The Chinese Abacus 8

Division

Method

The initial digit of the dividend (placed near the middle of the abacus) is compared with the initial digit of the divisor and the multiples are noted on the abacus someplace in the left most rods. This multiple is multiplied by the initial digit of the divisor and subtracted from the initial digit of the dividend, resulting in a partial dividend. The second digit of the divisor is then multiplied by the multiple and subtracted from the second digit of the partial dividend.

(It is likely that this product of the multiplication is greater than the number represented by the first and second rods. If so, the initial multiple is reduced in value by one, and this new multiple multiplied by the initial digit of the divisor is added back to the initial digit of the dividend, and this forms our first revised partial dividend. This way, the revised multiple multiplied by the second digit of the divisor may be small enough to be subtracted from the the left most rods of the dividend. All will become apparent in the example below)

Once there are no digits left in the dividend, the answer will be all the multiple digits found.

• 3804 / 12 = ?
  Our first example.

             3804

3           -3      The first digit of the divisor is 1, 1x3=3 .: take 3 from first digit

             0804   is the first partial dividend

             -6     The second digit of the divisor is 2, 1x3=6 .: take six from second digit

              204   is the second partial dividend





32           -2     The divisors's first digit yield a factor of 2 (the multiple) .: take 2 

              004   is the third partial dividend

32*********** -4*** The divisor's second digit is 2, multiply by the multiple 2 to yield 4

                    Problem!! Can't take 4 off so backtrack

-1            104   Remove 1 from the multiple, and multiply it by the first divisor digit

                    So we arrive at this revised partial dividend

31            -2    Therefore most recent digit of the multiple, 1, is multiplied by the 

                    second digit of the dividor 2, and taken away

               84   is our next partial dividend.

318           -8    The first digit of the divisor, 1, has a multiple of 8 so subtract 8

               04   is our next partial dividend.

318********** -16** The divisor's second digit is 2, so 8x2=16, subtract 16

                    Problem again! Backtrack

 -1            14   And recover 1x1 (first divisor digit * recovered multiple)

317            14   is our revised partial dividend  

              -14   The second digit of the divisor is 2, therefore, 7x2=14, subtract 14

               00   This leaves nothing on the abacus.

Manipulating decimal division

E.g.

( 462.1 / 3.9 ) = ( 4621 / 39 ) = ( 46210 / 390 ) = ( 46.21 / 0.39 ) = ( 4.621 / 0.039 )

To some, this is not very satisfactory, because we may have two numbers with virtually endless number of decimal significant places. To ascertain the correct location of the decimal requires one to think about the number of digits and their decimal point positions.

11 / 11000       =    0.001

11 / 1100        =    0.010

11 / 110         =    0.100

11 / 11          =    1.000

11 / 1.1         =   10.000

11 / 0.11        =  100.000

11 / 0.011       = 1000.000



220 / 11000       =    0.02

220 / 1100        =    0.20

220 / 110         =    2.00

220 / 11          =   20.00

220 / 1.1         =  200.00

220 / 0.11        = 2000.00

220 / 0.011       =20000.00

Scientific notation.

A note about scientific notation also helps understand the manipulation of numbers. Scientists and engineers like to represent the number 3200 as 3.2 x 10³, that is read three point two times ten to the power of three. Quite often, they will multiply two numbers in scientific notation as follows:

62.42 x 0.0000019 =
( 6.242 x 10¹ ) x ( 1.9 x 10^-6 ) =
( 6.242 x 1.9 ) x ( 10 ^{[1 + (-6) ]} ) =
( 6.242 x 1.9 ) x ( 10 ^-5 ) =
( 11.8598 ) x ( 10 ^-5 ) =
( 1.18598 x 10 ¹) x ( 10 ^-5 ) =
( 1.18598 x 10 ^{[1 + (-5) ]} ) =
( 1.18598 x 10 ^-4) =
0.000118598

Basically, in this notation, you can deal with powers of 10 immediately whilst leaving the more fiddly decimals as another bit of the problem. This way you can do the estimates to the answer quickly by rounding up the decimal parts.

The indices or powers of 10 that you see as superscript are added in the case of multiplication. Decimals which are numerically lower than 1.0 are shown with a negative power of ten.

In division a similar trick is employed. However, in division, the indices are subtracted in the following way. Using the same numbers but changing to division:

62.42 / 0.0000019 =
( 6.242 x 10¹ ) / ( 1.9 x 10^-6 ) =
( 6.242 / 1.9 ) x ( 10 ^{[ 1 -(-6) ]} ) =
( 6.242 / 1.9 ) x ( 10 ⁷ ) =
( 3.285263157895 ) x ( 10 ⁷ ) =
32852631.57895

Like scientific notation, you must have a close look at the powers of tens involved when you have divided by integer translations of the decimals. So if you had 22.111 / 0.024 and you entered the dividend in as 22111 and used 24 as the divisor, be sure to cound the number of powers of ten you deal with either side of the decimal point. The list of of examples of integers divided by various powers of ten of a divisor is above for reference.