The tadpole text on the tombstone is no longer recognizable, Lanling doesn't know it, and the ancient bluebird doesn't know it.

Gu Qingniao was curious: "Since this person has left this tombstone, doesn't he want people to know who he is and what he left here?" But write an epitaph in tadpole script, who can read it? ”

"Then there's only one possibility." Lan Ling said: "It's that what this person wants to leave behind can only be understood by specialized people, I don't know who this specialized person is, but it's definitely not the two of us." ”

The ancient blue bird nodded, indicating that it seemed to be like this: "But how do we find the portal to that space now?" ”

Lan Ling thought for a while and said, "There must be any traces around, traces of energy or traces of stuckness, let's look for them." ”

So far, it can probably only be like this, the ancient blue bird and Lanling separated, walking along the edge of the pit in both directions, looking for some clues.

Soon, the ancient bluebird discovered that the pit did not seem to be a regular circle, but a polygon with many sides.

A flat shape consisting of three or more line segments connected one after the other is called a polygon. According to different standards, polygons can be divided into regular polygons and non-regular polygons, convex polygons and concave polygons.

The so-called "circumcision" is a method of using the area of a regular polygon inside a circle to infinitely approximate the area of a circle and obtain pi from this. "Round, one in the same length". This means that the circle has only one center, and each point on the circumference of the circle is at an equal distance from the center. As early as the pre-Qin period in China, the definition of a circle has been given in the Book of Ink, and in the 11th century BC, Shang Gao, a mathematician in the Western Zhou Dynasty of China, also discussed the relationship between circles and squares with Zhou Gong. After understanding the circle, people began to calculate the circle, especially the area of the circle. In the first chapter of the "Fang Tian" chapter, the ancient Chinese mathematics classic "Nine Chapters of Arithmetic" writes that "the half-circumferential radius is multiplied to obtain the product step", which is the formula we are now familiar with (2019).

In order to prove this formula, Liu Hui, a mathematician in the Wei and Jin dynasties in China, wrote the "Nine Chapters of Arithmetic Notes" in 263 AD, and wrote a note of more than 1,800 words behind this formula, which is the famous "circumcision" in the history of mathematics. Mathematical significance

"Circumcision" is to use "the area of the circle with a regular polygon" to approximate the "area of the circle" infinitely. Liu Hui described his "circumcision" and said: Cutting is fine, what is lost is little, and cutting is cut again, so that it cannot be cut, it is combined with the circle, and nothing is lost.

That is, by connecting the circle with a regular polygon to cut the circle, and making the circumference of the regular polygon infinitely close to the circumference of the circle, a more accurate pi is obtained.

Liu Hui invented the "circumcision" in order to find "pi". So what exactly does pi mean? It is actually the ratio of the circumference of a circle to the diameter of the circle. Luckily, this is a constant "constant"! With the help of it, we humans can make all kinds of calculations about circles and spheres. Without it, then we would be helpless with circles and spheres, etc. In the same way, the "accuracy" of the pi value is directly related to the accuracy and precision of our calculations. That's why humans demand pi, and they demand it accurately.

According to "Circumference / Diameter = Pi", then Circumference = Diameter of Circle * Pi = 2 * Radius * Pi (This is what we are familiar with Circumference = 2π

of the origin). Therefore, there is no need to memorize the "circumference formula" at all, as long as you have primary school knowledge and know the "meaning of pi", you can derive and calculate it by yourself. Everyone may know "pi and π", but its "meaning and function" are often overlooked, which is the meaning of circumcision.

Since "pi = circumference of the circle / diameter of the circle", where the "diameter" is straight, it is easy to measure; What is difficult to calculate precisely is the "circumference of a circle". Through Liu Hui's "circumcision", this problem was solved. As long as the circumference of the circle is carefully and patiently calculated, a more accurate "pi" can be obtained. - As we all know, in China, Zu Chongzhi finally completed this work.

