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Rod calculus or rod calculation was the mechanical method of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
ic computation with
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
in China from the
Warring States The Warring States period () was an era in ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded with the Qin wars of conquest ...
to
Ming dynasty The Ming dynasty (), officially the Great Ming, was an Dynasties in Chinese history, imperial dynasty of China, ruling from 1368 to 1644 following the collapse of the Mongol Empire, Mongol-led Yuan dynasty. The Ming dynasty was the last ort ...
before the counting rods were increasingly replaced by the more convenient and faster
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...
. Rod calculus played a key role in the development of Chinese mathematics to its height in
Song Dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the res ...
and
Yuan Dynasty The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fif ...
, culminating in the invention of
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
s of up to four unknowns in the work of
Zhu Shijie Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...
.


Hardware

The basic equipment for carrying out rod calculus is a bundle of
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
and a counting board. The counting rods are usually made of bamboo sticks, about 12 cm- 15 cm in length, 2mm to 4 mm diameter, sometimes from animal bones, or ivory and jade (for well-heeled merchants). A counting board could be a table top, a wooden board with or without grid, on the floor or on sand. In 1971 Chinese archaeologists unearthed a bundle of well-preserved animal bone counting rods stored in a silk pouch from a tomb in Qian Yang county in Shanxi province, dated back to the first half of
Han dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
(206 BC – 8AD). In 1975 a bundle of bamboo counting rods was unearthed. The use of counting rods for rod calculus flourished in the
Warring States The Warring States period () was an era in ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded with the Qin wars of conquest ...
, although no archaeological artefacts were found earlier than the Western Han Dynasty (the first half of
Han dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
; however, archaeologists did unearth software artefacts of rod calculus dated back to the
Warring States The Warring States period () was an era in ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded with the Qin wars of conquest ...
); since the rod calculus software must have gone along with rod calculus hardware, there is no doubt that rod calculus was already flourishing during the Warring States more than 2,200 years ago.


Software

The key software required for rod calculus was a simple 45 phrase positional decimal multiplication table used in China since antiquity, called the nine-nine table, which were learned by heart by pupils, merchants, government officials and mathematicians alike.


Rod numerals


Displaying numbers

Rod numerals is the only numeric system that uses different placement combination of a single symbol to convey any number or fraction in the Decimal System. For numbers in the units place, every vertical rod represent 1. Two vertical rods represent 2, and so on, until 5 vertical rods, which represents 5. For number between 6 and 9, a biquinary system is used, in which a horizontal bar on top of the vertical bars represent 5. The first row are the number 1 to 9 in rod numerals, and the second row is the same numbers in horizontal form. For numbers larger than 9, a decimal system is used. Rods placed one place to the left of the units place represent 10 times that number. For the hundreds place, another set of rods is placed to the left which represents 100 times of that number, and so on. As shown in the adjacent image, the number 231 is represented in rod numerals in the top row, with one rod in the units place representing 1, three rods in the tens place representing 30, and two rods in the hundreds place representing 200, with a sum of 231. When doing calculation, usually there was no grid on the surface. If rod numerals two, three, and one is placed consecutively in the vertical form, there's a possibility of it being mistaken for 51 or 24, as shown in the second and third row of the adjacent image. To avoid confusion, number in consecutive places are placed in alternating vertical and horizontal form, with the units place in vertical form, as shown in the bottom row on the right.


Displaying zeroes

In Rod numerals, zeroes are represented by a space, which serves both as a number and a place holder value. Unlike in
Hindu-Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
, there is no specific symbol to represent zero. In the adjacent image, the number zero is merely represented with a space.


Negative and positive numbers

Song A song is a musical composition intended to be performed by the human voice. This is often done at distinct and fixed pitches (melodies) using patterns of sound and silence. Songs contain various forms, such as those including the repetiti ...
mathematicians used red to represent positive numbers and black for negative numbers. However, another way is to add a slash to the last place to show that the number is negative.


