In Wei and Jin Dynasties, Liu Hui created the secant in order to deduce the formula of circular area and calculate the exact value of pi, which laid a theoretical foundation and provided a scientific algorithm for the study of pi. The so-called pi refers to the ratio of the circumference to the diameter of a circle. Before Liu Hui, China usually adopted the "ancient rate", that is, taking pi as 3, which is very inaccurate. In fact, it is the ratio of the circumference of a hexagon inscribed in a circle to the diameter of the circle, not the ratio of the circumference to the diameter of the circle. However, Liu Hui got inspiration from it: if the circumference is divided into twelve equal parts and made into a regular dodecagon inscribed circle, its area and perimeter are correspondingly closer to the circle than the regular hexagon inscribed circle, so it is more accurate to use the ratio of the perimeter of the regular dodecagon inscribed circle to the diameter of the circle as the approximate value of pi than "three-circumference diameter one". If we make a subdivision and make a circle inscribed with quadrangles, then we can get a more accurate approximate value of pi. "If you cut carefully, you will lose less. Cut it, cut it, so that it can't be cut, then it is in harmony with the circle, and nothing is lost. " Liu Hui starts from a regular hexagon with a circle in it, and multiplies it by the number of sides of the graph. The more sides there are, the closer the polygon area is to the circle area. This is the secant created by Liu Hui. Liu Hui cuts the regular hexagon into the regular dodecagon in the circle and gets the approximate value of pi 3. 14. When Liu Hui increased the number of sides of a regular polygon to 3072, the fractional value of pi was obtained, and the approximate value of decimal was 3. 14 16, accurate to four decimal places. Later generations called this number "emblem rate". This was a world-class achievement at that time. More than two hundred years later, Zu Chongzhi continued to calculate and got more accurate results:
3. 14 15926 < pi < 3. 14 15927。
Zu Chongzhi is the first person in the world to make the value of pi accurate to seven decimal places.
In addition, Zu Chongzhi also gives two low-precision fractional values of pi (called sparsity).
High accuracy (called confidentiality rate)
However, how did Zu Chongzhi make the value of pi accurate to seven decimal places, and how did he calculate the approximate score as pi? These problems are still mysteries in the history of mathematics. According to the analysis of mathematical historians, he probably adopted Liu Hui's "secant circle". If the analysis is correct, then Zu Chongzhi needs to divide a regular hexagon into a regular 12288 polygon and a regular 24576 polygon, and then calculate the perimeter and area of each polygon in turn. This calculation is quite huge. Nine digits must be repeated at least 130 times, including nearly 50 times of power sum roots. Any slight mistake will lead to the failure of calculation. We can see that Zu Chongzhi has a solid mathematical foundation and a rigorous and realistic scientific attitude. The value of pi obtained by Zu Chongzhi was broken by Arab mathematicians in 1427 years later.
Rounding technique
It is an outstanding creation of Shen Kuo, a scientist in the Northern Song Dynasty, in Meng Qian Bi Tan, and gives the approximate relationship between the chord, vector and arc length of the bow. Knowing the circle is developed on the basis of the arc field technique contained in the square field chapter of Nine Chapters Arithmetic. The so-called technique of knowing the diameter of the circle and the height of the bow (that is, the vector) and finding the bow bottom (that is, the chord) and the bow arc. The approximate value calculated by "arc field technology" is not very accurate. Only "circle technology" can get the approximate value, but it is much more accurate.
Shen Kuo's approximate formula of arc length:
Where d is the radius of arc, c is the chord of arc field, and v is the vector of arc field.
Gravity difference technology
The last problems in the Pythagorean chapter of Nine Chapters of Arithmetic are the measurement of cities, the height of mountains and the depth of wells. This measurement method is called "gravity difference technique". Liu Hui, a mathematician in the Three Kingdoms period, wrote a volume of gravity difference, which was attached to the Pythagorean chapter in Nine Chapters of Arithmetic. It was not until the early Tang Dynasty that this part was extracted from Nine Chapters of Arithmetic and became an independent work. Because its first question is about measuring the height and distance of an island, it changed the name of "gravity difference" to "island calculation".
The first question in the island calculation book
Today, an island has hope. The two tables are three feet high and separated by thousands of steps, so that the back table is in line with the front table, but there are 123 steps from the front table. People look at the ground, take the peak of the king, and combine the bottom table, but take 127 steps from the back table. People look at the land, take the peak of the king, and also refer to the bottom table. Ask about the height of the island and the geometry of the table.
This topic proposes that there is an island, and I don't know its height and offshore distance. How to measure the height and offshore distance of the island is discussed.