Development HistoryEdit

Since the pre-Qin period, ancient China has always used the value of "three diameter one" (that is, the ratio of the circumference to the diameter of the circle is 3:1) to calculate the circle. However, the results of the calculation with this value are often very incorrect. As Liu Hui said, the circumference of a circle calculated by "Wednesday Diameter One" is actually not the circumference of the circle but the circumference of the regular hexagon within the circle, and its value is much smaller than the actual circumference of the circle. Zhang Heng of the Eastern Han Dynasty was not satisfied with this result, and he began to obtain pi by studying the relationship between the circle and its inscribed square. This value is better than the "Wednesday Diameter One", but Liu Hui believes that the calculated circumference of the circle must be larger than the actual circumference and is not accurate. Guided by the idea of limits, Liu Hui proposed to use the "circumcision technique" to find pi, which was both bold and innovative, and rigorously demonstrated, thus pointing out a scientific road for the calculation of pi.

In Liu Hui's view, since the circumference of the circle calculated by "Wednesday Diameter One" is actually the circumference of the circle with a regular hexagon, there is a lot of difference from the circumference of the circle; Then we can divide the circumference of the circle into six arcs on the basis of the regular hexagon inside the circle, and then continue to divide each arc into two to make a circle with a regular dodecagon, and the circumference of the regular dodecagonal should be closer to the circumference than the circumference of the regular hexagon? If the circumference of the circle is further divided into a circle with a regular twenty-sided shape, then the circumference of the regular twenty-sided must be closer to the circumference than the circumference of the regular dodecagon. This shows that the finer the circumference is divided, the less error there is, and the closer the circumference of the circumference of the inscribed regular polygon is. This goes on until the circumference of the circle can no longer be divided, that is, when the circle is infinitely larger than the sides of the regular polygon, its circumference is exactly the same as that of the circumference.

According to this line of thought, Liu Hui calculated the area of the circle with regular polygons all the way to the regular 3072 polygons, and obtained the approximate values of pi of 3.1415 and 3.1416. This result was the most accurate data for pi calculations in the world at the time. Liu Hui was so confident in the new method of "circumcision" that he created that he extended it to all aspects of circular calculation, thus greatly advancing the development of mathematics since the Han Dynasty. Later, in the period of the Northern and Southern Dynasties, Zu Chongzhi continued to work the basis of Liu Hui, and finally made pi accurate to the seventh decimal place. In the West, this achievement was achieved by the French mathematician Veda in 1593, more than 1,100 years later than Zu Chongzhi. Zu Chongzhi also obtained two fractional values of pi, one is the "approximate rate" and the other is the "dense rate", of which this value was obtained by Otto of Germany and Antoniz of the Netherlands at the end of the 16th century in the West, both of which were 1,100 years later than Zu Chongzhi. History will never forget the significant contribution of the new method of "circumcision" created by Liu Hui to the development of ancient Chinese mathematics.

Basic algorithm editing

According to Liu Hui's records, before Liu Hui, when people verified the formula for the area of a circle, they used the area of the circle with a regular dodecagonal to replace the area of the circle. Applying the principle of complementarity between entry and exit, the circle is connected with regular dodecagons to form a rectangle, and the area formula of the rectangle is borrowed to demonstrate the circle area formula of the Nine Chapters of Arithmetic. Liu Hui pointed out that this rectangle is half of the circumference of the circle and the radius of the circle is the height of the rectangle, and its area is the area of the circle and the area of the circle is the area of the circle and the circle is connected to the regular dodecagon. This kind of argument "the diameter ratio is one and the arc circumference is three", that is, the "three diameter one" that is often said later, is of course not rigorous. He believes that there is a difference between the area of the regular polygon within the circle and the area of the circle, and it is impossible to prove the formula of the circle area in the "Nine Chapters of Arithmetic" by dividing and assembling it with a finite number of divisions and patches. Therefore, Liu Hui boldly introduced the idea of limits and infinitesimal segmentation into mathematical proofs. He began to cut the circle from the inside of the circle with a regular hexagon, "the cut is fine, and the loss is small, and the cut is cut again, so that it cannot be cut, then it is combined with the circumference, and nothing is lost." That is to say, if the number of sides of the circle is doubled, the difference between them and the area of the circle becomes smaller and smaller, and when the number of sides cannot be increased, the limit of the area of the circle with the regular polygon is the area of the circle. Liu Hui examines the area of the inscribed polygon, that is, its "power", and at the same time proposes the concept of "difference power". The "power of difference" is the difference between the last and previous cuts, which can be expressed as the area of the triangle in the shaded part of the diagram. At the same time, it is equal to the sum of the area of the two small yellow triangles. Liu Hui pointed out that in the process of approximating the area of the circle with a regular polygon within a circle, the radius of the circle has a residual diameter between the regular polygon and the circle. Multiply the coincidence by the side length of the regular polygon, i.e., 2 times the "power of difference", and add it to the regular polygon, and its area is greater than the area of the circle. This is an upper bound sequence of the area of the circle. Liu Hui believes that when the circle is connected with a regular polygon and the circle is the limit state of the combination, "then there is no residual diameter on the surface." If there is no residual diameter in the table, the power will not go out. That is, the coarse is gone, and the rectangle of the coarse is gone. Thus, the limit of this upper bound sequence of the area of the circle is also the area of the circle. So the inner and outer sequences tend to the same value, i.e., the area of the circle.

