Rod calculus

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Rod calculus or rod calculation is the method of mathematical computation with counting rods in China from The Warring States to Ming dynasty before the counting rods were replaced by more convenient and faster abacus.

Contents 1 Hardware 2 Software 3 Rod Numerals 3.1 Displaying Numbers 3.2 Displaying Zeroes 3.3 Negative and Positive Numbers 4 Addition 5 Subtraction 5.1 Without Borrowing 5.2 Borrowing 6 Multiplication 7 Division 7.1 Fractions 8 Others 9 See also 10 References

[edit] Hardware

The basic equipment for carrying out rod calculus is a bundle of counting rods and a counting board. The counting rods are usually made of bamboo sticks, about 12 cm- 15mm 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 archeologists 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). In 1975 a bundle of bamboo counting rods was unearthed.

The use of counting rods for rod calculus flourished in the Warring States, although no archeological artifacts were found earlier than the Western Han Dynasty(the first half of Han dynasty, however archeologist did unearth software artifacts of rod calculus dated back to the Warring States); 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.

[edit] 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.

[edit] Rod Numerals

[edit] Displaying Numbers

Rod Numerals is the only numeric system that uses different placement 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 image to the right, 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 image to the right. To avoid confusion, number in consecutive places are placed in alternating vertical and horizontal form, with the units place in vertical form,^[1] as shown in the bottom row on the right.

[edit] Displaying Zeroes

In Rod Numerals, zeroes are represented by a space, which serves both as a number and a place holder value. Unlike in Arabic Numerals, there is no specific symbol to represent zero. In the image to the right, the number zero is merely represented with a space.

[edit] Negative and Positive Numbers

Song 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. ^[2]

[edit] Addition

Rod calculus itself works on the principle of addition. Unlike Arabic Numerals, counting rods itself have additive properties. The process of addition involves mechanically moving the rods without the need of memorizing 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 image to the right presents the steps in adding 3748 to 289:

1. Place the augend 3748 in the first row, and the addend 289 in the second.
2. Take one rod from the 8 on top, put it together with the 9 on the bottom to form 10, which carries over so that the 4 in the tens place becomes 5.
3. Take two rods from the 5 in the tens place in the first row, add it to the 8 below to form 10. Carry over to the hundreds place which add one to 7 to form 8.
4. Take the 8 and the 2 in the hundreds place to form 10, carry one over to the thousands, which results in 4. The sum is shown as 4037.

The rods in the augend changes throughout the addition, while the rods in the addend at the bottom "disappears". Augend's rods in the first row changes throughout the addition. Addend's rods in the bottom "disappears" throughout the process.

[edit] Subtraction

[edit] Without Borrowing

In situation in which no borrowing is needed, one only needs to take the number of rods in the subtrahend from the minuend. The result of the calculation is the difference. The image on the left shows the steps in subtracting 23 from 54.

[edit] Borrowing

In situations in which borrowing is needed such as 4231-789, the steps are shown on the right.

1. Place the minuend 4231 on top, the subtrahend 789 on the bottom. Calculate from the left to the right.
2. 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.
3. 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.
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 row subtracted, the 3442 in the top row is then, the result of the calculation

[edit] Multiplication

Sun Tzu described in detail the method of multiplication in his Calculation Classic. On the right are the steps to calculate 38×76:

1. 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.
2. Start calculating from the highest place of the multiplicand (in the example, calculate 30×76, and then 8×76). Using the multiplication table 3 times 7 is 21. Place 21 in rods in the middle, with 1 aligned with the units place of the multiplier (on top of 6). 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.
3. Move the multiplier one place to the right. Change 7 to horizontal form, 6 to vertical.
4. 8×76 = 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.
5. 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.
6. Sum the 2380 and 508 in the middle, which results in 2388, the product.

[edit] Division

The image on the right shows the step to calculating 309÷7：

1. Place the dividend, 309, in the middle row and the divisor, 7, in the bottom row. Leave space for the top row.
2. Move the divisor, 7, one place to the left, changing it to horizontal form.
3. Using the 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. 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

[edit] Fractions

If there is a remainder in a division, both the divisor and the remainder must be left in place with one on top of another. In Liu Hui's The Nine Chapters on the Mathematical Art, the number on top is called "shi", while the one on bottom is called "fa". In Sun Tzu's Calculation Classic, the number on top is called "zi" or "fenzi" (lit., son), and the one on the bottom is called "mu" or "fenmu" (lit., mother). Fenzi and Fenmu are also the modern Chinese name for numerator and denominator, respectively. As shown on the right, 1 is the numerator, 7 is the denominator, which leaves the fraction . The quotient of the division is 44 +

[edit] Others

Rod calculus not only could do the four basic types of calculation, it could also be used to find square roots, cube roots, nth roots, roots of polynomials, or solve simultaneous equations

[edit] See also

Counting rods
Chinese mathematics

[edit] References

• 吴文俊 主编 《中国数学史大系·第四卷》第一章 《孙子算经》 第三节 算筹与筹算 北京师范大学出版社 ISBN：7303049258
• 李约瑟 原著 柯林·罗南改编《中华科学文明史》卷2 第一章 数学
• Lam Lay Yong（兰丽蓉） Ang Tian Se（洪天赐）， Fleeting Footsteps， World Scientific ISBN 981-02-3696-4
• Ho Peng Yoke， Li， Qi and Shu ISBN 0-486-41445-0