Rod calculus 1

Rod calculusRod calculus or rod calculation is the mechanical method of algorithmic computation with counting rods in Chinafrom the Warring States to Ming dynasty before the counting rods were replaced by the more convenient and fasterabacus. Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty andYuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of ZhuShijie.

Japanese counting board with grids

Rod calculus facsimile from the Yongle encyclopedia

Hardware

The basic equipment for carrying outrod calculus is a bundle of countingrods and a counting board. Thecounting rods are usually made ofbamboo sticks, about 12 cm- 15 cm inlength, 2mm to 4 mm diameter,sometimes from animal bones, or ivoryand 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 archaeologistsunearthed a bundle of well preservedanimal bone counting rods stored in asilk pouch from a tomb in Qian Yangcounty in Shanxi province, dated backto the first half of Han dynasty(206 BC– 8AD). In 1975 a bundle of bamboocounting rods was unearthed.

The use of counting rods for rodcalculus flourished in the WarringStates, although no archaeologicalartefacts were found earlier than theWestern Han Dynasty (the first half ofHan dynasty; however, archaeologistsdid unearth software artefacts of rodcalculus dated back to the WarringStates); since the rod calculus softwaremust have gone along with rodcalculus hardware, there is no doubtthat rod calculus was already flourishing during the Warring States more than 2,200 years ago.

Rod calculus 2

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

Rod Numerals

Displaying Numbers

representation of the number 231

Rod Numerals is the only numeric system that uses differentplacement combination of a single symbol to convey any numberor fraction in the Decimal System. For numbers in the units place,every vertical rod represent 1. Two vertical rods represent 2, andso on, until 5 vertical rods, which represents 5. For numberbetween 6 and 9, a biquinary system is used, in which a horizontalbar on top of the vertical bars represent 5. The first row are thenumber 1 to 9 in rod numerals, and the second row is the samenumbers in horizontal form.

For numbers larger than 9, a decimal system is used. Rods placedone place to the left of the units place represent 10 times thatnumber. For the hundreds place, another set of rods is placed tothe left which represents 100 times of that number, and so on. Asshown in the image to the right, the number 231 is represented inrod numerals in the top row, with one rod in the units placerepresenting 1, three rods in the tens place representing 30, andtwo 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 placedconsecutively in the vertical form, there's a possibility of it being mistaken for 51 or 24, as shown in the second andthird row of the image to the right. To avoid confusion, number in consecutive places are placed in alternatingvertical and horizontal form, with the units place in vertical form,[1] as shown in the bottom row on the right.

Displaying Zeroes

In Rod Numerals, zeroes are represented by aspace, which serves both as a number and aplace holder value. Unlike in ArabicNumerals, there is no specific symbol torepresent zero. In the image to the right, thenumber zero is merely represented with aspace.

Negative and Positive Numbers

Song mathematicians used red to representpositive numbers and black for negativenumbers. However, another way is to add aslash to the last place to show that the number is negative.[2]

Rod calculus 3

Decimal fractionThe Mathematical Treatise of Sun Zi used decimal fraction metrology. The unit of length was 1 chi,1 chi=10cun,1cun=10fen,1fen=10li,1li=10hao,1hou=10hu.1 chi2cun3fen4li5hao6shi7hu is laid out on counting board as

where is the unit measurement chi.Southern Song dynasty mathematicial Qin Jiushao extended the use of decimal fraction beyond metrology. In hisbook Shu shu Jiuzhanghe formally expressed 1.1446154 day as

日

He marked the unit with a word “日”(day) underneath it。[3]

Addition

Rod calculus addition 3748+289=4037

Rod calculus itself works on the principle ofaddition. Unlike Arabic Numerals, counting rodsitself have additive properties. The process ofaddition involves mechanically moving the rodswithout the need of memorising an additiontable. This is the biggest difference with ArabicNumerals, as one cannot mechanically put 1 and2 together to form 3, or 2 and 3 together to form5.

The image to the right presents the steps inadding 3748 to 289:1. Place the augend 3748 in the first row, and

the addend 289 in the second.2.2. Calculate from LEFT to RIGHT, from the 2

of 289 first.3.3. Take away two rod from the bottom add to 7 on top to make 9.4.4. 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.5.5. 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 7.6.6. Result 3748+289=4037The rods in the augend change throughout the addition, while the rods in the addend at the bottom "disappear".