The method of approximating pi by using the circumscribed or tangent regular polygon of the circle is that when the number of sides of the regular polygon increases, its side length gradually approximates the circumference. As early as the 5th century B.C., the ancient Greek scholar Antiphon devised a method in order to study the problem of turning a circle into a square shape: first make a circle with a regular quadrilateral, and then make a circle with a regular octagon on this basis, and then double the number of its sides one by one to obtain a regular 16-sided shape, a regular 32-sided shape, and so on, until the side length of the regular polygon is so small that it coincides with the circumference part of the circle where they are located, and he thinks that the problem of square the circle can be completed. By the 3rd century B.C., the ancient Greek scientist Archimedes used the exhaustion method in his book "On Spheres and Cylinders" to establish such a proposition: as long as the number of sides is sufficient, the difference between the area of the circle inscribed regular polygon and the area of the inscribed regular polygon can be arbitrarily small. Archimedes also used the method of regular polygon circumcision in the book "The Measurement of Circles" to obtain the value of pi less than three and one-seventh and greater than three and seventies tenths, and also said that the ratio of the area of the circle to the area of the inscribed square is 11:14, that is, the pi is equal to 22/7. In 263 A.D., the Chinese mathematician Liu Hui put forward the theory of "cutting the circle" in the "Nine Chapters of Arithmetic Notes", he started from the circle with a regular hexagon, and doubled the number of sides each time until the circle was connected with a regular 96 sides, and calculated that the pi was 3.14 or 157/50, which was called the Hui rate by later generations. A more precise value of pi is also recorded in the book: 3927/1250 (equal to 3.1416). Liu Hui asserted that "if you cut it finely, you will lose it, and if you cut it again, you will not be able to cut it, but it will be combined with the circle, and nothing will be lost." Its ideas coincide with the ancient Greek law of exhaustion. Circumcision has long been used in the history of pi calculations. In 1610, the German mathematician Curran used a 2^62 polygon to calculate pi to 35 decimal places. In 1630, Greenberg used an improved method to calculate to 39 decimal places, which became the best result of circumcision to calculate pi. After the invention of the analytical method, circumcision gradually replaced circumcision, but circumcision has been praised as the earliest scientific method for calculating pi.

Trigonometric arithmetic for pi

Trigonometric arithmetic for pi

π=lim(

→∞）1/2*si

(360°/

)*

Thought Value Edit

There are two important ideas in proving this formula for the area of a circle, one of which is what we are talking about now. Then the second step, and the more critical step, is to divide the regular polygon that is combined with the circumference, that is, the regular polygon that cannot be cut again, into infinitesimal divisions, and then divide it into infinitely many small isosceles triangles with the center of the circle as the vertex and each side of the polygon as the base, and this base radius is twice the area of the small triangle, and all these base radii should be twice the area of the circle. Then it is equal to the circumference of the circle multiplied by the radius equal to the area of the two circles. Therefore, the area of a circle is equal to the radius of half a circumference, so Liu Hui said that the radius of half a circumference is the power of the circle. Then his original words are "multiply one side by the radius, and cut it, and each time doubles itself." Therefore, it is the power of the circle by the radius of half a circumference". In the end, the formula for the area of the circle is fully proved, and the formula for the area of the circle is proved to be inaccurate. With the proof of the formula for the area of a circle, Liu Hui also created a scientific procedure for finding an exact approximation of pi. Before Liu Hui, the ancient Greek mathematician Archimedes had also studied the problem of solving pi.

