How did π come from?

All numbers that can be expressed by fractions are rational numbers, and irrational numbers cannot be expressed by fractions.

Calculation process of pi

Han xuetao

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, mathematicians at all times and all over the world have devoted their wisdom and labor to this goal. 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, it should be said that finding the value of pi was the number one problem in mathematics. In order to get the value of pi, mankind has gone through a long and tortuous road, and its history is interesting. 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 martial arts work Shu Jing Zhou Pian recorded the conclusion that the diameter of the 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 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."

Another guess is to use the continued fraction method.

Because the multiphase subtraction technique for finding the greatest common divisor of two natural numbers has been very popular since the publication of Nine Chapters Arithmetic, it should be natural to use this tool to find approximate fractions. So it was suggested that Zu Chongzhi might use this tool to express 3. 14 159265 as a continued fraction and get its asymptotic scores: 3,22/7,333/106,355//kloc-0.

Finally, 355/ 1 13 with high accuracy but small denominator is taken as the approximate value of pi. As for the concrete solution of the asymptotic fraction of pi above, it is omitted here. You might as well ask yourself in the way we introduced earlier. Dr Needham of Britain holds this view. When he talked about Zu Chongzhi's secret rate in the geometry compilation of Chapter 19 of the History of Science and Technology in China, he said: "The score of secret rate is a continued fraction asymptotic number, so it is an extraordinary achievement."

Let's review the achievements made abroad.

1 150 years, Indian mathematician Bashgaro calculated π = 3927/1250 = 3.1416 for the second time. 1424, Kathy, an astronomer and mathematician in Central Asia, wrote The Theory of Circles, calculated the perimeters of 3× 228 = 805, 306, 368 regular polygons with circumscribed sides, and calculated the π value. His result is:

π=3. 14 159265358979325

There are seventeen exact figures. This is the first time that a foreign country has broken Zu Chongzhi's record.

16th century French mathematician Veda used Archimedes method to calculate π approximation, and used 6×2 16 regular polygon to calculate π value, accurate to 9 decimal places. He still adopts Archimedes' method, but David has a more advanced tool than Archimedes: decimal position system. /kloc-At the beginning of the 7th century, Rudolf, a German, spent almost his whole life studying this problem. He also combined the new decimal system with the early Archimedes method, but he didn't start with a regular hexagon and double its sides. Starting from a square, he has always deduced a regular polygon with 262 sides, about 461000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 This gives a 35-digit decimal. In order to commemorate his extraordinary achievements, the Germans called pi "Rudolph". However, it takes a lot of calculation to find its value by geometric method. If this calculation continues, the life of poor mathematicians will not be greatly improved. In Rudolph, it can be said that it was the pinnacle, and the classical method led mathematicians to go far. To move forward, we must make a breakthrough in methods.

Mathematical analysis appeared in the17th century, which solved many helpless problems in elementary mathematics. The calculation history of π has also entered a new stage.

Analysis period

During this period, people began to get rid of the complicated calculation of polygon perimeter and use infinite series or infinite connected product to calculate π.

1593, David gave it.

This unusual formula is the earliest analytical expression of π. Even today, we are still amazed at the beauty of this formula. It shows that π value can be calculated by a series of addition, multiplication, division and square root with only the number 2.

Then all kinds of expressions appeared. As described in Wallis 1650:

1706, McKinley established an important formula, which is now named after him:

Using the series expansion in the analysis, he calculated 100 digits after the decimal point.

This method is much simpler than the 35-bit decimal method that poor Rudolph spent most of his life digging up. Obviously, the series method declared the classical method obsolete. Since then, the calculation of pi is like a marathon, recording one after another:

In 1844, Dasey used the formula:

Count to 200.

After19th century, similar formulas have emerged continuously, and the number of π digits has also increased rapidly. 1873, Xie Ke used a series of methods and series formulas of McKin to calculate π to 707 decimal places. It took him 20 years to get this unprecedented record. After his death, people carved this value, which condensed his lifelong efforts, on his tombstone to celebrate his tenacious will and perseverance. So he left the fruit of his life's efforts on his tombstone: 707 digits after π decimal point. This amazing result became the standard for the next 74 years. For the next half century, people were convinced of his calculation, or even if they doubted it, there was no way to test whether it was correct. So much so that in the courtyard of the Discovery Hall of the Paris World Expo in 1937, the π value he calculated was still engraved conspicuously.

A few years later, the mathematician Ferguson had doubts about his calculation results. His suspicion is based on the following conjecture: although there is no law to follow in the value of π, the probability of each number should be the same. When he counted the results of the shack, he found that the figures were too uneven. So doubt is wrong. He used the most advanced computing tools he could find at that time, from May 1944 to May 1945, and counted for a whole year. 1946, Ferguson found the 528th dislocation (it should be 4, but wrong should be 5). Xie Sike's value 100 has been completely written off, completely written off poor Xie Sike and his wasted fifteen years.

In this regard, someone once laughed at him and said: In addition to recording the works of Archimedes, Fermat and others, the history of mathematics will squeeze out one or two lines to describe the fact that Xie Ke calculated π to 707 decimal places 1873 years ago. So he may feel that his life has not been wasted. If so, his purpose has been achieved.

