61-633: Jiuzhang may refer to: Jiuzhang suanshu , or The Nine Chapters on the Mathematical Art , Chinese mathematics book, composed from the 10th–2nd century BCE Shu shu Jiuzhang , or Mathematical Treatise in Nine Sections , 13th century Chinese mathematical text by Qin Jiushao Jiu Zhang , collection of poems attributed to Qu Yuan Nine Chapter Law , or Jiuzhang Lü, law of
122-406: 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
183-413: 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
244-462: 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
305-452: 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
366-406: 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
427-420: 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
488-459: 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
549-490: A manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide formal proofs in the Euclidean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang ( fl. 165 BCE – d. 142 BCE) and Geng Shouchang (fl. 75 BCE – 49 BCE) (see armillary sphere ) with the initial arrangement and commentary on
610-450: 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 (
671-462: A number of more or less independent short sections of text drawn from a number of sources. The Zhoubi Suanjing , a mathematics and astronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong . The title of the book has been translated in a wide variety of ways. In 1852, Alexander Wylie referred to it as Arithmetical Rules of
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#1732783457076732-554: 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);
793-471: A section, several parts of an article, or an entire treatise. In this light, many scholars of the history of Chinese mathematics compare the significance of The Nine Chapters on the Mathematical Art on the development of Eastern mathematical traditions to that of Euclid's Elements on the Western mathematical traditions. However, the influence of The Nine Chapters on the Mathematical Art stops short at
854-436: 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
915-503: 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
976-456: 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 =
1037-427: 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
1098-398: 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
1159-583: Is also the mathematical proof given in the treatise for the Pythagorean theorem . The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and Japan . Its influence on mathematical thought in China persisted until the Qing dynasty era. Liu Hui wrote a detailed commentary in 263. He analyses the procedures of The Nine Chapters step by step, in
1220-427: 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
1281-617: Is represented by a Cartesian coordinate system in analytic geometry , Euclidean distance satisfies 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
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#17327834570761342-448: 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
1403-604: Is the basis for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics. The "equations" discussed in the Fang Cheng chapter are equivalent to today's simultaneous linear equations. The solution method called "Fang Cheng Shi" is best known today as Gaussian elimination. Among the eighteen problems listed in the Fang Cheng chapter, some are equivalent to simultaneous linear equations with two unknowns, some are equivalent to simultaneous linear equations with 3 unknowns, and
1464-627: The Ten Computational Canons . The full title of The Nine Chapters on the Mathematical Art appears on two bronze standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles. The title is also mentioned in volume 24 of the Book of the Later Han as one of the books studied by Ma Xu (馬續). Based on this known knowledge, his younger brother Ma Rong (馬融) places
1525-576: The Zhangjiashan Han bamboo texts . From documentary evidence this tomb is known to have been closed in 186 BCE, 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 Suàn shù shū is however much less systematic than The Nine Chapters ; and appears to consist of
1586-405: 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
1647-521: The geometry clauses of the Mozi of the 4th century BCE. This is no longer the case. The Suàn shù shū (算數書) or Writings on Reckonings is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of texts known as
1708-479: The 10th–2nd century BCE, its latest stage being from the 1st century CE. This book is one of the earliest surviving mathematical texts from China , the others being the Suan shu shu (202 BCE – 186 BCE) and Zhoubi Suanjing (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with
1769-400: The Han dynasty Jiuzhang (quantum computer) , a model quantum computer developed by University of Science and Technology of China Zhao Jiuzhang , Chinese scientist Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Jiuzhang . If an internal link led you here, you may wish to change the link to point directly to
1830-534: The Mathematical Art can be regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms in The Nine Chapters on the Mathematical Art are very detailed. Through these discussions, one can understand the achievements of the development of ancient Chinese mathematics. Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but also general quadratic and cubic equations. It
1891-418: The Mathematical Art for the first time. Later in 1994, Lam Lay Yong used this title in her overview of the book, as did other mathematicians including John N. Crossley and Anthony W.-C Lun in their translation of Li Yan and Du Shiran's Chinese Mathematics: A Concise History (Li and Du 1987). Afterwards, the name The Nine Chapters on the Mathematical Art stuck and became the standard English title for
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1952-404: The Mathematical Art gives a certain discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. Generally speaking, it has the prototype of the real number system used in modern mathematics. The geometric figures included in The Nine Chapters on the Mathematical Art are mostly straight and circular figures because of its focus on the applications onto