Liu Hui lived in the era of the warlords in the society, especially at that time was the separation of Wei, Shu and Wu, so at this time China's society, politics and economy had undergone great changes, especially in the ideological circles, the literati and scholars debated with each other, so it became the wind of debate at that time, a group of literati and scholars found a piece, just like our college debate, a positive and a negative, put forward a proposition to debate each other, and when debating, people had to study and discuss the technology of debate and the law of thinking. Therefore, in this period of people's ideological emancipation, it should be said that there was no time after the Spring and Autumn Period and the Warring States Period, when people's research on the laws of thinking was particularly developed, and some people believe that people's abstract thinking ability at this time far exceeded that of the Spring and Autumn Period and the Warring States Period. Liu Hui stated in the preface to the Nine Chapters of Arithmetic Notes that he took exploring the roots of mathematics as his highest task in mathematical research. He noted that the purpose of "Nine Chapters of Arithmetic" is to "analyze the reasoning and use the diagram". "Analysis" was synonymous with the arguments that scholars had with each other at that time. Through the analysis of mathematical principles, Liu Hui established a theoretical system of traditional Chinese mathematics. As we all know, ancient Greek mathematics achieved very high levels of achievement and established a rigorous deductive system. However, Liu Hui's "circumcision" was the first time in human history that limits and infinitesimal segments were introduced into mathematical proofs, becoming an immortal chapter in the history of human civilization.

Liu Hui (c. 225 – c. 295), a native of Zouping City, Binzhou, Shandong Province [1], was a great mathematician during the Wei and Jin dynasties and one of the founders of classical Chinese mathematical theory. He has made great contributions to the history of Chinese mathematics, and his masterpieces "Nine Chapters of Arithmetic Notes" and "Island Arithmetic" are the most valuable mathematical heritage in China. Liu Hui is quick in thought and flexible in his methods, advocating both reasoning and intuitiveness. He was the first person in China to explicitly advocate the use of logical reasoning to prove mathematical propositions. Liu Hui's life was a life of hard work for mathematics. Although he had a low status, he had a noble personality. He is not a mediocre man who sells his fame and reputation, but a great man who never tires of learning, and he has left a valuable wealth to the Chinese nation.

The Nine Chapters of Arithmetic was written at the beginning of the Eastern Han Dynasty and has a total of 246 solutions. In many aspects, such as solving simultaneous equations, fractional four-rule operations, positive and negative number operations, and calculation of the volume area of geometric figures, etc., are among the world's advanced. Liu Hui annotated "Nine Chapters of Arithmetic Notes" in the fourth year of Cao Wei Jingchu.

However, because the solution method is relatively primitive and lacks the necessary proof, Liu Hui has made supplementary proofs. In these proofs, his creative contribution in many ways is shown. He was the first person in the world to come up with the concept of decimal decimal numbers and use decimal decimal numbers to represent the cube roots of irrational numbers. In algebra, he correctly proposed the concept of positive and negative numbers and the rules of addition and subtraction, and improved the solution of linear equations. In terms of geometry, the "circumcision" is proposed, which is a method of exhausting the circumference of a circle with inscribed or inscribed regular polygons. He scientifically found the result of pi π=3.1416 by circumcision. He used the circumcision technique, starting from the circle with a diameter of 2 feet with a regular hexagon, and then obtained a regular 12-sided shape, a regular 24-sided ......, the finer the cut, the smaller the difference between the area of the regular polygon and the area of the circle, in his original words, "the cut is fine, the loss is small, the cut is cut again, so that it cannot be cut, then it is combined with the circumference and nothing is lost." He calculated the area of 3072 polygons and verified this value. The scientific method of calculating pi proposed by Liu Hui has established China's leading position in the calculation of pi in the world for more than 1,000 years.

This method is consistent with the later method of finding the approximation of irrational roots, which is not only necessary for the accurate calculation of pi, but also facilitates the production of decimal decimals; In the linear system of equation solution, he created a simpler method of multiplication and elimination than direct division, which is basically the same as the current solution. For the first time in the history of Chinese mathematics, he proposed the "indefinite equation problem". He also established the difference series before

terms and formulas; Many mathematical concepts have been proposed and defined: such as power (area); equations (systems of linear equations); Positive and negative numbers and so on. Liu Hui also put forward a number of generally correct judgments as a premise for proof. Most of his reasoning and proofs are logical and rigorous, thus basing the Nine Chapters of Arithmetic and his own solutions and formulas on necessity. Although Liu Hui did not write a self-contained work, the mathematical knowledge used in his notes on the Nine Chapters of Arithmetic had in fact formed a unique theoretical system that included concepts and judgments, and was linked by mathematical proofs.

