108-477: Xiahou Yang Suanjing ( Xiahou Yang's Mathematical Manual ) is a mathematical treatise attributed to the fifth century CE Chinese mathematician Xiahou Yang. However, some historians are of the opinion that Xiahou Yang Suanjing was not written by Xiahou Yang. It is one of the books in The Ten Computational Canons , a collection of mathematical texts assembled by Li Chunfeng and used as
216-406: A + b ) 2 {\displaystyle (a+b)^{2}} as well as 2 a b + c 2 {\displaystyle 2ab+c^{2}} , with 2 a b {\displaystyle 2ab} representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length
324-413: A b + a 2 + b 2 {\displaystyle 2ab+a^{2}+b^{2}} . Since both squares have the area of ( a + b ) 2 {\displaystyle (a+b)^{2}} it follows that the other measure of the square area also equal each other such that 2 a b + c 2 {\displaystyle 2ab+c^{2}} = 2
432-462: A b + a 2 + b 2 {\displaystyle 2ab+a^{2}+b^{2}} . With the area of the four triangles removed from both side of the equation what remains is a 2 + b 2 = c 2 . {\displaystyle a^{2}+b^{2}=c^{2}.} In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of
540-452: A + b = c , there exists a triangle with sides a , b and c as a consequence of the converse of the triangle inequality . This converse appears in Euclid's Elements (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle
648-406: A and b . These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner. In this new position, this left side now has a square of area ( a + b ) 2 {\displaystyle (a+b)^{2}} as well as 2
756-420: A to give the equation This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy . The converse of the theorem is also true: Given a triangle with sides of length a , b , and c , if a + b = c , then the angle between sides a and b is a right angle . For any three positive real numbers a , b , and c such that
864-688: A 10th order equation. Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa (詳解九章算法), although it was described earlier around 1100 by Jia Xian . Although the Introduction to Computational Studies (算學啓蒙) written by Zhu Shijie ( fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics . Ceyuan haijing ( Chinese : 測圓海鏡 ; pinyin : Cèyuán Hǎijìng ), or Sea-Mirror of
972-481: A circle so that eventually the area of a higher-order polygon will be identical to that of the circle. From this method, Liu Hui asserted that the value of pi is about 3.14. Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles. In
1080-470: A counting board and included the use of negative numbers as well as fractions. The counting board was effectively a matrix , where the top line is the first variable of one equation and the bottom was the last. Liu Hui 's commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available. Hui is believed by most to be a mathematician shortly after
1188-459: A creator of mathematics, although debate about this continues. The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c , as shown in the lower part of the diagram. This results in a larger square, with side a + b and area ( a + b ) . The four triangles and the square side c must have the same area as the larger square, giving A similar proof uses four copies of
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#17327721046391296-502: A form of algebraic geometry based on tiān yuán shù . His book; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians. Things grew quiet for
1404-588: A good fraction approximate for pi; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, "the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe". Along with his son, Zu Geng, Zu Chongzhi applied
1512-545: A mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His Ts'e-yuan hai-ching ( Sea-Mirror of the Circle Measurements ) includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used
1620-450: A measure of a + b {\displaystyle a+b} and which contain four right triangles whose sides are a , b and c , with the hypotenuse being c . In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c . Each outer square has an area of (
1728-558: A new revision would be published in 1776 to correct various errors as well as include a version of The Nine Chapters from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng. The final version of Dai Zhen's work would come in 1777, titled Ripple Pavilion , with this final rendition being widely distributed and coming to serve as the standard for modern versions of The Nine Chapters . However, this version has come under scrutiny from Guo Shuchen, alleging that
1836-533: A number of more or less independent short sections of text drawn from a number of sources. The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art. An example of the elementary mathematics in the Suàn shù shū , the square root is approximated by using false position method which says to "combine the excess and deficiency as
1944-554: A right triangle with sides a , b and c , arranged inside a square with side c as in the top half of the diagram. The triangles are similar with area 1 2 a b {\displaystyle {\tfrac {1}{2}}ab} , while the small square has side b − a and area ( b − a ) . The area of the large square is therefore But this is a square with side c and area c , so This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction);
2052-436: A small central square. Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a and b , which must have the same area as the initial large square. The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in
2160-534: A sophisticated use of hexagrams . Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers . Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations , and negative numbers with counting rods . Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also
2268-757: A standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering. Emperor Qin Shi Huang ordered many men to build large, life-sized statues for the palace tomb along with other temples and shrines, and the shape of the tomb was designed with geometric skills of architecture. It is certain that one of the greatest feats of human history, the Great Wall of China , required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion. Qin bamboo cash purchased at
