1, estimation of -pi in experimental period;
An ancient Babylonian stone tablet (about BC 1900 to BC 1600) clearly recorded that pi = 25/8 = 3. 125. Rhind papyrus, an ancient Egyptian cultural relic of the same period, also shows that pi is equal to the square of score 16/9, which is about 3. 1605. Egyptians seem to have known pi earlier. ?
British writer john tyler (1781–1864) wrote in his masterpiece The Great Pyramid: Why was it built and who built it? ) It is pointed out that the pyramid of khufu built around 2500 BC is 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 Brahman of Sa tabata, an ancient Indian religious masterpiece written from 800 to 600 BC, shows that pi is equal to 339/ 108, which is about 3. 139.
2. Geometric method period-the calculation of pi began to take the initiative and became scientific;
(1) As an ancient kingdom of geometry, ancient Greece made a particularly outstanding contribution 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.
(2) There is a record in China's ancient book "Zhou Bi Shu Jing" (about 2nd century BC) that "Jing Yi and Wednesday" means to take it.
During the Han Dynasty, Zhang Heng concluded that
that is
(about 3. 162). This value is not accurate, but it is easy to understand.
(3) In 263 AD, China mathematician Liu Hui used the secant method to calculate pi. He first connected the regular hexagon from the circle, and then divided it step by step until the circle connected the regular hexagon of 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 cut the circle into 1536 polygon, find out the area of 3072 polygon, and get a satisfactory pi.
(4) Around 480 AD, Zu Chongzhi, a mathematician in the Northern and Southern Dynasties, further got the result accurate to 7 decimal places, giving the insufficient approximation 3. 14 15926 and the surplus approximation 3. 14 15927, and also getting two approximate fractional values, density ratio.
Peace treaty rate
The secret rate is a good approximation of the score, so it is necessary to get it.
Get a ratio
A slightly accurate approximation.
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.
(5) Around 530 AD, the Indian mathematician Ayabata calculated that pi was about
Brahmagupta used another method to derive the arithmetic square root of pi equal to 10.
(6)1At the beginning of the 5th century, the Arabic mathematician Kathy got the accurate decimal value of pi 17, 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.
3, analysis period-scientific deduction of pi:
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, McKin's calculated π value exceeds the decimal mark of 100, and he uses the following formula:
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.
4, the computer age-scientific and efficient calculation of pi:
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:
1, pi day International:
20 1 1 year, the International Mathematical Association officially announced that March 14 every year would be designated as the International Mathematics Festival, which originated from the pi of Zu Chongzhi, an ancient mathematician in China. ?
Pi day can be traced back to1March, 98814th, a physicist at the Science Museum in San Francisco, Larry Shaw. He organized museum employees and participants to make 3 and 1/7 laps around the museum monument (22/7, one of the approximate values of π) and eat fruit pies together. Later, the San Francisco Science Museum inherited this tradition and held a celebration on this day every year.
In 2009, the US House of Representatives formally passed a non-binding resolution, which designated March 14 as "pi day" every year. The resolution holds that "because mathematics and natural science are interesting and indispensable parts of education, and learning π is a fascinating way to teach children geometry and attract them to learn natural science and mathematics ... π is about 3. 14, so March 14 is the most appropriate day to commemorate pi day."
2, the application of pi in various disciplines:
(1) Geometry:
(2) Algebra:
π is an irrational number, that is, it cannot be expressed as the ratio of two integers, which was proved by Swiss scientist johann heinrich lambert in 176 1. In 1882, Lin Deman proved that π is a transcendental number, that is, π cannot be the root of any integer coefficient polynomial.
The transcendence of pi denies the possibility of turning a circle into a square, because all rulers can only draw algebraic numbers, and transcendental numbers are not algebraic numbers.
(3) number theory:
The probability that two arbitrary natural numbers are coprime is
Take any integer, and the probability that the integer has no repeated prime factor is
On average, any integer is available.
This method is written as the sum of two perfect numbers.
(4) Probability theory:
Suppose we have a floor paved with parallel equidistant wood grains, throw a needle with a length less than the spacing of wood grains at random, and find the probability that the needle intersects one of the wood grains. This is Buffon's needle throwing problem. 1777, Buffon solved the problem himself-the probability value is 1/π.
(5) Statistics:
Probability density function of normal distribution;
(6) physics:
Heisenberg uncertainty principle;
Field equation of relativity;
Baidu encyclopedia -pi