Rod calculus 4

Subtraction

Without Borrowing

In situation in which no borrowing isneeded, one only needs to take the numberof rods in the subtrahend from the minuend.The result of the calculation is thedifference. The image on the left shows thesteps in subtracting 23 from 54.

Borrowing

In situations in which borrowing is neededsuch as 4231-789, the steps are shown onthe right.1.1. Place the minuend 4231 on top, the

subtrahend 789 on the bottom. Calculatefrom the left to the right.

2.2. Borrow 1 from the thousands place for aten in the hundreds place, minus 7 fromthe row below, the difference 3 is addedto the 2 on top to form 5. The 7 on thebottom is subtracted, shown by the space.

3.3. Borrow 1 from the hundreds place,which leaves 4. The 10 in the tens placeminus the 8 below results in 2, which is

added to the 3 above to form 5. The top row now is 3451, the bottom 9.4.4. 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 rowsubtracted, the 3442 in the top row is then, the result of the calculation

Rod calculus 5

Multiplication

38x76=2888

al Uqlidis (952 AD)multiplication, avariation of Sun zi multiplication

Sun Tzu described in detail the algorithm of multiplication in TheMathematical Classic of Sun Zi. On the right are the steps to calculate 38×76:

1. Place the multiplicand on top, the multiplier on bottom. Line up the unitsplace of the multiplier with the highest place of the multiplicand. Leaveroom in the middle for recording.

2. Start calculating from the highest place of the multiplicand (in theexample, calculate 30×76, and then 8×76). Using the multiplication table 3times 7 is 21. Place 21 in rods in the middle, with 1 aligned with the tensplace of the multiplier (on top of 7). Then, 3 times 6 equals 18, place 18 asit is shown in the image. With the 3 in the multiplicand multiplied totally,take the rods off.

3.3. Move the multiplier one place to the right. Change 7 to horizontal form, 6to vertical.

4.4. 8×7 = 56, place 56 in the second row in the middle, with the units placealigned with the digits multiplied in the multiplier. Take 7 out of themultiplier since it has been multiplied.

5.5. 8×6 = 48, 4 added to the 6 of the last step makes 10, carry 1 over. Take off8 of the units place in the multiplicand, and take off 6 in the units place ofthe multiplier.

6.6. Sum the 2380 and 508 in the middle, which results in 2888, the product.

Division

10th century al-Uqlidis division

Sunzi division 309/7 = 441/7

.

Rod calculus 6

al Khwarizmi division of 825ADwas identical to Sunzi division

algorithm

11th century Kushyar ibnLabban division, a replica of

Sunzi division

The animation on the left shows the steps for calculating 309/7 = 441/7.1.1. Place the dividend, 309, in the middle row and the divisor, 7, in the bottom row.

Leave space for the top row.2.2. Move the divisor, 7, one place to the left, changing it to horizontal form.3. 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.4.4. 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 theremainder in the middle row in place of the dividend in this step. The result is thequotient is 44 with a remainder of 1

The Sunzi algorithm for division was transmitted in toto by al Khwarizmi to Islamic country from Indian sources in825AD. Al Khwarizmi's book was translated into Latin in 13th century, The Sunzi division algorithm later evolvedinto Galley division in Europe. The division algorithm in Abu'l-Hasan al-Uqlidisi's 925AD book Kitab al-Fusul fial-Hisab al-Hindi and in 11th century Kushyar ibn Labban's Principles of Hindu Reckoning were identical toSunzu's division algorithm.

Fractions

If there is a remainder in a place value decimal rodcalculus division, both the remainder and the divisormust be left in place with one on top of another. In LiuHui's notes to Jiuzhang suanshu (2nd century BCE), thenumber on top is called "shi" (实), while the one atbottom is called "fa" (法). In Sun Tzu's CalculationClassic, the number on top is called "zi" (子) or "fenzi"(lit., son of fraction), and the one on the bottom iscalled "mu" (母) or "fenmu" (lit., mother of fraction).Fenzi and Fenmu are also the modern Chinese name for numerator and denominator, respectively. As shown on theright, 1 is the numerator remainder, 7 is the denominator divisor, formed a fraction 1/7. The quotient of the division309/7 is 44 + 1/7. Liu Hui's used a lot of calculations with fraction in The Sea Island Mathematical Manual.

This form of fraction with numerator on top and denominator at bottom without a horizontal bar in between, wastransmitted to Arabic country in a 825AD book by al Khwarizmi via India, and in use by 10th century Abu'l-Hasanal-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithematic Key".