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#17327721046392376-590: A symbol for zero he had difficulties expressing the number). Northern Song dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule. Four outstanding mathematicians arose during the Song dynasty and Yuan dynasty , particularly in the twelfth and thirteenth centuries: Yang Hui , Qin Jiushao , Li Zhi (Li Ye), and Zhu Shijie . Yang Hui, Qin Jiushao, Zhu Shijie all used
2484-503: A theory of proportions, a topic not discussed until later in the Elements , and that the theory of proportions needed further development at that time. Albert Einstein gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of
2592-573: A time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with Zhu Shijie 's two books Suanxue qimeng and the Jade Mirror of the Four Unknowns . In one case he reportedly gave a method equivalent to Gauss 's pivotal condensation. Qin Jiushao ( c. 1202 – 1261)
2700-629: A tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty . While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of
2808-456: A triangle, CDE , which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC . Therefore, the ratios of their sides must be the same, that is: This can be rewritten as y d y = x d x {\displaystyle y\,dy=x\,dx} , which is a differential equation that can be solved by direct integration: giving The constant can be deduced from x = 0, y =
2916-455: A zenith in the 13th century during the Yuan dynasty with the development of tian yuan shu . As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while
3024-427: Is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y , the side AC of length x and the side AB of length a , as seen in the lower diagram part. If x is increased by a small amount dx by extending the side AC slightly to D , then y also increases by dy . These form two sides of
3132-398: Is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality ). The following statements apply: Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language: where α
3240-427: Is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be congruent , proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting
3348-424: Is done through successive approximation, the same as division, and often uses similar terms such as dividend ( shi ) and divisor ( fa ) throughout the process. This process of successive approximation was then extended to solving quadratics of the second and third order, such as x 2 + a = b {\displaystyle x^{2}+a=b} , using a method similar to Horner's method. The method
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3456-679: Is really known about his life. Today, the only sources are found in Book of Sui , we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927), which would remain the most accurate approximation of π available for the next 900 years. He also applied He Chengtian's interpolation for approximating irrational number with fraction in his astronomy and mathematical works, he obtained 355 113 {\displaystyle {\tfrac {355}{113}}} as
3564-508: Is right." It can be proved using the law of cosines or as follows: Let ABC be a triangle with side lengths a , b , and c , with a + b = c . Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length c = √ a + b , the same as the hypotenuse of the first triangle. Since both triangles' sides are
3672-450: Is subdivided into (1) "ordinary division"; (2) "division by ten, hundred, and so on," especially intended for work in mensuration; (3) "division by simplification" (yo ch'ut). The last problem in the section is as follows: Fractions are also mentioned, special names being given to the four most common ones, as follows: In the second section there are twenty-eight applied problems relating to taxes, commissions, and such questions as concern
3780-422: Is taken to be equal to three in both texts. However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used. Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment. The Chinese did not focus on theoretical proofs based on geometry or algebra in
3888-713: The Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300 and 250 BCE. The Zhoubi Suanjing contains an in-depth proof of the Gougu Theorem , a special case of the Pythagorean theorem ) but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips , dated c. 305 BCE , has revealed some aspects of pre-Qin mathematics, such as
3996-455: The Book on Numbers and Computation and Huainanzi are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to
4104-453: The Chinese calendar and astronomy . Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham writes: Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs , one of which passed through the summer solstice point...By such methods he was able to obtain
4212-484: The Greek philosopher Pythagoras , born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry , Euclidean distance satisfies
4320-463: The Han dynasty , as diophantine approximation being a prominent numerical method , the Chinese made substantial progress on polynomial evaluation . Algorithms like regula falsi and expressions like simple continued fractions are widely used and have been well-documented ever since. They deliberately find the principal n th root of positive numbers and the roots of equations . The major texts from
4428-526: The Horner - Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove " Pascal's Triangle ", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on
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4536-609: The Indian mathematician , Aryabhata , were translated into the Chinese mathematical book of the Kaiyuan Zhanjing , compiled in 718 AD during the Tang dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry , binomial theorem , and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics . Yi Xing ,
4644-405: The altitude from point C , and call H its intersection with the side AB . Point H divides the length of the hypotenuse c into parts d and e . The new triangle, ACH, is similar to triangle ABC , because they both have a right angle (by definition of the altitude), and they share the angle at A , meaning that the third angle will be the same in both triangles as well, marked as θ in