Liu Hui proposed in the circumcision technique that "the cutting is fine, the loss is small, and the cutting is so that it cannot be cut, then it is combined with the circle and nothing is lost", which can be regarded as a masterpiece of the ancient Chinese concept of limits. In the book "Island Calculations", Liu Hui carefully selected nine measurement problems, which attracted the attention of the West at that time because of their creativity, complexity and representativeness. Liu Hui is quick in thought and flexible in his methods, advocating both reasoning and intuitiveness. He was the first person in China to explicitly advocate the use of logical reasoning to demonstrate mathematical propositions.

Liu Hui's mathematical achievements are broadly twofold:

The first is to sort out the ancient Chinese mathematical system and lay its theoretical foundation, which is embodied in the "Nine Chapters of Arithmetic Notes". It has actually become a relatively complete theoretical system:

(1) The same and different types of numbers are used to explain the operation rules of general fraction, reduction fraction, four operations, and simplification of complex fractions; In the commentary on the opening of the square, he discussed the existence of irrational square roots in the sense of inexhaustible prescriptions, introduced new numbers, and created a method of infinitely approximating irrational roots with decimal fractions.

(2) In terms of the theory of formula calculus, he first gave a relatively clear definition of rate, and then established the unified theoretical basis of number and formula operation based on the three basic operations of universal multiplication, generalization, and qi, and he also used "rate" to define the "equation" in ancient Chinese mathematics, that is, the augmented matrix of linear equations in modern mathematics.

(3) In terms of Pythagorean theory, the Pythagorean theorem and the calculation principle of solving Pythagorean shape were demonstrated one by one, the theory of similar Pythagorean shape was established, the Pythagorean measurement technique was developed, and the theory of similarity with Chinese characteristics was formed through the analysis of typical figures such as "Hook in the middle and horizontal" and "Straight in the strand".

Area and volume theory

Liu Hui's principle is proposed by the principle of complementing each other in and out, and the limit method of "circumcision", and solves the problem of calculating the area and volume of various geometries and geometries. The theoretical value of these aspects still shines today.

Severe Difference Surgery

In his self-written "Island Calculations", he proposed the method of heavy difference, using the method of height and distance measurement such as double table, connecting cable and tired moment. He also used the method of "analogy and derivation" to develop the gravity difference technique from two measurements to "three looks" and "four looks". In the 7th century, India and Europe only began to study the problem of two surveys in the 15th ~ 16th century. Liu Hui's work not only had a profound impact on the development of ancient Chinese mathematics, but also established a lofty historical position in the history of mathematics in the world. In view of Liu Hui's great contributions, many books have called him "the Newton of Chinese mathematics".

His masterpiece "Nine Chapters of Arithmetic Notes" is a commentary on the book "Nine Chapters of Arithmetic". The Nine Chapters of Arithmetic is one of the oldest mathematical treatises in China that has survived to the present day, and it was written during the Western Han Dynasty. The completion of this book has gone through a historical process, and the various mathematical problems collected in the book, some of which were circulated before the Qin Dynasty, have been deleted and revised by many people for a long time, and finally sorted out by mathematicians in the Western Han Dynasty. The content of the definitive version that is now circulating was formed before the Eastern Han Dynasty. Nine Chapters of Arithmetic is one of the most important classical mathematical works in China, and its completion laid the foundation for the development of mathematics in ancient China, and occupies an extremely important position in the history of Chinese mathematics. The present biography of the Nine Chapters of Arithmetic collects a total of 246 application problems and solutions to various problems, which are respectively subordinated to the nine chapters of Fangtian, Corn, Decay, Shaoguang, Shanggong, Average Loss, Surplus and Deficit, Equation, and Pythagorean.

