Pi is a very famous number. This figure has aroused the interest of laymen and scholars since it was recorded in writing. Pi, as a very important constant, was originally used to solve the problem of calculating circles. Based on this, it is an extremely urgent problem to get its approximate value as accurately as possible. This is also the case. For thousands of years, as the goal of mathematicians, generations of mathematicians at home and abroad have devoted their wisdom and labor to this. Looking back at history, the process of human understanding π reflects one aspect of the development of mathematics and computing technology. To some extent, the study of π reflects the mathematical level of this region or era. Cantor, a German mathematician historian, said: "The accuracy of calculating pi in a country in history can be used as an indicator to measure the level of mathematical development in this country at that time." Until the beginning of19th century, finding the value of pi should be said to be the number one problem in mathematics. In order to get the value of pi, human beings have gone through a long and tortuous road with an interesting history. We can divide this calculation process into several stages.
experimental period
It is the first step to estimate the value of π through experiments. This estimation of π value is basically based on observation or experiment, and is based on the actual measurement of the circumference and diameter of a circle. In the ancient world, the value π = 3 was actually used for a long time. The earliest written record is a chapter in the Christian Bible, in which pi is considered to be 3. The event described in this passage took place around 950 BC. Other countries, such as Babylonia, India, China, etc. , have long used the rough, simple and practical value of 3. Before Liu Hui in China, "Circle Diameter One and Wednesday" was widely circulated. China's first book, The Classic of Weekly Parallel Calculations, recorded the conclusion that the diameter of a circle on Wednesday is one. In China, carpenters have two formulas handed down from generation to generation: they are called: "The diameter of three sides is one, the square is five, and the diagonal is seven", which means that the circle with the diameter of 1 is a square with a circumference of about 3 and a side length of 5, and the diagonal length is about 7. This reflects the early people's rough estimation of π and √2 irrational numbers. During the Eastern Han Dynasty, the government also clearly stipulated that pi should be 3 as the standard for calculating the area. Later, people called it "ancient rate".
Early people also used other crude methods. For example, in ancient Egypt and ancient Greece, the grain was placed on a circle, and the value was obtained by comparing the number of grains with the number of squares. Or saw it into a circle and a square with a balance weight board, and compared the values by weighing ... Therefore, a slightly better pi value can be obtained. For example, the ancient Egyptians used 4 (8/9)2 = 3. 1605, which took about 4,000 years. In India, in the 6th century BC, π= √ 10 = 3. 162. At the turn of the Eastern and Western Han Dynasties in China, the new dynasty Wang Mang ordered Liu Xin to make a measuring instrument-the two lakes in Lv Jia. Liu Xin needs the value of pi in the process of manufacturing standard containers. To this end, he also obtained some non-uniform approximations about pi by doing experiments. Now, according to the inscriptions, the calculated values are 3. 1547, 3. 1992, 3. 1498 and 3.438+0, which are higher than the ancient three-week rate. The results of human exploration have little influence on production when estimating the round field area, but they are not suitable for making utensils or other calculations.
Geometric method period
The experimental method of calculating π value by intuitive speculation or physical measurement is quite rough.
First of all, Archimedes provided a scientific basis for the calculation of pi. He was the first person to make scientific research on this constant. He first proposed a method to make the value of π accurate to any accuracy through mathematical process rather than measurement. Thus, the second stage of pi calculation began.
The circumference of a circle is larger than the inscribed regular quadrangle and smaller than the circumscribed regular quadrangle, so 2 √ 2 < π < 4.
Of course, this is a terrible example. It is said that Archimedes used a regular 96-sided polygon to calculate his range.
Archimedes' method of finding a more accurate approximation of pi is embodied in one of his papers, The Determination of Circle. In this book, Archimedes used the upper and lower bounds to determine the approximate value of π for the first time. He proved geometrically that "the ratio of the circumference of a circle to the diameter of a circle is less than 3+( 1/7) and greater than 3+(171)", and he also provided an estimate of the error. Importantly, this method can theoretically get more accurate values of pi. By about 150 AD, the Greek astronomer Ptolemy had obtained π = 3. 14 16, which was a great progress since Archimedes.
Circumcision. Constantly use Pythagorean theorem to calculate the side length of regular N-polygon.
In China, mathematician Liu Hui first got a more accurate pi. Around 263 AD, Liu Hui put forward the famous secant technique, and got π = 3. 14, which is usually called "emblem rate". He pointed out that this was an approximation. Although he proposed that the circle cutting was later than Archimedes', its method was indeed more beautiful than Archimedes'. Circumtangent only uses inscribed regular polygons to determine the upper and lower bounds of pi, which is much simpler than Archimedes using inscribed regular polygons and circumscribed regular polygons at the same time. In addition, some people think that Liu Hui provided a wonderful sorting method in the cyclotomy, so that he got that PI = 3927/1250 = 3.1416 has four significant figures through simple weighted average. And this result, as Liu Hui himself pointed out, needs to be cut into 3072 polygons if this result is obtained through the calculation of circle cutting. This finishing method has a very good effect. This magical finishing technology is the most wonderful part of ring cutting, but unfortunately it has been buried for a long time because people lack understanding of it.