It may be normal for people who make unremitting efforts in all corners of the earth to feel incomprehensible. However, the irony of this point is too cruel. People's abilities are different, so we can't ask everyone to be like Fermat and Gauss. But not being a great mathematician doesn't mean that we can't make our own limited contribution to this society. Everyone has his own strengths. As an energetic calculator, Xie Sike is willing to devote most of his life to this work for free, and finally add a brick to the world's knowledge treasure house. Shouldn't we be infected by his unremitting efforts and get some inspiration and education from it?

1948 65438+ 10 In October, Ferguson and Ronchi published π with 808 correct decimals. This is the highest record of manual calculation of π.

Computer age

1946, the world's first computer ENIAC was successfully manufactured, marking the beginning of the computer age in human history. The emergence of computers has led to a fundamental revolution in the field of computing. 1949, ENIAC calculated 2035 (2037) decimal places according to Machin formula, including preparation and sorting time, which only took 70 hours. With the rapid development of computers, their records are often broken.

ENIAC: the beginning of an era

1973, someone calculated pi to 100 after the decimal point, and printed the result into a 200-page book, which is the most boring book in the world. 1989 broke through the 1 billion mark, and1995 broke through 6.4 billion in June. On September 30th, 1999, Abstracts reported that Yasumasa Kanada, a professor at the University of Tokyo, got the decimal value of 20665438+5843 million. If these numbers are printed on A4-sized copy paper with 20,000 numbers printed on each page, the paper will be piled up to 500-600 meters. From the latest report: Yasumasa Kanada used a supercomputer to calculate the number of digits 124 1 1 billion after the decimal point of pi, rewriting his own record set two years ago. It is reported that Professor Jintian, in cooperation with employees of Hitachi, used a supercomputer with the current computing power ranking 26th in the world and used a new computing method. It took more than 400 hours to calculate these new figures, which is six times more than the 26 1 1 decimal places he calculated in September 1999. The first trillion digits of pi after the decimal point are two, and the first trillion digits are five. If you read a number every second, it will take about 40 thousand years to finish reading.

However, it won't be particularly unexpected to break the record now, no matter how advanced it is. In fact, it is of little practical significance to calculate the value of π too accurately. A dozen π values used in modern science and technology are enough. If Rudolph's π value of 35 decimal places is used to calculate the circumference of a circle that can surround the solar system, the error is less than one millionth of the proton diameter. We can also quote the words of American astronomer simon newcomb to illustrate the practical value of this calculation:

"Ten decimal places are enough to make the circumference of the earth accurate within an inch, and thirty decimal places can make the circumference of the entire visible universe accurate to an amount that even the most powerful microscope can't distinguish."

So why do mathematicians, like mountaineers, strive to climb upwards and constantly seek rather than stop exploring π? Why is its decimal value so attractive?

There are probably human curiosity and the mentality of being ahead of others, but there are also many other reasons.

The wonderful relationship between Pentium and pi ...

1 can now be used to test or examine the performance of supercomputers, especially the operation speed and the stability of the calculation process. This is crucial to the improvement of the computer itself. Just a few years ago, when Intel introduced Pentium, it found that it had a little problem, which was discovered by running π calculation. This is one of the reasons why ultra-high precision π calculation is still of great significance today.

2. Calculation methods and ideas can lead to new concepts and ideas. Although the operation speed of the computer is beyond anyone's imagination, mathematicians still need to write programs to guide the computer to run correctly. In fact, exactly speaking, when we divide the calculation history of π into an electronic computer period, this does not mean the improvement of calculation methods, but only a big leap in calculation tools. Therefore, how to improve the calculation technology, study a better calculation formula, make the formula converge faster, and quickly reach greater accuracy is still an important topic faced by mathematicians. In this regard, Ramanuyan, a talented mathematician in India in this century, has achieved some good results. He found many formulas that can calculate π approximation quickly and accurately. His insights opened the way to calculate π approximation more effectively. Now he has worked out the formula for calculating π value by computer. As for the story of this legendary mathematician, we don't want to introduce it in this little book. However, I hope everyone can understand that the story of π is about the victory of human beings, not the victory of machines.

3. Another question about the calculation of π is: Can we continue the calculation indefinitely? The answer is: no! According to Judarovsky's estimation, the most we can count is 1077. Although we are still far from this limit, it is a boundary after all. In order not to be bound by this limitation, a new breakthrough in calculation theory is needed. The calculation mentioned above, no matter what formula is used, must be calculated from scratch. Once a number in front is wrong, the value in the back is completely meaningless. Remember the regrettable Schacks? He is the most painful lesson in history.

4. So, some people think, is it possible to start from the beginning, but from the middle? The basic idea is to find the parallel algorithm formula. 1996, finally found the parallel algorithm formula of pi, but it is the formula of 16, so it is easy to get the value of 1000 billion bits, but it is only 16. Whether there is a parallel calculation formula of 10 is still a big problem in mathematics in the future.