4752-500: The "School of Computations". Wang Xiaotong was a great mathematician in the beginning of the Tang dynasty , and he wrote a book: Jigu Suanjing ( Continuation of Ancient Mathematics ), where numerical solutions which general cubic equations appear for the first time. The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan , who died in 630. The table of sines by
4860-569: The Cavalieri's principle to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi. His work, Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that Zhui Shu contains the formulas and methods for linear , matrix algebra , algorithm for calculating
4968-408: The Chinese had the concept of negative numbers . By the Tang dynasty study of mathematics was fairly standard in the great schools. The Ten Computational Canons was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (李淳風 602–670), as the official mathematical texts for imperial examinations in mathematics. The Sui dynasty and Tang dynasty ran
5076-468: The Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science. Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics
5184-441: The Circle Measurements , is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by Li Zhi (or Li Ye) (1192–1272 AD). He used Tian yuan shu to convert intricated geometry problems into pure algebra problems. He then used fan fa , or Horner's method , to solve equations of degree as high as six, although he did not describe his method of solving equations. "Li Chih (or Li Yeh, 1192–1279),
5292-515: The Han dynasty. The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them. Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art . Calculating the square and cube roots of numbers
5400-456: The Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint. For instance, throughout The Nine Chapters on the Mathematical Art , the value of pi is taken to be equal to three in problems regarding circles or spheres. In his commentary, Liu Hui finds a more accurate estimation of pi using the method of exhaustion . The method involves creating successive polygons within
5508-577: The Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275). The Jade Mirror of the Four Unknowns was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses
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#17327721046395616-518: The Italian Jesuit Matteo Ricci (1552–1610). After the overthrow of the Yuan dynasty , China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of botany and pharmacology . Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes: At the end of the 16th century, Chinese autochthonous mathematics known by
5724-465: The Mathematical Art also deals with solving a system of two equations with two unknowns with the false position method. To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or zi (which are the values given for the excess and deficit) with the major terms mu . To solve for the lesser of the two unknowns, simply add the minor terms together. Chapter Eight of The Nine Chapters on
5832-502: The Mathematical Art deals with solving infinite equations with infinite unknowns. This process is referred to as the "fangcheng procedure" throughout the chapter. Many historians chose to leave the term fangcheng untranslated due to conflicting evidence of what the term means. Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Problems were done on
5940-452: The Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. The theorem can be generalized in various ways: to higher-dimensional spaces , to spaces that are not Euclidean , to objects that are not right triangles, and to objects that are not triangles at all but n -dimensional solids. In one rearrangement proof, two squares are used whose sides have
6048-479: The Song dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendar science and astronomical calculations. The polymath and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Joseph W. Dauben notes that in Shen's "technique of intersecting circles" formula, he creates an approximation of
6156-555: The antiquarian market of Hong Kong by the Yuelu Academy , according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise. In the Han dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called rod calculus , consisting of only nine symbols with a blank space on the counting board representing zero. Negative numbers and fractions were also incorporated into solutions of
6264-460: The arc of a circle s by s = c + 2 v / d , where d is the diameter , v is the versine , c is the length of the chord c subtending the arc. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). Gauchet and Needham state Guo used spherical trigonometry in his calculations to improve
6372-400: The area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. Another by rearrangement is given by the middle animation. A large square is formed with area c , from four identical right triangles with sides a , b and c , fitted around
6480-452: The area of the square, that is The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of 1 2 {\displaystyle {\frac {1}{2}}} , which is removed by multiplying by two to give the result. One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus . The triangle ABC
6588-499: The average scholar, then, tianyuan seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art , he omitted Tian yuan shu and the increase multiply method. Pythagorean theorem#History In mathematics , the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between
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#17327721046396696-403: The book The Pythagorean Proposition contains 370 proofs. This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C , as shown on the figure. Draw
6804-588: The calculation by calculating pi to be 3.141024. Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle. Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century. There is no explicit method or record of how he calculated this estimate. Basic arithmetic processes such as addition, subtraction, multiplication and division were present before
6912-536: The development of place-value systems and place-value systems and the associated Galley division in the West. European sources learned place-value techniques in the 13th century, from a Latin translation an early-9th-century work by Al-Khwarizmi . Khwarizmi's presentation is almost identical to the division algorithm in Sunzi , even regarding stylistic matters (for example, using blank spaces to represent trailing zeros);