Liu Hui's solution is:
Set two columns with the height of h feet, and the distance between the two columns is d feet, so that the two columns are in a straight line with the island. A foot back from the front column, eyes rested on the ground, and the top of the observation pole was in a straight line with the top of the mountain. Then step back from the post at the back with foot B, and you can see that the top of the post is in a straight line with the top of the mountain.
Ask the height of the island and the offshore distance of the island:
The height of the island
The distance of the island
Because this calculation requires two differences, namely D and b-a, it was called "double difference technique" in ancient times.
Solution: a = 127 step, b = 127 step, h = 3 feet = 30 feet = 5 steps, d = 1000 step.
Island height (1 Li = 300 steps)
Daoyuan
Residual deficiency technology
The residual method has a long history in the history of mathematics development in China, and it is an original method to solve problems. Nowadays, the method of finding the approximate value of the real root of an equation in higher mathematics is developed from the ancient residual technique. Later mathematicians didn't pay much attention to it, but after it spread to Central Asia and Europe, it was widely used to solve algebraic problems hundreds of years before the development of European algebra, so the residual technique has a glorious position in the history of mathematics in the world.
The skills to solve this kind of problem in "Nine Chapters Arithmetic" are equivalent to the formula:
Number of people:
Price:
Cheng Dawei's lyrics are:
If a mathematician wants to know more,
The two families use each other and become things.
Consolidated surplus, insufficient real number (dividend),
Divide the remaining points by method (divisor) and divide things into prices by method.
Divide the number of people by law
Example: Today, there are (people) shopping, and people give eight and do three; People out of seven, less than four; How many people and things are geometric?
Answer: seven people; Price fifty-three
Solution:
Price = number of people =
equation
Two thousand years ago, there was a famous mathematical work in ancient China called Nine Chapters Arithmetic, in which one chapter was called Equation, which was about a system of linear equations with multiple variables. /kloc-Around the 0/7th century, European algebra was first introduced to China, and then' equation' was translated into' equation'. /kloc-In the middle of the 9th century, modern western mathematics was introduced to China again. 1859, Li, a mathematician in the Qing Dynasty, and Willie Yali, a British missionary, jointly translated Elementary Algebra, in which the translation of' equation' borrowed the word' equation' in ancient China, so the word' equation' originally meant' equation with unknowns'. 1873, mathematician Hua of Qing Dynasty and English missionary Lan Yahe translated Algebra. They translated "equality" into "equality". What they mean is to distinguish between' equation' and' equation', the latter still refers to the meaning in' nine chapters of arithmetic', while' equation' refers to' equation with unknowns'. Until 1934, mathematicians in China investigated the mathematical terms one by one and determined that "equation" and "equation" had the same meaning. So far, "equation" and "equation" are synonyms, which have been used up to now.
Yang Hui Triangle-Binomial Array
Yang Hui, a mathematician in the Song Dynasty, wrote a book "Detailed Explanation of Nine Chapters Algorithm" in A.D. 126 1 year, which recorded a "root diagram of square root method". People call it "Yang Hui Triangle", which is a triangular array arranged by numbers. This triangle is called "Pascal's triangle" in the west, but it was the seventeenth century that the French mathematician Pascal made it. According to Yang Hui's Source Map of Methods: "Jia Xian used this technique in the book of calculation". Jia Xian was a mathematician in the Northern Song Dynasty at the beginning of 1 1 century, more than two centuries before Yang Hui, so this triangle should be called "Jia Xian Triangle".
"Jiaxian Triangle" is actually a coefficient table of binomial a+b multiplied by square. See the following five sentences in Draw a Square:
"The large piece on the left is a product, and the large piece on the right is an even calculation. Hiding in China is cheap, so taking advantage of it takes advantage of the business side, and life is actually removed. 」
The first three sentences explain the structure of Jiaxian Triangle, and the last two sentences explain the role of each coefficient in the method of establishing and releasing locks.
A rectangular piece of land, the east-west length is called width, and the north-south length is called Mao. North and south extend up and down. )
"The product number on the left" means that the top-down line "111111"on the left is a constant term coefficient in binomial expansion;
"Even number on the right" means that "111111"from top to bottom on the right is the highest term coefficient in the expansion;
"People hiding in China are cheap" means that the middle number is the corresponding coefficient of each item;
"Quotient multiplied by cheap, but divided by real number" means that the obtained quotient is multiplied by the coefficients of each term when solving the equation, and then the real number is subtracted.
After Yang Hui, Zhu Shijie's "Meet Siyuan" has the same picture.