Decimal fraction

The Mathematical Treatise of Sunzi used decimal fraction metrology. The unit of length was 1 ''chi'', 1 ''chi'' = 10 ''cun'', 1 ''cun'' = 10 ''fen'', 1 ''fen'' = 10 ''li'', 1 ''li'' = 10 ''hao'', 10 hao = 1 shi, 1 shi = 10 ''hu''. 1 ''chi'' 2 ''cun'' 3 ''fen'' 4 ''li'' 5 ''hao'' 6 ''shi'' 7 ''hu'' is laid out on counting board as :::: where is the unit measurement ''chi''.
Southern Song dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the res ...
mathematician Qin Jiushao extended the use of decimal fraction beyond metrology. In his book '' Mathematical Treatise in Nine Sections'', he formally expressed 1.1446154 day as :::::: ::::::日 He marked the unit with a word “日” (day) underneath it.


Addition

Rod calculus works on the principle of addition. Unlike
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such a ...
, digits represented by counting rods have additive properties. The process of addition involves mechanically moving the rods without the need of memorising an addition table. This is the biggest difference with Arabic numerals, as one cannot mechanically put 1 and 2 together to form 3, or 2 and 3 together to form 5. The adjacent image presents the steps in adding 3748 to 289: #Place the augend 3748 in the first row, and the addend 289 in the second. #Calculate from LEFT to RIGHT, from the 2 of 289 first. #Take away two rod from the bottom add to 7 on top to make 9. #Move 2 rods from top to bottom 8, carry one to forward to 9, which becomes zero and carries to 3 to make 4, remove 8 from bottom row. #Move one rod from 8 on top row to 9 on bottom to form a carry one to next rank and add one rod to 2 rods on top row to make 3 rods, top row left 7. #Result 3748+289=4037 The rods in the augend change throughout the addition, while the rods in the addend at the bottom "disappear".


Subtraction


Without borrowing

In situation in which no borrowing is needed, one only needs to take the number of rods in the
subtrahend Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
from the
minuend Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
. The result of the calculation is the difference. The adjacent image shows the steps in subtracting 23 from 54.


Borrowing

In situations in which borrowing is needed such as 4231–789, one need use a more complicated procedure. The steps for this example are shown on the left. #Place the minuend 4231 on top, the subtrahend 789 on the bottom. Calculate from the left to the right. #Borrow 1 from the thousands place for a ten in the hundreds place, minus 7 from the row below, the difference 3 is added to the 2 on top to form 5. The 7 on the bottom is subtracted, shown by the space. #Borrow 1 from the hundreds place, which leaves 4. The 10 in the tens place minus the 8 below results in 2, which is added to the 3 above to form 5. The top row now is 3451, the bottom 9. #Borrow 1 from the 5 in the tens place on top, which leaves 4. The 1 borrowed from the tens is 10 in the units place, subtracting 9 which results in 1, which are added to the top to form 2. With all rods in the bottom row subtracted, the 3442 in the top row is then, the result of the calculation


Multiplication

''
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...
'' described in detail the algorithm of multiplication. On the left are the steps to calculate 38×76: #Place the multiplicand on top, the multiplier on bottom. Line up the units place of the multiplier with the highest place of the multiplicand. Leave room in the middle for recording. #Start calculating from the highest place of the multiplicand (in the example, calculate 30×76, and then 8×76). Using the
multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essenti ...
3 times 7 is 21. Place 21 in rods in the middle, with 1 aligned with the tens place of the multiplier (on top of 7). Then, 3 times 6 equals 18, place 18 as it is shown in the image. With the 3 in the multiplicand multiplied totally, take the rods off. #Move the multiplier one place to the right. Change 7 to horizontal form, 6 to vertical. #8×7 = 56, place 56 in the second row in the middle, with the units place aligned with the digits multiplied in the multiplier. Take 7 out of the multiplier since it has been multiplied. #8×6 = 48, 4 added to the 6 of the last step makes 10, carry 1 over. Take off 8 of the units place in the multiplicand, and take off 6 in the units place of the multiplier. #Sum the 2380 and 508 in the middle, which results in 2888: the product.


