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Pi is one of the important constants in mathematics. It refers to a mathematical constant representing the ratio of the circumference to the diameter of a circle, which is expressed by the Greek letter π. π is also equal to the ratio of the square of the area and radius of a circle, and the approximate value is about 3. 14 159265359, which is the key value for accurately calculating the geometric shapes such as the circumference of a circle, the area of a circle and the volume of a sphere. It is the first special constant recognized by human beings. There was a record of "three times a week" in ancient China. At that time, pi was considered to be constant. Since 1737, Euler has become a universal symbol after expressing pi.
Pi formula:
The cycle rate () is generally defined as the ratio of the circumference (c) to the diameter (d) of a circle, or directly defined as the half circumference of a unit circle. According to the properties of similar graphs, the value of is the same for any circle, thus defining a constant.
formula
Note: it is meaningful to define it as half the circumference of the unit circle, because from the perspective of modern mathematics, the circumference c of a circle with a diameter of d and a radius of r is given by the following integral:
formula
that is
formula
In the above formula, the substitution method of definite integral can obtain:
formula
Where is the circumference of the unit circumference (r= 1 in the expression of c). If defined, it is consistent with the well-known perimeter formula.
formula
While the area s of a circle with radius r is given by the following integral:
formula
Order, which can be obtained from the substitution method of definite integral:
formula
formula
Where is the area of the unit circle (r= 1 in the expression of s). Using component integration,
formula
formula
So,
formula
Therefore, we get the relation:
formula
In this way, the well-known formula of circular area is obtained. The second method is to make a square with the radius of the circle as the side length, and then set the ratio of the area of the circle to the area of the square as, that is, the ratio of the area of the circle to the square of the radius.
A brief history of pi calculation
1. 1? 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. [ 1]
1.2? 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.
There is a record in China's ancient book "Parallel Calculation of Classics in Weeks" (about 2nd century BC) that "Daoyi is Wednesday", which means to take it. Zhang Heng concluded in Han Dynasty (about 3. 162). Although this value is not accurate, 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 cut the circle into 1536 polygon, find out the area of 3072 polygon, and get a satisfactory pi.
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 excessive approximation of 3. 14 15927, and also got two approximate fractional values, namely density ratio and shrinkage ratio. Density is a good approximation of a fraction, and only by getting it can we get a slightly more accurate approximation. [2] (see Diophantine approximation)
In the next 800 years, the π value calculated by Zu Chongzhi is the most accurate. Among them, The Secret Rate was not obtained by the German Valentinus Osso until the West 1573, and was published in the works of the Dutch engineer Antoine in 1625, and was called Metis & apos；; in Europe; No. 10.
Around 530 AD, Indian mathematician Ayabata calculated that pi was about. 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.
1.3? 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, McKin's calculated π value exceeds the decimal mark of 100, and he uses the following formula: [3].
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.
1.4? 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
circumference ratio
Digital integrator and computer) are open 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.
Now, how many decimal places can be calculated has become an important index to measure the operation speed, memory capacity and overall ability of a computer.