7020-572: The division by army officers of loot and food (silk, rice, wine, soy sauce, vinegar, and the like) among their soldiers. The third section contains forty-two problems. The translations of some of these problems are given below. Chinese mathematics Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers , more than one numeral system ( binary and decimal ), algebra , geometry , number theory and trigonometry . Since
7128-434: The divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method. The Nine Chapters on the Mathematical Art dates archeologically to 179 CE, though it is traditionally dated to 1000 BCE, but it
7236-469: The du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree). Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and
7344-415: The edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself. Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout
7452-590: The extreme end of a line is a point. Much like Euclid 's first and third definitions and Plato 's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus , the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved." It stated that two lines of equal length will always finish at
7560-412: The figure. By a similar reasoning, the triangle CBH is also similar to ABC . The proof of similarity of the triangles requires the triangle postulate : The sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate . Similarity of the triangles leads to the equality of ratios of corresponding sides: The first result equates the cosines of the angles θ , whereas
7668-488: The first known decimal multiplication table . The abacus was first mentioned in the second century BC, alongside 'calculation with rods' ( suan zi ) in which small bamboo sticks are placed in successive squares of a checkerboard. Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars , circa 213–210 BC. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created
7776-592: The first to develop negative numbers, algebraic geometry , and the usage of decimals. Math was one of the Six Arts students were required to master during the Zhou dynasty (1122–256 BCE). Learning them all perfectly was required to be a perfect gentleman, comparable to the concept of a " renaissance man ". Six Arts have their roots in the Confucian philosophy . The oldest existent work on geometry in China comes from
7884-648: The great mathematical texts of the period. The mathematical texts of the time, the Book on Numbers and Computation and Jiuzhang suanshu solved basic arithmetic problems such as addition, subtraction, multiplication and division. Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order. Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns. The value of pi
7992-400: The hypotenuse (see Similar figures on the three sides ). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have
8100-500: The legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well. In outline, here is how the proof in Euclid 's Elements proceeds. The large square
8208-452: The lower sections. The first chapter contains 19 problems, the second chapter contains 29 problems and the last chapter contains 44 problems. As in all the older Chinese books, no technical rules are given, and the problems are simply followed by the answers, occasionally with brief explanations. In the first section the five operations of addition, subtraction, multiplication, division, and square and cube roots are given. The work on division
8316-422: The lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called dissection . This shows the area of the large square equals that of the two smaller ones. As shown in the accompanying animation, area-preserving shear mappings and translations can transform
8424-400: The mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha , while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. Yi Xing was famed for his genius, and was known to have calculated the number of possible positions on a go board game (though without
8532-1067: The method of fan fa , today called Horner's method, to solve these equations. There are many summation series equations given without proof in the Mirror . A few of the summation series are: 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 3 ! {\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}} 1 + 8 + 30 + 80 + ⋯ + n 2 ( n + 1 ) ( n + 2 ) 3 ! = n ( n + 1 ) ( n + 2 ) ( n + 3 ) ( 4 n + 1 ) 5 ! {\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}} The Mathematical Treatise in Nine Sections ,
8640-471: The modern Gaussian elimination and back substitution . The version of The Nine Chapters that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from Yongle Encyclopedia , he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations. His finished work would be first published in 1774, but
8748-414: The modern sense of proving equations to find area or volume. The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life. The Book on Numbers and Computation is approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1984 when archaeologists opened
8856-410: The official mathematical for the imperial examinations . Though little is known about the period of the author, there is some evidence which more or less conclusively establishes the date of the work. These suggest 468 CE as the latest possible date for the work to be written and 425 CE as the earliest date. The treatise is divided into three parts and these are spoken of as the higher, the middle and
8964-480: The period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions . The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra . The achievement of Chinese algebra reached
9072-473: The philosophical Mohist canon c. 330 BCE , compiled by the followers of Mozi (470–390 BCE). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms
9180-491: The problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. Later, Liu Hui attempted to improve
9288-529: The proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as
9396-465: The same area as one of the two squares on the legs. For the formal proof, we require four elementary lemmata : Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. The proof is as follows: This proof, which appears in Euclid's Elements as that of Proposition 47 in Book ;1, demonstrates that