Rod calculus 7

Addition

rod calculus fraction addition

1/3 + 2/5

•• Put the two numerators 1 and 2 on the left side of counting board, put thetwo denominators 3 and 5 at the right hand side

•• Cross multiply 1 with 5, 2 with 3 to get 5 and 6, replace the numeratorswith 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: 1/3 + 2/5 = 11/15

Subtraction

subtraction of two rod numeral fractions

8/9 − 1/5

•• Put down the rod numeral for numerators 1 and 8 at left hand side of acounting board

•• Put down the rods for denominators 5 and 9 at the right hand side of acounting 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: 8/9 − 1/5 = 31/45

Multiplication

rod calculus fraction multiplication

31/3 × 52/5

• Arrange the counting rods for 31/3 and 52/5 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: 31/3 × 52/5 = 18

Rod calculus 8

Highest common factor and fraction reduction

highest common factor

The algorithm for finding the highest common factor of twonumbers and reduction of fraction was laid out in Jiuzhangsuanshu. The highest common factor is found by successivedivision with remainders until the last two remainders areidentical. The animation on the right illustrates the algorithm forfinding the highest common factor of 32,450,625/59,056,400 andreduction of a fraction.

In this case the hcf is 25.Divide the numerator and denominator by 25. The reducedfraction is 1,298,025/2,362,256.

Interpolation

π in fraction

Calendarist and mathematician He Chengtian used fraction interpolation method, called"harmonisation of the divisor of the day" to obtain a better approximate value than theold one by iteratively adding the numerators and denominators a "weaker" fraction witha "stronger fraction".[4] Zu Chongzhi's legendary π = 355/113 could be obtained with HeChengtian's method[5]

System of linear equations

system equations

Chapter Eight Rectangular Arrays ofJiuzhang suanshu provided analgorithm for solving System of linearequations by method of elimination:[6]

Problem 8-1: Suppose we have 3bundles of top quality cereals, 2bundles of medium quality cereals, anda bundle of low quality cereal withaccumulative weight of 39 dou. Wealso have 2, 3 and 1 bundles ofrespective cereals amounting to 34dou; we also have 1,2 and 3 bundles ofrespective cereals, totaling 26 dou.Find the quantity of top, medium, andpoor quality cereals. In algebra, thisproblem can be expressed in three

system equations with three unknowns.3x+2y+z=392x+3y+z=34

Rod calculus 9

x+2y+3z=26This problem was solved in Jiuzhang suanshu with counting rods laid out on a counting board in a tabular formatsimilar to a 3x4 matrix:

quality left column center column right column

top

medium

low

shi

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:

quality left column center column right column

top

medium

low

shi

The amount of on bundle of low quality cereal =

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

One bundle of medium cereal=4 dou >

Rod calculus 10

Extraction of Square rootAlgorithm for extraction of square root was described in Jiuzhang suanshu and with minor difference in terminologyin The Mathematical Classic of Sun Zi.

extraction of square root of 234567 in The Mathematical Classic of Sun Zi

extraction of sq root by Kushyaribn Labban

The animation shows the algorithm for rodcalculus extraction of an approximation ofthe square root

from the algorithm in chap 2 problem 19 ofThe Mathematical Classic of Sun Zi:

Now there is a square area 234567,find one side of the square.[7]

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 the4th row named xia fa

• Estimate the first digit of square root tobe counting rod numeral 4, put on the toprow (shang) hundreds position,

• Multiply the shang 4 with xiafa 1, put theproduct 4 on 3rd row named fang fa

• Multiply shang with fang fa deduct theproduct 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 thevertical 8 into horizontal 8 after movedright.

• 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 North Song dynasty mathematician Jia Xian developed an additive multiplicative algorithm for square rootextraction, in which he replaced the traditional "doubling" of "fang fa" by adding shang digit to fang fa digit, withsame effect.

Rod calculus 11

Extraction of cubic root

Jia Xian's additive multiplicative method of cubic root extraction

Jiuzhang suanshu vol iv "shaoguang"provided algorithm for extraction of cubicroot.

〔 一 九 〕 今 有 積 一 百 八 十 六

萬 八 百 六 十 七 尺 。 問 為 立 方

幾 何 ？ 答 曰 ： 一 百 二 十 三 尺

。

problem 19: We have a 1860867 cubic chi,what is the length of a side ? Answer:123chi.North Song dynasty mathematician Jia Xianinvented a method similar to simplified formof Horner scheme for extraction of cubicroot. The animation at right shows Jia Xian'salgorithm for solving problem 19 in Jiuzhang suanshu vol 4.