The Nine Chapters of Arithmetic is the result of social development and the long-term accumulation of mathematical knowledge, and it brings together the fruits of the labor of mathematicians in different periods. Liu Hui, a mathematician during the Three Kingdoms, believed: "There are nine numbers in the Zhou metric system, and if there are nine numbers, then the "Nine Chapters" is the same. …… Zhang Cang, the Pinghou of the Northern Han Dynasty, and Geng Shouchang, the Great Si Nong, are all good fortune tellers. Due to the remnants of the old text, Cang and others are called deletions and supplements. Therefore, the purpose of the school is different from the ancient, and the discussion is more similar. According to Liu Hui's research results, "Nine Chapters of Arithmetic" originated from the "Nine Numbers" of the Zhou Dynasty, and the "Nine Chapters of Arithmetic" he saw was deleted and supplemented by Zhang Cang and Geng Shouchang in the Western Han Dynasty on the basis of the pre-Qin posthumous texts, including a large number of supplements in the Western Han Dynasty. According to the analysis of historical documents and unearthed cultural relics, Liu Hui's statement is credible. The various algorithms contained in the Nine Chapters of Arithmetic were supplemented and revised by the mathematicians of the Han Dynasty on the basis of the mathematics handed down before the Qin Dynasty to meet the needs of the time. According to Liu Hui's research, Zhang Cang and Geng Shouchang were both major mathematicians who participated in the revision work. According to the "Historical Records: The Biography of Prime Minister Zhang Cheng", Zhang Cang (about 250 ~ 152 BC) experienced the Qin and Han dynasties, and he was awarded the title of Marquis of Beiping in the sixth year of Emperor Gao (201 BC) for his meritorious service in attacking Tibetan tea. "Since the Qin period, it is the history of the column, and the books will be counted tomorrow. He also makes good use of the arithmetic calendar. "He also wrote" 18 books, talking about the law of yin and yang. Geng Shouchang's birth year is unknown, Emperor Xuan of the Han Dynasty was an official to the Great Si Nong Zhongcheng, "to be good for calculation, to be able to business utilitarianism" to favor the emperor (see "Hanshu Food and Goods"). In astronomy, he advocated the theory of Hun Tian, and in the second year of Ganlu (52 BC), he played "to measure the sun and the moon with a circular instrument, and to test the fortune of the heavens" (see "Later Han Shu and Chronicles of the Law"). Zhang Cang and Geng Shouchang were both famous mathematicians and held high positions, so it was natural for them to preside over the revision of the "Arithmetic" handed down by the pre-Qin dynasty. According to Liu Hui's account, the "Nine Chapters of Arithmetic" annotated by him was finally deleted by Geng Shouchang. We believe that the date of Geng Shouchang's deletion and revision of "Nine Chapters of Arithmetic" can be set as the age when this book was completed.

Ma Yuan Biography has a negative record of Ma Xu (about 70~141) "Boguan Group Membership, Good Nine Chapters of Arithmetic". In addition, there are also accounts of Zheng Xuan (127~200), Liu Hong and others in the history books of "nine chapters of arithmetic". It can be seen that the book was an important textbook for learning mathematics at that time, and the inscription on a copperplate in the second year of Guanghe in the Eastern Han Dynasty (179) stipulates: "Dasi Nong Yi Wuyin (138?) Edict,...... It is especially for the states to make copper buckets, oblique, and weighed. According to the Huang Zhonglu calendar, the "Nine Chapters of Arithmetic" is based on the average length, weight, and size, and the seven politics of Qi, so that the sea is the same. This shows that the book was not only widely circulated in the Eastern Han Dynasty, but also that the mathematical problems involved in the development of weights and measures should also be based on the algorithms in the book. Xu Shang and Du Zhi may have been the first mathematicians to study the book after the book was written. Xu Shang and Du Zhi were both mathematicians in the late Western Han Dynasty. "Hanshu Art and Literature" contains 26 volumes of "Xu Shang Arithmetic" and 16 volumes of "Du Zhi Arithmetic". Both books were written before Yin Xian proofread mathematical works in the third year of Emperor Cheng of the Han Dynasty (26 BC). The time when Xu Shang and Du Zhi's works were completed is not far from the time when Geng Shouchang deleted and supplemented the "Nine Chapters of Arithmetic", and their mathematical works should have been completed on the basis of studying the "Nine Chapters of Arithmetic".