2013-629: The Nine Sections. With only a slight variation, the Japanese historian of mathematics Yoshio Mikami shortened the title to Arithmetic in Nine Sections . David Eugene Smith , in his History of Mathematics (Smith 1923) , followed the convention used by Yoshio Mikami . Several years later, George Sarton took note of the book, but only with limited attention and only mentioning the usage of red and black rods for positive and negative numbers. In 1959, Joseph Needham and Wang Ling (historian) translated Jiu Zhang Suan shu as The Nine Chapters on
2074-534: The Pythagorean Theorem, the book divides it into four main categories: Gou Gu mutual seeking, Gou Gu integer, Gou Gu dual capacity, Gou Gu similar. Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including famously the triple 3,4,5. Gou Gu dual capacity discusses algorithms for calculating
2135-530: The advancement of modern mathematics due to its focus on practical problems and inductive proof methods as opposed to the deductive, axiomatic tradition that Euclid's Elements establishes. However, it is dismissive to say that The Nine Chapters on the Mathematical Art has no impact at all on modern mathematics. The style and structure of The Nine Chapters on the Mathematical Art can be best concluded as "problem, formula, and computation". This process of solving applied mathematical problems can now be considered
2196-539: The agricultural fields. In addition, due to the needs of civil architecture, The Nine Chapters on the Mathematical Art also discusses volumetric algorithms of linear and circular 3 dimensional solids. The arrangement of these volumetric algorithms ranges from simple to complex, forming a unique mathematical system. Regarding the direct application of the Gou Gu Theorem, which is precisely the Chinese version of
2257-458: The algorithm of equations, the rules of addition and subtraction of positive and negative numbers are given. The subtraction is "divide by the same name, benefit by different names. The addition is "divide by different names, benefit from each other by the same name. Among them, "division" is subtraction, "benefit" is addition, and "no entry" means that there is no counter-party, but multiplication and division are not recorded. The Nine Chapters on
2318-415: The approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms . Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by Liu Hui in the 3rd century. The book was later included in the early Tang collection,
2379-400: 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
2440-452: 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
2501-408: The areas of the inscribed rectangles and other polygons in the circle, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles. The methods of completing the squares and cubes as well as solving simultaneous linear equations listed in The Nine Chapters on
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2562-403: 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
2623-463: The book, yet Han dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century The Nine Chapters is an anonymous work, and its origins are not clear. Until recent years, there was no substantial evidence of related mathematical writing that might have preceded it, with the exception of mathematical work by those such as Jing Fang (78–37 BCE), Liu Xin (d. 23), and Zhang Heng (78–139) and
2684-696: The book. Contents of The Nine Chapters are as follows: The Nine Chapters on the Mathematical Art does not discuss natural numbers, that is, positive integers and their operations, but they are widely used and written on the basis of natural numbers. Although it is not a book on fractions, the meaning, nature, and four operations of fractions are fully discussed. For example: combined division (addition), subtraction (subtraction), multiplication (multiplication), warp division (division), division (comparison size), reduction (simplified fraction), and bisector (average). The concept of negative numbers also appears in "Nine Chapters of Arithmetic". In order to cooperate with
2745-488: The date of composition to no later than 93 CE. Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when The Nine Chapters reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss (1777–1855). There
2806-412: 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
2867-400: 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
2928-464: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Jiuzhang&oldid=1214119179 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Jiuzhang suanshu The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from
2989-500: 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
3050-422: 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
3111-404: The most complex example analyzes the solution to a system of linear equations with up to 5 unknowns. The word jiu , or "9", means more than just a digit in ancient Chinese. In fact, since it is the largest digit, it often refers to something of a grand scale or a supreme authority. Further, the word zhang , or "chapter", also has more connotations than simply being the "chapter". It may refer to
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#17327834570763172-581: 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
3233-465: 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
3294-408: 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
3355-405: 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
3416-414: 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
3477-539: 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
3538-602: 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 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
3599-413: 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
3660-401: The standard approach in the field of applied mathematics. Pythagorean theorem In mathematics , the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between 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
3721-422: 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
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