It is called "the seven power diagrams of ancient law"
Multiplication and opening method
That is, the numerical solution of the higher order equation, the coefficients of any higher order expansion can be obtained. The numerical solution of higher order equation is one of the most important contents in China's traditional mathematics, which has a long history and outstanding achievements. There are clear and standardized steps in the Nine Chapters of Arithmetic in Hanshu, and there are also records of solving quadratic equations with one variable. Since then, Zu Chongzhi and his son have studied the solution of cubic equation in their works in the Northern and Southern Dynasties, and Wang Xiaotong in the Tang Dynasty has studied the solution of cubic equation in "Jigu Shujing". In the Northern Song Dynasty, Liu Yi founded the Positive and Negative cholesky decomposition. Jia Xian wrote The Fine Grass of the Nine Chapters Algorithm of Huangdi, some of which were adopted by Yang Hui as Detailed Explanation of the Nine Chapters Algorithm, which kept Jia Xian's outstanding mathematical achievements: multiplication and division; Jia Xian developed the method of multiplication and division, founded the origin of cholesky decomposition, and solved the common problem of opening the higher power. Cholesky decomposition's root graph is a numerical table with numbers arranged in a triangle, which is called Jia Xian Triangle. The coefficients in binomial expansion are given.
Large derivative technique
Is to solve the problem of simultaneous linear congruence groups. This kind of problem has a long history in ancient Chinese mathematics, which can be traced back at least to the calculation of Shangyuan accumulated years in the calendar of Han Dynasty. The mathematical model of "I don't know how to count things" in Sun Tzu's calculation shows that this method was quite mature in the Southern and Northern Dynasties. In the13rd century, Qin gave a complete method and extended it to the most general situation. This method is called "the technique of finding the total number by great derivation", and the method of solving a congruence problem in ancient China is usually called "the technique of finding a congruence by great derivation". In Europe, it ranks in Euler (1707- 1783), Lagrange (1736- 18 13) and Gauss (1777-6550). China's outstanding creation of ancient mathematics is called "China's Remainder Theorem" by local scholars, and it should be called "Sun Tzu's Theorem" by Chinese mathematical history circles.
Tianyuanshu
Tianyuan refers to the unknown in the problem, and "setting Tianyuan XXX" is equivalent to the current meaning of "setting X as XXX". This general method of establishing a unary algebraic equation with only one solid unknown is called "astronomy". The origin of "Tianyuan Shu" is about1around the beginning of the third century, and the name and age of the creator cannot be verified. There are Measuring the Round Sea Mirror, Siyuan Jade Mirror by Zhu Shijie in Song Dynasty and Arithmetic Enlightenment.
To the left of the word "yuan" is the coefficient of the linear term.
The upper layer is the quadratic and cubic term coefficients in turn, and the lower layer is the constant term, as shown in the equation on the right.
Quaternary technology
It is a set of algebraic methods to deal with the problems of multivariate high-order systems (up to four unknowns) in ancient China. It is an extension of one-dimensional equation containing only one element to higher-order simultaneous equations of binary, ternary and quaternary. Because there can be as many as four unknowns, later generations call the technology of expanding celestial bodies "four elements". The four elements in the "Four Elements Technique" are equivalent to X, Y, Z and W now, and each term of the equation has its corresponding fixed position in the formulation.
Multivariate linearity means that different unknowns are represented by different "elements",
There are Tianyuan, Geo-yuan, Human-yuan and Matter-yuan. And then put the word "Tai" in the middle of each yuan, with Tianyuan below, Matter Yuan above, Ground Yuan on the left and Man Yuan on the right.
The equation 2 x+6 y+3 z+7 w shown on the right is 0.
Appeal difference
That is, interpolation method is an important achievement with world significance in the history of Chinese mathematics. The calendar of the Han Dynasty used an interpolation method, the Sui and Tang Dynasties initiated the second interpolation method, and the meta-mathematician Wang Xun used the third interpolation method and applied it to many problems in the calendar. On this basis, Zhu Shijie went a step further, and regarded the sum of overlapping products as a relative inverse operation, and established four interpolation formulas by using the results of triangular overlapping system, which was superior to similar results in the West.
Stacking operation
That is, the summation of higher-order arithmetic progression. Some discrete objects with the same shape and size are piled into a regular platform. How to calculate the number of these objects?
Volume formulas of various platforms and quasi-platforms have been drawn in Nine Chapters of Arithmetic, but the problem of accumulation of discrete objects was formally put forward by Shen Kuo and solved satisfactorily. This achievement constitutes the beginning of the research on stacking technology in China. Then someone studied it. In the Southern Song Dynasty, Yang Hui gave three superposition formulas in Detailed Explanation of Nine Chapters Algorithm and Algorithm Changing Background:
Triangular stack
fourfold
Square stack (where n is the number of stack layers)
Later, with the great development of Zhu Shijie in Yuan Dynasty, he made a systematic and in-depth study on the overlapping problem in Meet with Siyuan, and made great achievements, making it a topic that mathematicians have paid attention to for hundreds of years.
In Zhu Shijie's many series summation problems, a series of important formulas can be summarized:
This summation formula is collectively called triangular crib formula.
