Systematic Treatise on Arithmetic (1592) - Page 15

General Source of Computational Methods, Volume 2

Question: If drying wet grain yields nine dou of dried grain, how much [original] wet grain is accounted for?

Answer: 1,111 shi, 1 dou, 1 sheng, 1 he, 1 shao, and 1 chao. The answer reflects a repeating decimal (1111.111...) resulting from division by 9. These units represent a decimal system: 1 shi = 10 dou; 1 dou = 10 sheng; 1 sheng = 10 he; 1 he = 10 shao; 1 shao = 10 chao.

Method: Set the wet grain as the Dividend Dividend (shi): The number to be divided. and use the nine dou of dried grain as the Divisor Divisor (fa): The number by which the dividend is divided.. Apply the division mnemonics to find the result.

Dividend (Shi)
Units	Tens	Hundreds	Thousands	Ten-Thousands
Call The verbal rhyme used to perform the calculation.
9 into 9 is 81	9 into 7 is 63	9 into 6 is 54	9 into 5 is 45	9 into 4 is 36
Change 9 to 8; add 1 to the lower position	Change 7 to 6; add 3 to the lower position	Change 6 to 5; add 4 to the lower position	Change 5 to 4; add 5 to the lower position	Change 4 to 3; add 6 to the lower position
Positioning
This is the dou position; move forward to increase the units
Shi Unit of volume
The upper result is the total shi. When dried 9 dou meets a shi, stop. The next position down is the dou.

Divisor is 9 dou. Operate.

The table illustrates "Guichu" (Fixed-position division), a method of abacus calculation where the operator uses rhymes to manipulate beads. For example, "Change 9 to 8, add 1 below" is a mnemonic for dividing by 9. Because 1/9 is 0.1 with a remainder of 1, the operator adjusts the current rod and passes the remainder to the next rod.