Nine Chapters of Arithmetic not only occupies an important position in the history of Chinese mathematics, but also makes important contributions to the development of mathematics in the world. The fractional theory and its complete algorithms, proportional and proportional allocation algorithms, area and volume algorithms, and solutions to various application problems have been described in considerable detail in the chapters of Fangtian, Corn, Decay, Shanggong, and Even-Loss. The opening methods in the chapters of Shaoguang, Surplus and Deficiency, Equations, Pythagorean Method, Surplus and Deficiency (Double Hypothesis Method), the Concept of Positive and Negative Numbers, the Method of Solving Linear Simultaneous Equations, and the General Formula of Integer Pythagorean Strings are all outstanding achievements in the history of world mathematics. The biography "Nine Chapters of Arithmetic" has annotations by Liu Hui and Tang Li Chunfeng. Liu Hui was an outstanding mathematician in ancient China, who lived in the Wei State during the Three Kingdoms era. "Sui Shu Law Chronicles" on the measurement of the past dynasties to quote the Shang Gong chapter notes, said: "Wei Chen Liuwang Jingyuan four years (263) Liu Hui annotated the "Nine Chapters". "His life is not well known. Liu Hui's annotations to the "Nine Chapters" not only made important achievements in sorting out the ancient mathematical system and perfecting the theory of ancient arithmetic, but also put forward a variety of creative ideas and inventions. Liu Hui has made outstanding contributions to arithmetic, algebra, and geometry. For example, he used the ratio theory to establish the theoretical basis for the unity of numbers and formulas, and he applied the principle of complementarity and the limit method to solve many area and volume problems, and established a unique style of area and volume theory. He gave rigorous proof of many of the conclusions in the Nine Chapters, and some of his methods have been a great inspiration for later generations, even for modern mathematics.

Pi (Pi) is the ratio of the circumference of a circle to its diameter, generally represented by the Greek alphabet π, and is a mathematical constant that is prevalent in mathematics and physics. π is also equal to the ratio of the area of the circle to the square of the radius, and is the key value for accurately calculating the circumference of the circle, the area of the circle, the volume of the sphere, and other geometric shapes. In analytics, π can be strictly defined as satisfying SI

x = the smallest positive real number x of 0.

Pi, represented by the Greek letter π (pronounced pài), is a constant (approximately equal to 3.141592654) that represents the ratio of the circumference to the diameter of a circle. It is an irrational number, i.e., an infinite non-cyclic decimal. In daily life, 3.14 is usually used to approximate pi. A decimal place of 3.141592654 is sufficient for general calculations. Even the most sophisticated calculations for engineers or physicists can be taken to a few hundred decimal places at best. [1]

In 1965, British mathematician Joh Wallis

Wallis published a mathematical treatise in which he derives a formula that finds that pi is equal to the product of the multiplication of infinite fractions. In 2015, scientists at the University of Rochester discovered the same formula for pi in quantum mechanical calculations of the energy levels of hydrogen atoms [2].

On March 14, 2019, Google announced that pi is now at 31.4 trillion decimal places.

Experimental period

An ancient Babylonian stone plaque (circa 1900-1600 BC) clearly states that pi = 25/8 = 3.125. [4] Ancient Egyptian artifacts from the same period, Rhi

d Mathematical Papy

us) also indicates that pi is equal to the square of the fraction 16/9, which is approximately equal to 3.1605. [4] The Egyptians seem to have known pi much earlier. British writer Joh

Taylo

(1781–1864) in his famous book The Pyramid (The G

eat Py

amid: Why was it built, a

d Who Built It?"), the Pyramid of Khufu, built around 2500 BC, is related to pi. For example, the ratio of the circumference and height of a pyramid is equal to twice the ratio of pi and exactly equal to the ratio of the circumference and radius of the circle. The ancient Indian religious masterpiece Satapatha B

ahma

a) shows that pi is equal to the fraction 339/108, which is approximately equal to 3.139. [5]

Geometric period

Ancient Greece, as an ancient geometric kingdom, made a particularly prominent contribution to pi. The great Greek mathematician Archimedes (287–212 BC) pioneered the theoretical calculation of pi as a precedent in human history. Starting from the unit circle, Archimedes first used the inscribed regular hexagon to find the lower bound of pi as 3, and then used the external regular hexagon and found that the upper bound of pi was less than 4 with the help of the Pythagorean theorem. Next, he doubles the number of sides of the inner and outer regular hexagons into inscribed regular 12 and external regular 12 sides, respectively, and then improves the lower and upper bounds of pi with the help of the Pythagorean theorem. He gradually doubles the number of sides of the inner and outer regular polygons until the inscribed regular polygon and the outer regular 96 polygons are inscribed. Finally, he found the lower and upper bounds of pi to be 223/71 and 22/7, respectively, and took their average value of 3.141851 as the approximate value of pi. Archimedes used iterative algorithms and the concept of two-sided numerical approximation, which can be called the originator of "computational mathematics".

