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.
Once there are no digits left in the dividend, the answer will be all the multiple digits found.
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.Our answer is therefore 317
E.g.
( 462.1 / 3.9 ) = ( 4621 / 39 ) = ( 46210 / 390 ) = ( 46.21 / 0.39 ) = ( 4.621 / 0.039 )And so on. We note that each differs by factors of 10 from each other, because of the way we have move the decimal point. Once one number has been changed, the other must follow in the same manner. This keeps the ratios constant and allows us to manipulate the numbers before we load them onto the abacus.
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
62.42 x 0.0000019 =
( 6.242 x 10^{1} ) 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 10^{1} ) / ( 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.
© Dylan W.H.Sung 1996-onwards