Division

. The animation on the left shows the steps for calculating . #Place the dividend, 309, in the middle row and the divisor, 7, in the bottom row. Leave space for the top row. #Move the divisor, 7, one place to the left, changing it to horizontal form. #Using the Chinese multiplication table and division, 30÷7 equals 4 remainder 2. Place the quotient, 4, in the top row and the remainder, 2, in the middle row. #Move the divisor one place to the right, changing it to vertical form. 29÷7 equals 4 remainder 1. Place the quotient, 4, on top, leaving the divisor in place. Place the remainder in the middle row in place of the dividend in this step. The result is the quotient is 44 with a remainder of 1 The Sunzi algorithm for division was transmitted in toto by
al Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
to Islamic country from Indian sources in 825AD. Al Khwarizmi's book was translated into Latin in the 13th century, The Sunzi division algorithm later evolved into Galley division in Europe. The division algorithm in Abu'l-Hasan al-Uqlidisi's 925AD book ''Kitab al-Fusul fi al-Hisab al-Hindi'' and in 11th century
Kushyar ibn Labban Abul-Hasan Kūshyār ibn Labbān ibn Bashahri Daylami (971–1029), also known as Kūshyār Daylami ( fa, کوشیار دیلمی), was an Iranian mathematician, geographer, and astronomer from Daylam, south of the Caspian Sea, Iran. Career His ...
's Principles of Hindu Reckoning were identical to Sunzu's division algorithm.


Fractions

If there is a remainder in a place value decimal rod calculus division, both the remainder and the divisor must be left in place with one on top of another. In
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
's notes to Jiuzhang suanshu (2nd century BCE), the number on top is called "shi" (实), while the one at bottom is called "fa" (法). In ''
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...
'', the number on top is called "zi" (子) or "fenzi" (lit., son of fraction), and the one on the bottom is called "mu" (母) or "fenmu" (lit., mother of fraction). Fenzi and Fenmu are also the modern Chinese name for
numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
and
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, respectively. As shown on the right, 1 is the numerator remainder, 7 is the denominator divisor, formed a fraction . The quotient of the division is 44 + . Liu Hui's used a lot of calculations with fraction in
Haidao Suanjing ''Haidao Suanjing'' (; ''The Sea Island Mathematical Manual'') was written by the Chinese mathematician Liu Hui of the Three Kingdoms era (220–280) as an extension of chapter 9 of ''The Nine Chapters on the Mathematical Art''. L. van. He ...
. This form of fraction with numerator on top and denominator at bottom without a horizontal bar in between, was transmitted to Arabic country in an 825AD book by
al Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
via India, and in use by 10th century Abu'l-Hasan al-Uqlidisi and 15th century
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer ...
's work "Arithematic Key".


Addition

*Put the two numerators 1 and 2 on the left side of counting board, put the two denominators 3 and 5 at the right hand side *Cross multiply 1 with 5, 2 with 3 to get 5 and 6, replace the numerators with the corresponding cross products. *Multiply the two denominators 3 × 5 = 15, put at bottom right *Add the two numerators 5 and 6 = 11 put on top right of counting board. *Result:


Subtraction

*Put down the rod numeral for numerators 1 and 8 at left hand side of a counting board *Put down the rods for denominators 5 and 9 at the right hand side of a counting board *Cross multiply 1 × 9 = 9, 5 × 8 = 40, replace the corresponding numerators *Multiply the denominators 5 × 9 = 45, put 45 at the bottom right of counting board, replace the denominator 5 *Subtract 40 − 9 = 31, put on top right. *Result:


Multiplication

3 × 5 *Arrange the counting rods for 3 and 5 on the counting board as shang, shi, fa tabulation format. *shang times fa add to shi: 3 × 3 + 1 = 10; 5 × 5 + 2 = 27 *shi multiplied by shi:10 × 27 = 270 *fa multiplied by fa:3 × 5 = 15 *shi divided by fa:


Highest common factor and fraction reduction

The algorithm for finding the highest common factor of two numbers and reduction of fraction was laid out in Jiuzhang suanshu. The highest common factor is found by successive division with remainders until the last two remainders are identical. The animation on the right illustrates the algorithm for finding the highest common factor of and reduction of a fraction. In this case the hcf is 25. Divide the numerator and denominator by 25. The
reduced fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). ...
is .


