Pi

Number pí (greek sign pí, π , according to greek periféreia, perimeter) expresses ratio between the perimeter of a circle (in plane) and its radius. This ratio is the same for all  circles, approximately 3,14159 26535 89793 23846 26433 83279… and this number π  is also called  Ludolf’s number. Ludolf‘s number is designated to Ludolf van Ceulen, who calculated Ludolf‘s number with accuracy of  35 decimals. A lot of mathematical and physical patterns and formulas use the number π .

This number was already known in ancient times. In Babel imperium (2000 B.C.) the approximite figure used was ( 25/8). Egyptians at those times, used the figure 22/7 (3,14285).

Builders of pyramids had a different way of measuring height and distance on Earth. Height was measured by a stick of defined lenght (king’s elbow). On the other hand, distance was measured by rolling a wheel with radius of the ‚king’s elbow‘. Then, Ludolf’s number arises as a quotient of height of a pyramid and the lenght of side of its base without them even thinking of Ludolf’s number. The Indian text Shatapatha brahmana defines π  as 339/108 which is 3,139.

The first person to estimate the figure π close to its real value was Archimedes. He found out that a circle, which is inscribed and outscribed to two polygons at the same time is equal to their radius and diameter respectively, and when he used a 96-agon he determined that: 223/71 < π < 22/7.

After this determination, people tried to find out the exact value of π until 480 in China. It was a Chinese mathematician Zu Chongzhi who calculated π as 335/113 and showed that 3.1415926 < π < 3.1415927. It was the most accurate value of π for the next 900 years. There were only ten decimals of π known until the second millenium. In 1400, another big step in the studies of π came about when Madhava of Sangamagrama used these progressions for the first time:

which is the same as:

Madhava was able to calculate π as 3.14159265359 (11 decimals). This „record“ was broken in 1424 by the Persian astronomist Jamshid al-Kashi, who determinated π to 16 decimals.

We can say that in the new age, the most famous mathematican in this area was in 1615 a Dutch university profesor in Leidene – Ludolph van Ceulen, who calculated this figure to 35 decimals. He was so proud of his calculations that he had his „numbers“ written on his tombstone. At these times, in Europe, attempts to determine π began. The first was Viete‘s formula, found in 1593 by Francois Viete :

Another famous pattern was discovered by John Wallis:

It was only in 1655, when another mathematican, who tried to determine value of π was John Machin and he calculated it to 100 decimals:

with another formula:

These types of pattern are called Machin‘s formulas and were used in a lot of succesful calculations and they still stay the best way of calculation of π in the times of computers. A great achievement of Zacharius Dase , who ,in 1844, used Machin’s formula to calculate 200 decimals of  π  in his head is famous. However the „record holder“ in number of decimals is Wiliam Shanks, who calculated these numbers for 15 years. He determined 707 of them but he made a mistake and only 527 of them were correct.

In 1761, Johann Heinrich Lambert proved that this number is irrational, in 1794 a mathematican Adrien-Marie Legendre proved that π² is also irrational and a German mathematican Linderman later on (in 1882) proved that it is transcendential. When Leohnard Euler in 1735 pointed out Basel‘s trouble to find an exact value

which is  π ²/6, he proved a deeper connection between π and the prime numbers. Both of them, Legendre and Euler, agreed that π is a transcendential number.

Transcendentiality means that number is not a root of any algebric equation with whole coefficents. A theory about the normality of number π, which however has not been proved yet is also interesting.. Normality of a number pertains to how often a numeral or the group of numerals appears in the infinite order after the decimal mark. With statistic analysis of the first ten million of numerals in the decimal order of π was discovered that numerals are spread in order, which agrees with the theory of normality.

At the beggining of 20th century, when the first computers came to use, John von Neuman used the computer ENIAC to calculate 2037 decimals in 1949, caclculations took 70 hours. To fasten up the calculations the so-called „Fast Fourier Transform“ (FFT) was used, which let the computers to calculate arithmetic equations with long numbers extremely fast.. In 1958 F. Genuys got 10 000 numerals on IBM 704 and in 1961 D. Shanks with his colleagues determined 100 000 decimals on IBM 7090. A million decimals were calculated by J. Gilloud and M. Bouyer in 1973 on the computer CDC 7600 in less than a day. All of these calculations were based on  cyclo – metrical function arctg x and formulas, which were used by  J. Machin.

The next era of calculating number π with the help of computers was the usage of elegant Brant‘s and Salmin’s algoritms, which are shortening time needed for calculation dramatically. This way Y. Kanada and colleagues in 1984 calculated in 30 hours 16 000 000 numerals. Usage of integrated algoritm in 1987 provided Y. Kanada with the desired outcame of 201 326 000 decimals using a super computer Hitachi. This computing effort took only 6 hours. Apart from that, other theoretical results were obtained. However it was not the last word. Brothers David and Gregory Chundovsky, both from the University of Columbia in New York in 1989 got over a billion of numerals, exactly 1 011 196 691 decimals after the decimal mark in number π. It is interesting that authors used the idea of a legendary Indian mathematician S. I. Ramanujan from 1914

Ludolf’s number p is a mathematical root, which has a huge influence and is used in a lot of different areas of math, physics and other technological sciences. Number p is fascinating amateur mathematicians and arithmeticians, some of them are trying to figure out the problem in calculation of area of circle and triangle. Number p is so mysterious and secret that it even appears not to be from our world.

In the conclusion we would like to indicate that Simon Plouffe with two of his colleagues from the University of Simon Fraser in Canadian British Columbia made up mathematical procedure that enables us to determine the decimals of number p, without knowing the numeral on the decimal on the position before. This procedure was improved a few times so we can get even more decimals using super computers. One of them, in 2002 determined Ludolf‘s number exactly to 1241 billion decimals. However even this amount is definitely not finite.

p is a really interesting number even these days.