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Chinese Abacus, Art of Calculation
 
Copyright © 2002 Teddy Fung
    
         
 
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Methods: There are many methods to perform division with the Chinese Abacus. I will introduce one of the simpler methods and leave the others for an advanced topic in the future.

Guess Approximate Quotient Method (GAQM): The foundation of GAQM is based on multiplication and subtraction. The key point is to set the quotient position and guess the AQ.

(One Digit Divisor) -- Positions of Dividend - For easy reference, we named the Abacus columns from left to right as A, B, C, D, E, F, G, H, I, and J. I use the C column ( 3rd column from the left) for a reference column. As with multiplication, once you learn the method you can pick any column you like. There are two situation for determining the position of the dividend.

Situation One: When the highest digit of the dividend is equal to or greater than the highest digit of the divisor, we start the highest digit of the dividend from column C (the reference column).

Situation Two: When the highest digit of the dividend is smaller than the highest digit of the divisor, we start from column D (one column to the right of the reference column.)

Position of Quotient - In Situation One above, the quotient will be accumulated towards the right from column A (2 columns on the left of the highest digit of the dividend), otherwise the quotient will be accumulated towards the right from column B (1 column to the left of the highest digit of the dividend.)

Position of Divisor - This is the same as the OCSM multiplication, we don't put the divisor in the Abacus unlike in some other methods, but as a beginner you can put the divisor on the far right side of the Abacus for convenience. After you become familiar with the process, you don't need to do that.

To guess the AQ - Looking at the highest digit of the dividend, guess the AQ. (If the divisor is greater than the highest digit of the divident, you will need to look at more than one of the highest digits of the dividend.) Then add the AQ to the quotient position (according to the rules mentioned before).

Product & Subtration - After we guess the AQ and add it to the quotient position, we find the product of the AQ and the divisor then subtract the 10-digit of the product in the next column to the right of the quotient, then subtract the unit-digit of the product in the second column to the right of the quotient (same as the OCSM, the product here always has 2 digits).

Direction & Sequence - Start from the highest digit of the dividend (or start with the first and second highest digits of the dividend, if it is not Situation One above,) then guess the AQ. Then add the AQ to the quotient position (according to the rules mentioned before), then subtract the product of the AQ and divisor from the dividend (according to the rules mentioned before). After that the dividend will leave a virtual remainder on the Abacus.We will repeat this process on the virtual remainder until we arrive at the real quotient and the real remainder (if it has one). Very confusing, right? OK, let's do an example to make it clear. We'll try 2,250 / 6 = ? and please follow the instructions to do this example. First, we set 2250 starting from column D and going towards column G (because this is not Situation One above, start the dividend one column to the right of the reference column). Second, we determine the number of digits in the quotient. Because this is not Situation One, the quotient will have three digits (4-1=3). After determining the number of digits in the quotient, we guess the AQ. Because the highest digit of 2250 is 2, smaller than divisor 6, we consider 22 (first and second highest digits of the dividend) for the AQ. We guess AQ is 3 (because 3x6=18 and 4x6=24 is too big). Then add AQ 3 in column C (this is not Situation One, so add the AQ to the column just left of the highest digit of the dividend). Then subtract the product, which is 18 (AQ x Divisor=18) by digit 1 in column D and digit 8 in column E. Now the figure in column C should be 3, this is the first virtual quotient. The figure in column E, F, and G should be 450, this is the first virtual remainder of the original dividend after the first division process. We keep going by dividing 450 by 6. We guess the AQ is 7 (because 7x6=42) and add AQ 7 in column D (do you still remember that since this is not Situation One, we add the AQ in the column just left of the highest digit of the dividend?), then substract the product 42 (7x6=42) of the AQ and the divisor by digit 4 in column E and digit 2 in column F. Now the figure in columns C and D should be 37. This is the second virtual quotient. The figure in columns F and G should be 30. This is the second virtual remainder after the second division. Now the highest digit of 30 is smaller than 6, therefore, the AQ position is in column E (since this is not Situation One, add the AQ one column to the left of the highest digit of the dividend). Finally, we guess the AQ is 5 and add 5 in column E, then substract the product 30 (5x6=30) by digit 3 in column F and digit 0 in column G. Now the figure in columns C, D, and E should be 375 and the figure in columns F, G, H, and I should be all zeros (this means there is no remainder). At the beginning, we already determined the quotient is a 3-digit figure, therefore, 375 is the final quotient and there is no zero following it. We got 2250 / 6 = 375. Not that hard, right? OK, we should do one more example with the Cyber Chinese Abacus. Go to the next page.

  
       
  Division

Concept: Division is the process of dividing a dividend by a divisor in order to arrive at the quotient. As a formula, division can be expressed as Dividend / Divisor = Quotient + Remainder. From Fig. 4 in the section "Concept of Chu Suan," we learned that division is also a process of splitting up the dividend into a number of groups, each containing an equal number of beads. The number of beads equals the divisor. For example, 120 / 20 = 6 means 120 beads can be grouped (subtracted) into 6 smaller groups of 20 beads (the remainder is 0 in this case). This is a process of continuous subtraction that is just opposite of multiplication.

Positions and Unit Digit: I will show how to determine the positions of dividend, divisor and quotient in later examples. There are a few ways to determine the position of the unit digit of the quotient. I will introduce one of them and leave the other for an advanced topic in the future. When comparing the highest digits of the dividend and the divisor, we are faced with 3 possible scenarios. In the first case: the highest digit of the dividend is greater than the highest digit of the divisor, then the number of digits of the quotient will be equal to the difference of their number of digits plus 1 (dividend's # of digits - divisor's # of digits +1). In the second case: the highest digit of the dividend is smaller than the highest digit of the divisor, the number of digits of the quotient will be equal to the difference of their number of digits (dividend's # of digits - divisor's # of digits). In the third case: the highest digit of the dividend is equal to the highest digit of the divisor. If the divisor is a one-digit figure, then follow the first case scenario. If the divisor is a multi-digit figure, then we compare its second highest digit to the second highest digit of the dividend, and so on. If all of them are equal, then follow the first case scenario. If there is one digit of the divisor smaller than the dividend's then follow the second case scenario. Once we know the number of digits of the quotient, we can determine the position of its unit digit.

Approximate Quotient (AQ): This is a special term used in Chinese Abacus division. To calculate quickly on the Abacus, people usually guess the AQ first, with the help of the 99M Table, then correct the erorr during the division process and finally arrive at the real quotient. In this way, the calculation can be sped up. Therefore, it is important to guess the right AQ when doing the division.

 

    
         
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