I'm afraid you are more familiar with Zu Chongzhi's contribution. In this regard, the record of Sui Shu Law and Discipline is as follows: "At the end of the Song Dynasty, South Xuzhou engaged in Zu Chongzhi's more secret method. If the diameter of a circle is 100 million, the circumferential abundance is three feet, one foot, four inches, one minute, five millimeters, nine seconds, seven seconds, and three feet, one foot, four inches, five millimeters, nine millimeters, two seconds and six seconds, and the positive number is between the remainder and two limits. Density: circle diameter 1 13, circumference 355. Regarding the rate, the diameter of the circle is seven, and it is on the 22nd of the week. "
According to this record, Zu Chongzhi made two great contributions to Pi. The first is to find pi.
3. 14 15926 < π < 3. 14 15927
Secondly, two approximate fractions of π are obtained: the approximate rate is 22/7; The encryption rate is 355/ 1 13.
The 8-digit reliable figure of π calculated by him was not only the most accurate pi at that time, but also kept the world record for more than 900 years. So that some historians of mathematics proposed to name this result "ancestor rate".
How did this result come about? Tracing back to the source, Zu Chongzhi's extraordinary achievement is based on the inheritance and development of Liu Hui's secant technique. Therefore, when we praise Zu Chongzhi's achievements, we should not forget that his achievements were achieved by standing on the shoulders of Liu Hui, a great mathematician. It has been estimated by later generations that if this result is obtained simply by calculating the side length of the polygon inscribed in the circle, then it is necessary to calculate the polygon inscribed in the circle to get such an accurate value. Did Zu Chongzhi use other clever methods to simplify the calculation? This is unknown, because the seal script, which records its research results, has long been lost. This is a very regrettable thing in the history of mathematics development in China.
Zu Chongzhi commemorative stamps issued by China
Zu Chongzhi's research achievements are world-renowned: the wall of the Discovery Palace Science Museum introduces the pi obtained by Zu Chongzhi, the corridor of the Moscow University Auditorium is inlaid with a marble statue of Zu Chongzhi, and there is a crater named after Zu Chongzhi on the moon. ...
People usually don't pay much attention to Zu Chongzhi's second contribution to π, that is, he approximated π with two simple fractions, especially density. However, in fact, the latter is more important in mathematics.
The approximation between density and π is good, but the form is simple and beautiful, and only the numbers 1, 3,5 are used. Professor Liang Zongju, a historian of mathematics, has verified that among all fractions with denominator less than 16604, there is no fraction closer to π than density. Abroad, westerners got this result more than 1000 years after Zu Chongzhi's death.
It can be seen that it is not easy to put forward the confidentiality rate. People naturally want to know how he got this result. How did he change pi from an approximate value expressed in decimals to an approximate fraction? This problem has always been concerned by mathematical historians. Due to the loss of literature, Zu Chongzhi's explanation is unknown. Later generations made various speculations about this.
Let's look at the works in foreign history first, hoping to provide some information.
1573, the German Otto reached this result. He used Archimedes' result 22/7 and Ptolemy's result 377/ 120, which is similar to the "synthesis" in the addition process: (377-22)/(120-7) = 355/113.
1585, the Dutch Antoine used Archimedes' method to get: 333/106 < π < 377/120, and took them as the mother approximation of π. The numerator and denominator were averaged respectively, and the result was obtained by the addition process: 3 ((15+).
Although both of them got Zu Chongzhi's secret information, their usage methods are all coupling, which makes no sense.
In Japan,17th century-He's important work, The Algorithm of Enclosing, Volume IV, established the zeroing technique, the essence of which is to find approximate scores by addition process. He took 3 and 4 as mother approximations, added them six times in a row to get the approximate rate of Zu Chongzhi, and added them 112 times to get the secret rate. The students improved this stupid step-by-step method and put forward the method of adding the approximate values of adjacent losses and gains (in fact, the addition process we mentioned earlier). Starting from 3 and 4, the sixth addition to the approximation rate, the seventh addition is 25/8, the nearest 22/7 addition is 47/ 15, and so on, just add 23 times.
In the History of Arithmetic in China (193 1), Mr. Qian Zongyan proposed that Zu Chongzhi adopt "Japanese adjustment method" or weighted addition process, which was initiated by He Chengtian. He conceived the process of Zu Chongzhi's secret rate: taking the emblem rate of 157/50 and the approximation rate of 22/7 as the mother approximation, he calculated the addition weight x=9, so (157+22× 9)/(50+7× 9) = 355/1. Mr. Qian said, "After inheriting heaven, it is also interesting to use its technology to create a secret rate."