In the ancient Chinese arithmetic book "Zhou Ji Sutra" (about the 2nd century B.C.), there is a record of "one path and three days", which means to take. [6] During the Han Dynasty, Zhang Heng derived, i.e. (about 3.162). This value is less accurate, but it is simple and easy to understand.

In 263 A.D., the Chinese mathematician Liu Hui used the "circumcision" to calculate pi, he first connected the regular hexagon from the circle, and then divided it one by one until the circle was connected with a regular 192 polygon. He said, "If you cut it finely, you will lose it, and if you cut it again, you will not be able to cut it, but it will be combined with the circumference and there will be nothing lost." It contains the idea of finding the limit. Liu Hui gave an approximate value of pi of π=3.141024, and after obtaining pi = 3.14, Liu Hui tested this value with the diameter and volume of the copper volume measurement standard Jia Lianghu made in the Han Dynasty in the Jin arsenal, and found that the value of 3.14 was still small. So continue to cut the circle to the 1536 polygon, find the area of the 3072 polygon, and get the pi that satisfies you.

Around 480 A.D., Zu Chongzhi, a mathematician in the Northern and Southern Dynasties, further obtained a result accurate to 7 decimal places, giving an under-approximation of 3.1415926 and an excess approximation of 3.1415927, and also obtained two approximate fractional values, density and approximate rate. The density ratio is a good approximation of fractions, and it is necessary to get a slightly accurate approximation of . [8] (See Diophantine approximation)

In the next 800 years, Zu Chongzhi calculated the most accurate π values. One of the dense rates in the West was not until 1573 by the German Otto (Vale

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Around 530 A.D., the Indian mathematician Ayebodo calculated that pi was about . Brahmagupta used another method to deduce the arithmetic square root of pi equal to 10.

At the beginning of the 15th century, the Arab mathematician al-Qasi obtained the exact decimal value of 17 digits of pi, breaking the record held by Zu Chongzhi for nearly a thousand years. German mathematician Rudolf van Koylen (Ludolph va

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In 1596, he calculated the π to 20 decimal places, and in 1610, he devoted his life to the last 35 decimal places, which he called the Rudolph number after him.

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It was opened at the Aberdeen Proving Ground. The following year, Ritweissner, von Newman, and Medropulis used the computer to calculate the 2,037 decimal places of the π. The computer did the job in just 70 hours, and after deducting the time it took to insert the punch card, it equated to an average of two minutes to calculate a single digit. Five years later, it took only 13 minutes for IBM NORC to calculate the 3,089 decimal places of π. In the 60s and 70s, as computer scientists from the United States, Britain and France continued to compete with each other, the value of π became more and more accurate. In 1973, Jea

Guilloud and Ma

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With the computer CDC 7600 found the millionth decimal place of π.

In 1976, a new breakthrough appeared. Saramine (Euge

e Salami

published a new formula, which is a quadratic convergence rule, which means that with each calculation, the significant figures are multiplied. Gauss had discovered a similar formula, but it was too complex to be feasible in the days before computers. This algorithm is known as the Brent-Saramine (or Saramin-Brent) algorithm, also known as the Gauss-Legendre algorithm.

In 1989, researchers at Columbia University in the United States used the Clay-2 type (C

ay-2) and the IBM-3090/VF giant electronic computer calculated the π value to 480 million decimal places, and then continued to calculate to 1.01 billion decimal places. January 7, 2010 – French engineer Fabrice Béla calculates pi to 2,700 billion decimal places. August 30, 2010 – Japanese computer wizard Shigeru Kondo has used a combination of home computers and cloud computing to calculate pi to 5 trillion decimal places.

On October 16, 2011, Shigeru Kondo, an employee of a company in Iida City, Nagano Prefecture, Japan, used his home computer to calculate pi to 10 trillion decimal places, breaking the Guinness World Record of 5 trillion places set by himself in August 2010. Shigeru Kondo, 56, uses a computer he built himself, and it took about a year to set a new record starting in October.