Polynomial equation

Qin Jiushao's "Horner" algorithm

North Song dynasty mathematician JiaXian invented Horner scheme forsolving simple 4th order equation ofthe form

South Song dynasty mathematicianQin Jiushao improved Jia Xian'sHorner method to solve polynomialequation up to 10th order. Thefollowing is algorithm for solving

inhisMathematicalTreatiseinNine

Sections vol 6 problem 2.[8]

This equation was arranged bottom up with counting rods on counting board in tabular form

Rod calculus 12

0 shang root

626250625 shi constant

0 fang coefficent of x

15245 shang lian positive coef of x^2

0 fu lian negative coef of x^2

0 xia lian coef of x^3

1 yi yu negative coef of X^4

Algorithm:1.1. Arrange the coefficents 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.2.2. Advance shang lian two ranks3.3. Advance yi yu three ranks4.4. Estimate shang=205.5. let xia lian =shang * yi yu6.6. let fu lian=shang *yi yu7.7. merge fu lian with shang lian8.8. let fang=shang * shang lian9.9. subtract shang*fang from shi10.10. add shang * yi yu to xia lian11.11. retract xia lian 3 ranks,retract yi yu 4 ranks12.12. The second digit of shang is 013.13. merge shang lian into fang14.14. merge yi yu into xia lian15.15. Add yi yu to fu lian, subtract the result from fang, let the result be denominator

16. find the highest common factor =25 and simplies the fraction

17. solution

Rod calculus 13

Tian Yuan shu

Tian yuan shu in Li Zhi:Yigu yanduan

Yuan dynasty mathematician Li Zhi developed rod calculus into Tian yuanshu

Example Li Zhi Ceyuan haijing vol II, problem 14 equation of one unknown:

元

Polynomial equations of four unknowns

facsimile of Zhu Shijie: Jade Mirror of Four Unknowns

Mathematician Zhu Shijie furtherdeveloped rod calculus to includepolynomial equations of 2 to fourunknowns.

For example, polynomials of threeunknowns:Equation 1:

太

Equation 2:

Equation 3:

太

Rod calculus 14

After successive elimination of two unknowns, the polynomial equations of three unknowns was reduced to apolynomial equation of one unknown:

Solved x=5;

References[1][1] Ronan and Needham, The Shorter Science and Civilisation in China, vol 2, Chapter 1, Mathematics[2] *Ho Peng Yoke， Li， Qi and Shu ISBN 0-486-41445-0[3][3] Lam Lay Yong, p87-88[4][4] Jean claude Martzloff, A History of Chinese Mathematics p281[5][5] Wu Wenjun ed Grand Series of History of Chinese Mathematics vol 4 p125[6][6] Jean-Claude Martzloff, A History of Chinese Mathematics, p249-257[7][7] Lay Lay Yong, Ang Tian Se, Fleeting Footsteps, p66-73[8][8] Jean Claude Martzloff, A History of Chinese Mathematics, p233-246

• Lam Lay Yong（ 兰 丽 蓉 ） Ang Tian Se（ 洪 天 赐 ） ， Fleeting Footsteps， World Scientific ISBN981-02-3696-4

•• Jean Claude Martzloff, A History of Chinese Mathematics ISBN 978-3-540-33782-9

Article Sources and Contributors 15

Article Sources and ContributorsRod calculus  Source: http://en.wikipedia.org/w/index.php?oldid=580849798  Contributors: Big Wang, Chris the speller, Cmprince, Dominus, Dthomsen8, EoGuy, Ethan, Fauzan, Gisling,Histrion, Hyacinth, JP.Martin-Flatin, John of Reading, Jujutacular, JustAGal, Leirbag.arc, LilHelpa, Mark viking, Nick Number, NielsenGW, Sevilledade, Short Kilo, Sun Creator, Thedoctar, 28anonymous edits

Image Sources, Licenses and ContributorsFile:Counting board.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Counting_board.jpg  License: Public Domain  Contributors: Shujutsu sangakuzyte 1795File:Yanghui Yongle Dadian.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Yanghui_Yongle_Dadian.JPG  License: Public Domain  Contributors: Yang Hui(1238－1298)File:Chounumerals.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Chounumerals.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: GislingFile:Rod231.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Rod231.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: GislingFile:Rodnumberwithzero.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Rodnumberwithzero.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:GislingFile:Counting rod v1.png  Source: 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