9504-408: The same lengths a , b and c , the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle. The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proved without assuming the Pythagorean theorem. A corollary of the Pythagorean theorem's converse
9612-528: The same place," while providing definitions for the comparison of lengths and for parallels ," along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided word recognition for circumference, diameter, and radius, along with the definition of volume. The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example,
9720-405: The second result equates their sines . These ratios can be written as Summing these two equalities results in which, after simplification, demonstrates the Pythagorean theorem: The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved
9828-451: The similarity suggests that the results may not have been an independent discovery. Islamic commentators on Al-Khwarizmi's work believed that it primarily summarized Hindu knowledge; Al-Khwarizmi's failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China. In the fifth century the manual called " Zhang Qiujian suanjing " discussed linear and quadratic equations. By this point
9936-689: The solution of equations, and the properties of right triangles. The Nine Chapters made significant additions to solving quadratic equations in a way similar to Horner's method . It also made advanced contributions to fangcheng , or what is now known as linear algebra. Chapter seven solves system of linear equations with two unknowns using the false position method, similar to The Book of Computations. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns. The Nine Chapters solves systems of equations using methods similar to
10044-414: The square on the hypotenuse. A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative ). Instead of a square it uses a trapezoid , which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half
10152-539: The square. In this way they also form two boxes, this time in consecutive corners, with areas a 2 {\displaystyle a^{2}} and b 2 {\displaystyle b^{2}} which will again lead to a second square of with the area 2 a b + a 2 + b 2 {\displaystyle 2ab+a^{2}+b^{2}} . English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements , and mentions
10260-413: The squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly. Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of
10368-402: The text. The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of 246 problems. It was later incorporated into The Ten Computational Canons , which became the core of mathematical education in later centuries. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation,
10476-441: The third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium. He was the first Chinese mathematician to calculate π =3.1416 with his π algorithm . He discovered
10584-407: The three sides of a right triangle . It states that the area of the square whose side is the hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a , b and the hypotenuse c , sometimes called the Pythagorean equation : The theorem is named for
10692-516: The time of the Duke of Zhou . Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal , such as the Song-era polymath Shen Kuo . Shang dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the I Ching , which greatly influenced written literature during the Zhou dynasty (1050–256 BC). For mathematics, the book included
10800-422: The two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow. Let A , B , C be the vertices of a right triangle, with a right angle at A . Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having
10908-475: The usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the infinitesimal calculus during the 3rd century CE. In the fourth century, another influential mathematician named Zu Chongzhi , introduced the Da Ming Li. This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little
11016-452: The value of π , formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics. A mathematical manual called Sunzi mathematical classic dated between 200 and 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. Intriguingly, Sunzi may have influenced
11124-536: Was not extended to solve quadratics of the nth order during the Han dynasty; however, this method was eventually used to solve these equations. The Book of Computations is the first known text to solve systems of equations with two unknowns. There are a total of three sets of problems within The Book of Computations involving solving systems of equations with the false position method, which again are put into practical terms. Chapter Seven of The Nine Chapters on
11232-446: Was not rediscovered on a large scale prior to the last quarter of the 18th century. Correspondingly, scholars paid less attention to mathematics; preeminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the 'increase multiply' method. Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods. To
11340-510: Was the first to introduce the zero symbol into Chinese mathematics." Before this innovation, blank spaces were used instead of zeros in the system of counting rods . One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?" Qin also solved
11448-423: Was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten." He also worked with magic circle . The embryonic state of trigonometry in China slowly began to change and advance during
11556-454: Was written by the wealthy governor and minister Ch'in Chiu-shao ( c. 1202 – c. 1261 ) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis. The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261–1275), who worked with magic squares of order as high as ten. "The same "Horner" device
11664-403: Was written perhaps as early as 300–200 BCE. Although the author(s) are unknown, they made a major contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure. There are no formal mathematical proofs within the text, just a step-by-step procedure. The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within
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