As an infinite sequence, mathematicians are interested in expanding π to hundreds of millions of bits, which can provide enough data to verify some theoretical questions raised by people and find many fascinating properties. For example, in the decimal system of π, there are 10 numbers, which are sparse and which are dense? Will some numbers appear more frequently than others in the digital expansion of π? Maybe they're not completely random? This idea is not boring. Only sharp-minded people ask such seemingly simple questions, and many people are used to it but disdain to ask.

6. Mathematician Ferguson first had this conjecture: in the numerical formula of π, the probability of each number appearing is the same. It was his conjecture that made a great contribution to discovering and correcting Cox's mistake in calculating π value. However, conjecture is not equal to reality. Ferguson wanted to test it, but there was nothing he could do. Later generations also want to verify it, but they also suffer from the fact that there are too few digits in the known π value. Even if the number of digits is too small, people have reason to doubt the correctness of the guess. For example, the number 0 rarely appears at the beginning. The first 50 bits only have 1 zero, which first appeared in the 32nd bit. But this phenomenon changed quickly with the increase of data: 100 has 8 zeros; There are 19 zeros within 200 digits; .....100000 digits has 999440 zeros; ..... There are 599,963,005 zeros in the 6 billion digits, accounting for almost 65,438+0/65,438+00.

What about the other figures? The results show that almost every one is110, and some are more or less. Although there are some deviations, they are all within110000.

7. People still want to know: Is there really no certain pattern for the digital expansion of π? We hope to find any possible model by studying the statistical distribution of numbers in decimal expansion-if there is such a model, it has not been found so far. At the same time, we also want to know: does the expansion of π include infinite style changes? Or, will there be any form of digital arrangement? Hilbert, a famous mathematician, once asked the following question in his unpublished notebook: Are there 10 9s connected in the ten series of π? Judging from the 6 billion figures calculated now, it has already appeared: six consecutive 9s are connected together. The answer to Hilbert's question seems to be yes. It seems that any arrangement of numbers should appear, just when. But it needs more π digits to provide tangible evidence.

8. In this regard, there are the following statistical results: eight eights appear in the six billion figures; Nine sevens; 10 6; Starting from the decimal places 7 10 150 and 3204765, there are seven 3s in succession; The eight numbers 14 142 135 appear continuously from the decimal point of 52638, which is exactly the first eight digits; Starting from the 2747956th place after the decimal point, the interesting sequence 8765432 10 appeared, but unfortunately a 9 was missing in front of it. There is also an interesting series 123456789.

If you continue to count, it seems that various types of numerical column combinations may appear.

Change: Other Calculation Methods of π

In the book Probability Arithmetic Experiment published by 1777, Buffon proposed to calculate π by experimental method. The operation of this experimental method is very simple: find a thin needle with a uniform thickness and a length of D, draw a set of parallel lines with an interval of L on a piece of white paper (for convenience, l = d/2 is often taken), and then throw the small needle on the white paper again and again at will. Repeat this for many times, count the number of times the needle intersects any parallel line, and you can get the approximate value of π. Because Buffon himself proved that the probability that a needle intersects an arbitrary parallel line is p = 2l/π d. Using this formula, we can get the approximate value of pi by probability method. In an experiment, he chose l = d/2, and then put the needle 22 12 times, in which the needle crossed the parallel line 704 times, so the approximate value of pi was 22 12/704 = 3. 142. When the number of experiments is quite large, a more accurate π value can be obtained.

In 1850, a man named Wolff got an approximate value of 3. 1596 after more than 5000 votes. At present, it is Italian Laslini who claims to get the best result by this method. At 190 1, he repeated the experiment and made 3408 injections. The approximate value of π is 3. 14 15929, which is so accurate that many people doubt the authenticity of his experiment. For example, L Badger of Ogden National Weber University in Utah, USA, strongly questioned this.

However, the importance of Buffon experiment is not to get a more accurate π value than other methods. The importance of Buffon's needle problem lies in that it is the first example of expressing probability problem in geometric form. This method of calculating π is not only amazing because it is novel and wonderful, but also pioneered the use of random numbers to deal with deterministic mathematical problems, and is the forerunner of solving deterministic calculations by contingency methods.

When calculating π value by probability method, it should also be mentioned that R Chatter found in 1904 that the probability of two randomly written numbers being coprime is 6/π 2. 1In April, 995, the British magazine Nature published an article, which introduced how Robert Matthews of the Department of Computer Science and Applied Mathematics of Aston University in Birmingham, England used the distribution of bright stars in the night sky to calculate pi. Matthews randomly selects one pair after another from 100 brightest stars, and analyzes and calculates the angular distance between their positions. He looked up 6.5438+00000 pairs of factors, and based on this, the value of π was about 3.654.38+02772. The relative error between this value and the real value is less than 5%.

Infinite mystery: Givenchy men's perfume π. The advertising slogan is: explore pi, explore the universe.

π was discovered through a wide range of channels, such as geometry, calculus, probability, and so on, which fully showed the strange beauty of mathematical methods. It is really surprising that π should communicate with this seemingly unrelated experiment.