Interpolation

Calendarist and mathematician He Chengtian ( 何承天) used fraction
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...
method, called "harmonisation of the divisor of the day" ( 调日法) to obtain a better approximate value than the old one by iteratively adding the numerators and denominators a "weaker" fraction with a "stronger fraction". Zu Chongzhi's legendary could be obtained with He Chengtian's method


System of linear equations

Chapter Eight ''Rectangular Arrays of Jiuzhang suanshu provided an algorithm for solving
System of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in t ...
by method of elimination: Problem 8-1: Suppose we have 3 bundles of top quality cereals, 2 bundles of medium quality cereals, and a bundle of low quality cereal with accumulative weight of 39 dou. We also have 2, 3 and 1 bundles of respective cereals amounting to 34 dou; we also have 1,2 and 3 bundles of respective cereals, totaling 26 dou. Find the quantity of top, medium, and poor quality cereals. In algebra, this problem can be expressed in three system equations with three unknowns. \begin 3x+2y+z=39 \\ 2x+3y+z=34 \\ x+2y+3z=26 \end This problem was solved in Jiuzhang suanshu with counting rods laid out on a counting board in a tabular format similar to a 3x4 matrix: Algorithm: * Multiply the center column with right column top quality number. * Repeatedly subtract right column from center column, until the top number of center column =0 * multiply the left column with the value of top row of right column * Repeatedly subtract right column from left column, until the top number of left column=0 * After applying above elimination algorithm to the reduced center column and left column, the matrix was reduced to triangular shape: The amount of one bundle of low quality cereal =\frac=2 \frac From which the amount of one bundle of top and medium quality cereals can be found easily: One bundle of top quality cereals=9 dou \frac One bundle of medium cereal=4 dou \frac>


Extraction of Square root

Algorithm for extraction of square root was described in Jiuzhang suanshu and with minor difference in terminology in
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...
. The animation shows the algorithm for rod calculus extraction of an approximation of the square root \sqrt\approx484\tfrac from the algorithm in chap 2 problem 19 of Sunzi Suanjing: :''Now there is a square area 234567, find one side of the square''. The algorithm is as follows: * Set up 234567 on the counting board, on the second row from top, named shi * Set up a marker 1 at 10000 position at the 4th row named xia fa * Estimate the first digit of square root to be counting rod numeral 4, put on the top row (shang) hundreds position, * Multiply the shang 4 with xiafa 1, put the product 4 on 3rd row named fang fa * Multiply shang with fang fa deduct the product 4x4=16 from shi: 23-16=7, remain numeral 7. * double up the fang fa 4 to become 8, shift one position right, and change the vertical 8 into horizontal 8 after moved right. * Move xia fa two position right. * Estimate second digit of shang as 8: put numeral 8 at tenth position on top row. * Multiply xia fa with the new digit of shang, add to fang fa . * 8 calls 8 =64, subtract 64 from top row numeral "74", leaving one rod at the most significant digit. * double the last digit of fang fa 8, add to 80 =96 * Move fang fa96 one position right, change convention;move xia fa "1" two position right. * Estimate 3rd digit of shang to be 4. * Multiply new digit of shang 4 with xia fa 1, combined with fang fa to make 964. * subtract successively 4*9=36,4*6=24,4*4=16 from the shi, leaving 311 * double the last digit 4 of fang fa into 8 and merge with fang fa * result \sqrt\approx484\tfrac North Song dynasty mathematician
Jia Xian Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty. Biography According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...
developed an additive multiplicative algorithm for square root extraction, in which he replaced the traditional "doubling" of "fang fa" by adding shang digit to fang fa digit, with same effect.


