Why is pi so difficult to calculate

The history of the number pi

The history of the number pi goes back thousands of years. Traces and references to Pi can be found in old papyrus rolls or carved in stone. Whether Egyptians, Babylonians or the ancient Greeks, they were all taken with the magic number π.

Detail from a papyrus roll in the British Museum

Pi in ancient times

Even the ancient Egyptians have about 2000 BC. Possessed quite useful approximations for pi. With a value of around 3.16, they were less than one percent off the correct numerical value of pi. Even in the Bible there is a passage from which an approximate value of 3 for pi can be read:

And he made the sea, poured, ten cubits from one edge of it to the other edge, round about and five cubits its height; and a measuring line of thirty cubits went round it

Eureka - Archimedes constant

The first real written derivation for pi goes to the Greek mathematician and physicist Archimedes (287-212 BC). In his honor and his work, Pi is also called "Archimedes constant" designated. Archimedes chose a geometric approach for the approximate calculation of PI.

He nested a unit circle (circle with radius 1) with regular polygons. Starting with a regular hexagon that once encompassed the circle and was once inscribed in it. Using the 12-, 24- and 48-corners, he finally got to the 96-corner. In this way he received a lower and an upper limit for the circumference and thus also for the number pi.

With that, Pi was already up 2 places after the comma precisely determined. Archimedes is said to have succeeded in nesting the limit values ​​more precisely.

Pi in the Middle Ages

The Middle Ages (5th-15th centuries) were probably the most boring chapter in the history of Pi. Especially related to Europe. Had in far China Liu Hui in the year 263 pi 5 digits after the comma calculated and succeeded around the year 480 To Chongzhi an improvement on 7 decimal places. In Europe, on the other hand, it takes around 1220 for Leonardo of Pisa, also known as Fibonacci, to use a method similar to Archimedes to determine Pi to 3 decimal places. We hadn't noticed anything about the Chinese developments here. Really poor. There was the achievement of the Persian astronomer Al-Khashi something outstanding, with the help of a 3 * 228 Ecks screwed the accuracy of pi to remarkable in 1430 16 decimal places up.

Pi in modern times

With the nesting of polygons a la Archimedes it succeeded Ludolph van Ceulen on the basis of a 2nd62-Eck the first 35 positions from PI to calculate. That earned him a lot of recognition and the nickname of the number pi for a long time Ludolph number a. The 35-digit sequence was only published posthumously in 1615 (or 1621).

Engraved on Ludolph van Ceulen's tombstone - Pi on 35 places

The Austrian Jesuit and astronomer took the hit when it came to calculating Pi by hand using the polygon method Christoph Grienberger, who with the help of a 1040 Ecks the circle number 38 places after the comma has calculated. The number was published in his mathematical work "Elementa Trigonometrica".

The designation with the Greek letter π for pi first appeared in the 17th century and was later picked up by the great mathematician Leonhard Euler and made popular through his publications. And ultimately it caught on worldwide.

The astronomer and mathematician John Machin, known for his formula named after him, succeeded in calculating pi in 1707 with the help of his formula 100 digits.

Johann Dase reached the in 1844 200 positions Border.

In 1855 Richter managed to get the number pi on 500 positions to calculate exactly. His record was replaced by William Shanks, whose record of 707 digits unfortunately turned out to be a fallacy decades later. Shanks had miscalculated from the 528th position.

Pi in the computer age

With the advent of the computer, movement came into play. In 1949 G. W. Reitwieser (USA) succeeded in calculating Pi on an ENIAC machine 2037 decimal places. From then on, the electronic arithmetic servants made the race in the competition for further Pi decimal places records. Since then, the human mind only provides the ideas, formulas and algorithms, while the computing work is done by the machines. It was better that way, as the story of William Shanks shows, whose error was not revealed until 1945 by D.F. Ferguson was discovered. Ferguson was helped by an electrical-mechanical desk calculator.

In 1958 F. Genuys made it with the help of John Machin's arctangent formula and an IBM 704 mainframe 10.000 he's cracking the mark.

On July 29th, 1969 the 100.000 he barrier, the IBM researchers Daniel Shanks and John W. Wrench needed an IBM 7090 and less than 9 hours of computing time.

In 1973, Guilloud and Boyer succeeded in calculating pi to over 1.000.000 Places after the comma. That was the time when the old arctangent series slowly reached their limits and new calculation methods had to be found. The fast FFT multiplication, new formulas and especially the Gauss AGM algorithm then brought a huge acceleration in the calculations of Pi.

So Yoshino Canada succeeded in 1982 10.000.000 Leaving the border behind him, together with his partner, he also got that in 1987 100.000.000 Brand. And it was Chudnovskys who did the in August 1989 Limit cracked.

The Pi places world record is currently blatant 50 trillion jobs. Published on January 29th, 2020 by the American Timothy Mullican. In doing so, he surpassed the old record set by the search engine giant Google, which had an accuracy of 31,415,926,535,897 digits after the decimal point.

Source: Papyrus / @ British Museum - Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)