In the19th century, Li's "stack-to-stack ratio" embodies the achievements of stack in history, but it has been further developed. On this basis, a series of outstanding achievements have been produced, such as Li Heng's identity and "pointed cone technique".
Vertical and horizontal diagrams
That is, the so-called magic square in modern times generally refers to a square composed of continuous natural numbers from 1 to n, and the sum of n numbers in each row, column and diagonal is the same. It appeared at the latest in the Warring States period and was called Luoshu or Jiugong, but it didn't develop further in the next thousand years.
Luo Shu is clearly a third-order Rubik's cube, and the sum of the three numbers in the diagonal is fifteen. According to the Northern Zhou Dynasty's notes on numbers: "Nine palaces, two or four shoulders, six or eight feet, three left and seven right, and five middle schools", it is the oldest third-order Rubik's cube in the world.
Luoshu
4 9 2
3 5 7
8 1 6
Yang Hui created the name of "vertical and horizontal diagram" in his "Algorithm for Extracting Odds from Ancient Stories", which included thirteen magic squares, including: Luo Shu number (third-order magic square) 1, Hua Sixteen diagram (fourth-order magic square) 2, Wu Wutu (fifth-order magic square) 2, Liu Liutu (sixth-order magic square) 2 and derivative diagram. Yang Hui not only memorized many magic squares, but also put forward general methods to make magic squares of odd order, 3 n order and even order, becoming the first mathematician to systematically discuss magic squares in the history of Chinese mathematics. In addition, there are many kinds of vertical and horizontal diagrams in the Arithmetic Collection written by Wang Wensu in Ming Dynasty, and there are 14 kinds of vertical and horizontal diagrams in Cheng Dawei's Arithmetic Unification. In the Qing Dynasty, Fang Zhongtong's Shu Du Yan was equipped with 14 kinds of vertical and horizontal maps after the "Nine-Nine Maps", which were basically the same as those in Yang Hui's works. Similar work in Europe was not systematically carried out until the sixteenth century.
46 8 16 20 29 7 49
3 40 35 36 18 4 1 2
44 12 33 23 19 38 6
28 26 1 1 25 39 24 22
5 37 3 1 27 17 13 45
48 9 15 14 32 10 47
1 43 34 30 2 1 42 4
Diffraction diagram (seventh-order Rubik's cube) (vertical and horizontal oblique 175)
3 1 76 13 36 8 1 18 29 74 1 1
22 40 58 27 45 63 20 38 56
67 4 49 72 9 54 65 2 47
30 75 12 32 77 14 34 79 16
2 1 39 57 23 4 1 59 25 43 6 1
66 3 48 68 5 50 70 7 52
35 80 17 28 73 10 33 78 15
26 44 62 19 37 55 24 42 60
7 1 8 53 64 1 46 69 6 5 1
Jiujiutu (ninth-order Rubik's Cube) (vertical and horizontal oblique 369)
On the right is Yang Hui's 99-99 diagram, which clearly shows that he tried to construct a general 3 n-order magic square on the basis of the third-order magic square:
This ninth-order Rubik's Cube is obviously divided into nine-order squares, and each number of the third-order matrix is composed of multiples of nine plus the numbers in the blue box in the figure. The structure is completely consistent, harmonious and symmetrical, regular and orderly, and it has reached a very beautiful state in mathematics. It embodies the higher theoretical level of Yang Hui's Rubik's Cube research.
1 20 2 1 40 4 1 60 6 1 80 8 1 100
99 82 79 62 59 42 39 22 19 2
3 18 23 38 43 58 63 78 83 98
97 84 77 64 57 44 37 24 17 4
5 16 25 36 45 56 65 76 85 96
95 86 75 66 55 46 35 26 15 6
14 7 34 27 54 47 74 67 94 87
88 93 68 73 48 53 28 33 8 13
12 9 32 29 52 49 72 69 92 89
9 1 90 7 1 70 5 1 50 3 1 30 1 1 10
Baizi diagram (magic square of tenth order) (vertical and horizontal oblique 505)
Pointing cone technique
In a.d. 1845, Li Jianli put forward a set of integral theory-sharp cone in his book "Interpretation" in a simple form:
Volume consists of area, and area consists of line segments.
Volume can be changed into area, and area can be changed into line segment.
right triangle
Why did China call the right triangle "Pythagorean" in ancient times?
It turns out that there was a wooden pole called "table" on the ground during astronomical survey in ancient China.
The watch casts a shadow on the ground, so the watch and the shadow form two right sides of a right triangle. In ancient China, the right triangle was called "Pythagorean", the right side of the table was called "Hook", the right side of the sun shadow was called "Chord", and the hypotenuse of Pythagorean was called "Chord".
The height of the sun can be roughly calculated by measuring the length of Pythagoras.