Extraction of cubic root

Jiuzhang suanshu vol iv "shaoguang" provided algorithm for extraction of cubic root. problem 19: We have a 1860867 cubic chi, what is the length of a side ? Answer:123 chi. North Song dynasty mathematician
Jia Xian Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty. Biography According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...
invented a method similar to simplified form of
Horner scheme In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Hor ...
for extraction of cubic root. The animation at right shows Jia Xian's algorithm for solving problem 19 in Jiuzhang suanshu vol 4. \sqrt 1860867)=123


Polynomial equation

North Song dynasty mathematician
Jia Xian Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty. Biography According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...
invented
Horner scheme In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Hor ...
for solving simple 4th order equation of the form ::::::: x^4=a South Song dynasty mathematician Qin Jiushao improved Jia Xian's Horner method to solve polynomial equation up to 10th order. The following is algorithm for solving :::::::::::::-x^4 +15245x^2-6262506.25=0 in his Mathematical Treatise in Nine Sections vol 6 problem 2.Jean Claude Martzloff, A History of Chinese Mathematics, p233-246 This equation was arranged bottom up with counting rods on counting board in tabular form Algorithm: #Arrange the coefficients in tabular form, constant at shi, coeffienct of x at shang lian, the coeffiecnt of X^4 at yi yu;align the numbers at unit rank. #Advance shang lian two ranks #Advance yi yu three ranks #Estimate shang=20 #let xia lian =shang * yi yu #let fu lian=shang *yi yu #merge fu lian with shang lian #let fang=shang * shang lian #subtract shang*fang from shi #add shang * yi yu to xia lian #retract xia lian 3 ranks, retract yi yu 4 ranks #The second digit of shang is 0 #merge shang lian into fang #merge yi yu into xia lian #Add yi yu to fu lian, subtract the result from fang, let the result be denominator #find the highest common factor =25 and simplify the fraction \frac #solution x=20\frac


Tian Yuan shu

Yuan dynasty mathematician
Li Zhi Li Zhi may refer to: *Emperor Gaozong of Tang (628–683), named Li Zhi, Emperor of China *Li Ye (mathematician) (1192–1279), Chinese mathematician and scholar, birth name Li Zhi *Li Zhi (philosopher) (1527–1602), Chinese philosopher from the M ...
developed rod calculus into Tian yuan shu Example Li Zhi Ceyuan haijing vol II, problem 14 equation of one unknown: -x^2-680x+96000=0 ::::::::::: :::::::::元 ::::::::


Polynomial equations of four unknowns

Mathematician
Zhu Shijie Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...
further developed rod calculus to include polynomial equations of 2 to four unknowns. For example, polynomials of three unknowns: Equation 1:-y-z-y^2*x-x+xyz=0 :::::::::::太 :::::::::::: :::::::::: Equation 2:-y-z+x-x^2+xz=0 ::::::::::: :::::::::::: :::::::::::: Equation 3:y^2-z^2+x^2=0; :::::::::::太 ::::::::::::: ::::::::::::: After successive elimination of two unknowns, the polynomial equations of three unknowns was reduced to a polynomial equation of one unknown: x^4-6x^3+4x^2+6x-5=0 :::::::::::: :::::::::::: :::::::::::: :::::::::::: :::::::::::: Solved x=5; Which ignores 3 other answers, 2 are repeated.


See also

*
Chinese mathematics Mathematics in China emerged independently 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 ( base 2 and base 10), algebra, geo ...
*
Counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...


References

*Lam Lay Yong (蓝丽蓉) Ang Tian Se (洪天赐), Fleeting Footsteps, World Scientific *Jean Claude Martzloff, A History of Chinese Mathematics {{ISBN, 978-3-540-33782-9 Chinese mathematics Mathematical tools Science and technology in China