about, (I Kings 7, 23)". There is some speculation involved in this passage, as some say that the diameter of ten cubits was the measurement from one side to the next, whereas the thirty-cubit circumference was of the inner lining of the vessel. If the sides were about 0.2254 cubits thick, this would lead to decent value of pi. Around 225 B.C., Archimedes obtained his incredibly accurate limits for π with the perimeters of inscribed and circumscribed polygons having 96 sides.

This was an extremely difficult task for Archimedes, as he didn't have Arabic numerals to work with. Lacking Arabic numerals, he had much difficulty in computing the essential square roots square roots. Tsu Ch " ung-chih's method is not known for sure, but may have been similar to Archimedes'. Following the path of polygon usage, in 1579 Francois Vieta used polygons of 393,216 sides to obtain π to 9 correct places. This style came to an end in 1610, when the German eccentric Lu dolph van Ceulen calculated π to 35 places by means of a polygon with 4 quintillion, 611 quadrillion, 686 trillion, 18 billion, 427 million, 387 thousand, 904 sides. Van Ceulen worked himself to death on this computation, at which point the 35 places he had calculated were inscribed on his tombstone.

In honor of this poor man, Germany still refers to π as the Ludolphian number sometimes. In 1593, fourteen years after his polygon attempts, Francois Vieta discovered the first non-geometrical method for finding pi, in the form of an infinite continued product involving 1/2 and the square root of 1/2. By the middle of the 17th century, infinite processes like Vieta's began to dominate the pi-computing field. The simplest series of all was given by Leibniz in 1682, (π /4) = 1 - 1/3 + 1/5 - 1/7 +... Predating Leibniz's eries was the discovery by James Gregory in 1670. Leibniz's eries is just a special case of this series, which was probably the most important series in the history of π : arctan x = x - (x 3/3) + (x 5/5) - (x 7/7) +...

Next, John Machin arrived on the scene. Machin was an English mathematician and professor of astronomy at Gresham College in London. Around 1700 he established his important formula: (π /4) = 4 arctan (1/5) - arctan (1/239). Machin's formula, together with Gregory's series, paved the road for almost every π thinker from the 18th century until the beginning of the computer era. Over the next couple hundred years, at least nine different men calculated the value of π over 100 places. Then we reach the anomaly: Zacharias Dase.

In only two months of 1840 the sixteen-year-old German calculated π to 200 places. Perhaps the most phenomenal mental calculator who ever lived, Dase once obtained the product of two 8-digit numbers in less than a minute, working in his head. Dase died at the age of 37 however, spending most of his life factoring and multiplying. Moving down the line of π calculators, we reach William Shanks. Shanks calculated π to 707 places! He started in 1850, when he was 38 years old, and in three years was just 100 places away from his 707.

He published his results of pi, to the 607th digit, and put it in the public eye. Over the next 20 years, he calculated π to this historic 707th place. Shanks was recognized as one of the most fantastic computers of his time. That was, until his mistake was discovered. In his calculation, Shanks made an error in the 528th place of his result. This was overlooked for years, until 1946 when Dr. D.F. Ferguson made a calculation of his own.

Ferguson used the modern formula (π /4) = 3 arctan (1/4) + arctan (1/20) + arctan (1/1985). It is a popular belief, and in the past was even more so, that π will eventually come out even, or that the digits will repeat themselves. It wasn't until 1882 that π was proven as irrational, not to mention transcendental. This was proven by Ferdinand Lindemann.

This is what led to the realization of Shanks' miscalculation. If the digits in the expansion of π occur at random and are not repetitive, this means that each digit 0-9 should appear about the same amount of times. However, in Shanks' computation of Pi, the number seven appears far fewer than any other number. Shanks' mistake in the 528th place created an error that produced less 7's. Since then, π has been restored, and passes all tests for randomness. The cool thing about π comes out now.

If it is true, and π is a completely random number, than every string of numbers must occur somewhere in π . The string of 123456789 is just as likely as 314159265. This means that my birthday, your birthday, every lottery number, and even my social security numbers are all somewhere in π . If you go with the basic code that a = 1 b = 2 c = 3 and so on, with zero as a space, anything and everything is in the long code of π . Because π is completely random, this whole paper is probably encoded somewhere along the long line of π . I could waste the rest of my time trying to decipher it and find it, but I think writing some more would be more entertaining, and prove itself much more rewarding.

At least in the immediate future. We " ve come an extremely long way in the calculations of π since the days of Shanks or Dase. I just asked Mathematica to compute the first 100,000 digits of π and it took a whopping... four seconds. That kind of puts Shanks to shame if you think about it.

Mathematica uses the Chudnovsky formula to compute π up to ten million digits. This formula comes from the New York's Chudnovsky brothers, who have computed 2 billion digits of π on a homebrew computer. The current record for memorization of π's digits belongs to Hiroyuki Goto, who required over nine hours to recite 42000+ digits. Since the days of the Machin formula there have been many more developments in calculating π . David Bailey, Peter Bor wein, and Simon Pouffe have come up with an exceptionally astonishing formula for computing any given hexadecimal (or binary) digit of pi without first computing the preceding digits. Before this, no one had even considered the possibility.

One of the simplest ways to calculate π is using Gregory's formula with Machin's formula. This method gets you about 1.4 correct decimal digits for each term of the series that you calculate. Another method is the Gauss-Legendre method, which yields about double the correct decimals per term. Then there is the first Ramanujan formula that comes from the theory of complex multiplication of elliptic curves. This is a linear formula, and the best part is that it yields about 14 correct decimals for every term you calculate. The Indian mathematician Srinivas a Ramanujan discovered this method, and a few others.

The Buffon's Needle is yet another method of getting π , which is much less effective and accurate, but a bit more fun. The idea is very simple. You take a level surface with a grid consisting of equally spaced parallel lines. Then, you take a pin or needle, of a certain length.

You then drop the needle on the surface, and two things can happen. Either the needle is touching or crossing a line, or it doesn't touch a line. You repeat this method and record how many times it touches or doesn't touch a line. So you record the number of random drops, and the number of times it touches a line. Amazingly, if you multiply the number of drops by two and divide the quantity by the number of times it crosses a line, this value approaches π ! This works because the probability that any given needle will cross a line is 2/π .

After numerous trials, the value of crossing needles divided by total needles approaches this probability. So π has gone from being 3, to the square root of ten, to an irrational transcendental 206,158,430,000-digit number that is still being worked on. What it was in the beginning, it still is now; the ratio of a circle's circumference to its diameter. π is a valuable entity in mathematics today, and most people don't understand the intricacy of it. Without the discovery and knowledge of π it would be extremely difficult to perfect things such as the wheel, or to measure true quantities, or study the galaxy. π has left its mark on the human race, especially the few of us who are mathematically inclined.

This is just the beginning of my journey with π .