Rod calculus
Rod calculus played a key role in the development of Chinese mathematics to its height in the Song dynasty and Yuan dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie.
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).
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 (206 BC – 8AD).
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.
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.
In Rod numerals, zeroes are represented by a space, which serves both as a number and a place holder value.
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.
Southern Song dynasty mathematician Qin Jiushao extended the use of decimal fraction beyond metrology.
Unlike Arabic numerals, digits represented by counting rods have additive properties.
The Sunzi algorithm for division was transmitted in toto by al Khwarizmi to Islamic country from Indian sources in 825AD.
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's Principles of Hindu Reckoning were identical to Sunzu's division algorithm.
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 via India, and in use by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithematic Key".
1/3 + 2/5 8/9 − 1/5 31/3 × 52/5 The algorithm for finding the highest common factor of two numbers and reduction of fraction was laid out in Jiuzhang suanshu.
The animation on the right illustrates the algorithm for finding the highest common factor of 32,450,625/59,056,400 and reduction of a fraction.
[5] Zu Chongzhi's legendary π = 355/113 could be obtained with He Chengtian's method[6] Chapter Eight Rectangular Arrays of Jiuzhang suanshu provided an algorithm for solving System of linear equations by method of elimination:[7] 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.
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: The amount of one bundle of low quality cereal
From which the amount of one bundle of top and medium quality cereals can be found easily: Algorithm for extraction of square root was described in Jiuzhang suanshu and with minor difference in terminology in Sunzi Suanjing.
The animation shows the algorithm for rod calculus extraction of an approximation of the square root
North Song dynasty mathematician Jia Xian 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.
Jiuzhang suanshu vol iv "shaoguang" provided algorithm for extraction of cubic root.
North Song dynasty mathematician Jia Xian invented a method similar to simplified form of Horner scheme for extraction of cubic root.
The animation at right shows Jia Xian's algorithm for solving problem 19 in Jiuzhang suanshu vol 4.
North Song dynasty mathematician Jia Xian invented Horner scheme for solving simple 4th order equation of the form 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 This equation was arranged bottom up with counting rods on counting board in tabular form Algorithm: Yuan dynasty mathematician Li Zhi developed rod calculus into Tian yuan shu Example Li Zhi Ceyuan haijing vol II, problem 14 equation of one unknown:
Mathematician Zhu Shijie further developed rod calculus to include polynomial equations of 2 to four unknowns.
