I. Trial period
An ancient Babylonian stone tablet (about BC 1900 to BC 1600) clearly recorded that pi = 25/8 = 3. 125. The mathematical papyrus of Rhind, an ancient Egyptian cultural relic of the same period, also shows that pi is equal to the square of the score 16/9, which is about 3. 1605.
Egyptians seem to have known pi earlier. John tyler (1781–1864), a British writer, pointed out in his masterpiece Pyramid that the pyramid of khufu built around 2500 BC was related to pi. For example, the ratio of the circumference to the height of a pyramid is equal to twice the pi, which is exactly equal to the ratio of the circumference to the radius of a circle.
The Brahma Book, an ancient Indian religious masterpiece written from 800 to 600 BC, shows that pi is equal to the score of 339/ 108, which is about 3. 139.
Second, the geometric method period
As an ancient geometric kingdom, ancient Greece made great contributions to pi. Archimedes (287–2 BC12), a great mathematician in ancient Greece, initiated the theoretical calculation of the approximate value of pi in human history. Starting from the unit circle, Archimedes first found that the lower bound of pi was 3 by inscribed regular hexagon, and then found that the upper bound of pi was less than 4 by pythagorean theorem.
Then, he doubled the number of sides of inscribed regular hexagon and circumscribed regular hexagon to inscribed regular hexagon 12 and circumscribed regular hexagon 12 respectively, and then improved the upper and lower bounds of pi with the help of Pythagorean theorem. He gradually doubled the number of sides inscribed with regular polygons and circumscribed with regular polygons until inscribed with regular polygons and circumscribed with regular polygons.
Finally, he found that the upper and lower bounds of pi were 223/7 1 and 22/7, respectively, and took their average value of 3. 14 185 1 as the approximate value of pi. Archimedes used the concepts of iterative algorithm and bilateral numerical approximation, which is the originator of computational mathematics.
In China's ancient book "Parallel Calculation of Classics in Weeks" (about 2nd century BC), it is recorded that "Dao is one and Wednesday", which means π=3. Did Zhang Heng get π in Han Dynasty? Divided by 16 is about five eighths, that is, π is about equal to the root number ten (about 3. 162). This value is not accurate, but it is easy to understand. ?
In 263 AD, China mathematician Liu Hui used the secant method to calculate pi. He first connected a regular hexagon from the circle and divided it step by step until the circle connected a regular hexagon 192. He said, "If you cut carefully, you will lose very little. If you cut it again, you can't cut it. Then you will be surrounded and there will be no loss. " , contains the idea of seeking the limit.
Liu Hui gave an approximate value of pi =3. 14 1024. After Liu Hui got pi = 3. 14, he checked this value with the diameter and volume of Jia Lianghu, a copper system made in the Han and Wang Mang dynasties in the gold armory, and found that the value of 3. 14 was still small. Then continue to tangent to the polygon 1536, find out the area of the polygon 3072, and get the satisfactory pi of 3927 divided by 1250, which is about 3. 14 16.
Around 480 A.D., Zu Chongzhi, a mathematician in the Northern and Southern Dynasties, further got the result accurate to seven decimal places, gave the insufficient approximation of 3. 14 15926 and the surplus approximation of 3. 14 15927, and also got two approximate fractional values, that is, the density of 355 divided by/kloc. The density ratio is a good fractional approximation. Only by dividing 52 163 by 16604 can we get a slightly more accurate approximation than dividing 355 by 1 13.
In the next 800 years, the π value calculated by Zu Chongzhi is the most accurate. In the west, the secret rate was not obtained by German Valentinus Osso until 1573, and it was published in the work of Dutch engineer Antoine in 1625, and it was called Metis in Europe.
Around 530 AD, Indian mathematician Ayabata calculated the root sign of pi as 9.8684. Brahmagupta used another method to derive the arithmetic square root of pi equal to 10.
17 At the beginning of the 5th century, the Arabic mathematician Cassie got the exact decimal value of pi17, which broke the record kept by Zu Chongzhi for nearly a thousand years. German mathematician ludolph van ceulen calculated the π value to 20 decimal places in 1596, and then devoted himself to it all his life, and calculated it to 35 decimal places in 16 10, and named it Rudolph number after him.
Third, the analysis period
During this period, people began to use infinite series or infinite continuous product to find π and get rid of the complicated calculation of secant. Various expressions of π value, such as infinite product, infinite continued fraction and infinite series, appear one after another, which makes the calculation accuracy of π value improve rapidly.
The first fast algorithm was put forward by the British mathematician John McKin. In 1706, Machin's calculation of π value exceeds 100 digits after the decimal point. He used the following formula: π/4 = 4 arctan1/5-arctan1/239, where arctan x can be calculated by Taylor series. A similar method is called "McKinley formula".
1789, the Slovenian mathematician Jurij Vega got the first 140 digits after π decimal point, of which only 137 digits were correct. This world record has been maintained for fifty years. He used the number formula proposed by Mei Qin in 1706.
By 1948, both D. F. Ferguson in Britain and Ronchi * * in the United States had published the 808-bit decimal value of π, which became the highest record for manually calculating pi.
Fourth, the computer age.
The appearance of electronic computer makes the calculation of π value develop by leaps and bounds. 1949, the world's first American-made computer ENIAC (Electronic Numerical Integrator and Computer) was put into use at Aberdeen proving ground. The following year, Ritter wiesner, Von Newman and Mezopolis used this computer to calculate the 2037 decimal places of π.
It took the computer only 70 hours to finish the work. Deducting the time of punching in and out is equivalent to calculating single digits in two minutes on average. Five years later, IBM NORC (Naval Weapons Research Computer) calculated the 3089 decimal places of π in only 13 minutes.
With the continuous progress of science and technology, the computing speed of computers is getting faster and faster. In the sixties and seventies, with the continuous computer competition among computer scientists in the United States, Britain and France, the value of π became more and more accurate. 1973, Jean Guilloud and Martin Bouyer discovered the millionth decimal of π with the computer CDC 7600.
1976 has made a new breakthrough. Salamin published a new formula, which is a quadratic convergence algorithm, that is, after each calculation, it will be multiplied by the significant number. Gauss had found a similar formula before, but it was so complicated that it was not feasible in the era without computers.
This algorithm is called Brent-Salamin (or Salamin-Brent) algorithm, also known as Gauss-Legendre algorithm.
1989, researchers at Columbia University in the United States used Cray-2 and IBM-3090/VF giant computers to calculate 480 million decimal places of π value, and then continued to calculate to10/100 million decimal places. 10/7-The French engineer Fabrice Bellard calculated pi to the nearest 2.7 trillion decimal places.
20 10 August 30th-Japanese computer genius Mau Kondo uses home computers and cloud computing to calculate pi to 5 trillion decimal places.
20 1 1, 10, the staff of Iida City, Nagano Prefecture, Japan used their home computers to calculate pi to 10 trillion digits after the decimal point, setting a Guinness World Record of 5 trillion digits created by themselves in August of 20 10. 56-year-old Mau Kondo used his own computer to calculate from June+10 in 5438, which took about 1 year, setting a new record.
Extended data
The sign of pi: π is the lowercase of the 6th Greek letter/kloc-0. The symbol π is also the first letter of Greek π ε ρ φ ρ ε ρ α (meaning periphery, region, circumference, etc. ).
1706, the British mathematician William Jones (1675- 1749) first expressed pi.
1736, the great Swiss mathematician Euler also began to express pi. Since then, π has become synonymous with pi.
Be careful not to mix π with its capital π, which means multiplication.
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