General Source of Computational Methods, Volume 2

A diagram illustrating a mathematical calculation using circles containing numbers that represent place values (thousands, hundreds, and tens). The circles are labeled with Chinese characters for "thousand" (千), "hundred" (百), and "ten" (十). Lines connect these circles to demonstrate the "Returning to the Original" (還原) calculation method, showing how digits are moved or transformed across columns on the abacus.


Three: twenty-seven Rhyme for 3 × 9 = 27	Two: eighteen Rhyme for 2 × 9 = 18	One as nine Rhyme for 1 × 9 = 09
Change 3 to 2; add 7 to the lower position	Change 2 to 1; add 8 to the lower position	Remove from this position; add 9 to the lower position

Returning to the Original: Using the Nine Returns method; details follow below.


9-1: add 1 below	9-2: add 2 below	9-3: add 3 below
9-4: add 4 below	9-5: add 5 below	9-6: add 6 below
9-7: add 7 below	9-8: add 8 below	Upon meeting 9: advance 1 to the tens

These rhymes, known as "Nine Returns" (jiugui), are the standard mental shortcuts for dividing any number by 9 on an abacus. For example, "9-1: add 1 below" means that 1 divided by 9 equals 0 with a remainder of 1, which is placed in the next column to the right.

Nine Returns (Division by Nine)

For any divisor from 2 to 9 units, use this method. Set the items being counted as the Dividend Dividend (shi): The total amount to be divided.. Set the price or the number of people sharing the items as the Divisor Divisor (fa): The number by which you are dividing..

Recite the "Nine Returns" rhymes, either advancing digits or doubling values, starting from the leading digit of the Dividend. Use the multiplication method Multiplication (yin): The inverse operation used here to check the division. to return the result to its original value to verify accuracy.

The "Nine Returns" system is a cornerstone of Chinese bead arithmetic (zhusuan). It allows a practitioner to perform division through a series of predefined additions and subtractions, significantly speeding up calculation compared to long division.

**Technical