ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual
DECIMALS & DIVISION (contributed by Edvaldo Siqueira)
The following is a method used by Professor Fukutaro Kato, a Japanese soroban teacher living in Brazil in the 1960's. The method was published in the Professor's book, *SOROBAN pelo Método Moderno* which, when translated from Portuguese means *SOROBAN by the Modern Method*
This is an explanation on how to apply "Professor Kato's Method" for placing correctly the *dividend & quotient* on the soroban, in order that the unit digit of the *quotient* always falls on a previously chosen unit rod. In other words, you always know where your unit rod is. Take the example 0.00008 ÷ 0.002 = 0.04 Because you've already chosen your unit rod, what looks like a rather complicated division problem involving lots of zeros becomes a much more manageable 8 ÷ 2 = 4
Bear in mind that the arithmetic operation mechanics should be carried out according Kojima's standard method for division.
For the sake of clarity, let's split the following in 4 short sections: A) HOW TO COUNT DIGITS; B) RULE TO FOLLOW; C) NUMBERS ON THE SOROBAN; and D) A FEW EXAMPLES.
A) HOW TO COUNT DIGITS
i) If the *dividend or the divisor* are integers or numbers with a whole part followed by decimal places, take into account in each one only those digits in their whole part. So disregard any decimal digit, after the point, if any.
3.105.......has 1 digit;
47.02.......has 2 digits;
308.41.....has 3 digits,
1000........has 4 digits, and so on
ii) When the number is a pure decimal, consider (count) only those trailing zeros (if any) before the first significant digit attributing to the total a negative sign. Disregard the zero before the point
0.00084.....has -3 digits
0.40087.....has 0 digits
0.01010.....has -1 digit
B) RULES TO FOLLOW
B1) Compute the Number: N = # digits on dividend MINUS [# digits on divisor + 1]
Note: Before computing N, please note that number 1 should always be added to the # of digits on the divisor and also always remember that the operation between parenthesis must me carried out FIRST. If you need to, refresh your memory revisiting that old math book that taught you how to algebraically add negative numbers, many years ago. Or ask your son for advice :). (See D, 3rd example). That is the burden of this method, I admit.
B2) Further rules for placing the quotient digit (never FORGET THIS ALSO)
i) If the digit of the dividend is *greater or equal* to the corresponding digit of the divisor, *skip one rod* to the left of the of this class of the dividend to put the corresponding quotient digit;
ii) If the digit of the dividend is *less than* the corresponding digit of the divisor, put the quotient digit *next to* the highest class you are working on the dividend.
C) NUMBERS ON THE SOROBAN<----A----B----C----D----E---*F*---G----H----I----J--> <...+6...+5...+4...+3...+2...+1....0...-1...-2...-3..>
Let's associate an orientating axis extending from right to left of the soroban rods with the axis positive number 1 corresponding to the *unit rod* (*F*). That correspondence is paramount to the correct application of this method! Obviously, the rod chosen to hold the unit can be anyone - towards the center of the soroban - provided it is marked with a dot. To the left of *F* (rods E, D, C ...) we have positive values; to its right, Zero and negative values, as shown. Imagine the above diagram extending to both sides as far as necessary to work comfortably according Kojima's. You will soon note that when the absolute value of N "large" (say, > 4), the other printed dots on the bar are handy in choosing the appropriate rod for setting the multiplicand highest order digit.
i) 625 ÷ 5 = 125
N = 3 - [1 + 1] = 1 ----> set the dividend (625) on FGH and place the quotient on DEF, (rule B2i). The "5" from the quotient will automatically be placed on rod *F* (the unit).
ii) 0,02 ÷ 0.8 = 0.025
N = -1 - [-0 + 1] = -2 ----> set dividend (2) on I and place the quotient digits (25 ) on HI (Rule B2ii). As *F* - which always holds the unit (0.) - and G are both empty, one should automatically read 0.025.
iii) 0.08 ÷ 0.002 = 40
N = -1 - [-2 + 1] = -1 - [-1] = -1 +1 = 0 ----> set dividend (8) (disregard the zeros) on G and the calculated quotient 40 on EF (rule B2i) - only its digit 4 will be set, but result reads 40.
iv) 86 ÷ 43 = 2
(Mentally) N = -1 ----> set dividend 86 on HI. As first digit of the dividend (8) is greater than first digit of the divisor (4), their quotient will skip one column to the left of this digit of the dividend to place the digit for the quotient. "8 / 4 = 2, which will be set at *F*, (rule B2i).
v) 860 ÷ 43 = 20
(Mentally) N = 0 ----> set dividend 860 on GHI. Quotient (after carrying out the whole division process) will be 2 on E (again rule B2i), AND a 0 (zero) on *F*, because this column always holds the unit digit. Final answer will be: 20
Note: Take some time to compare the quotients digits on the previous two examples. One sees for both only number 2. BUT they are placed at DIFFERENT rods. The former is at the very column *F*. The latter is at E. But this one has a unit digit after it, 0; in fact the former is 20 and the latter is 2 only. So, even without evaluating beforehand their order or magnitude, the student is able to reach the correct answer for each example.
Below find 3 links to pages that have been scanned from Fukutaro Kato's book. These pages will further illustrate how to place correctly the *dividend & quotient digit* on the soroban using the method.
With practice, one will be able to compute N mentally and soon you should be using this alternative method correctly and confidently. The bonus is at the same time you will learn how to divide decimal numbers on the soroban.
And a final advice: as you have observed Fukutaro Kato's method allows one to reach the exact result without having to evaluate beforehand the quotient order of magnitude, while operating both with integers or decimal numbers! But, if you are already familiar with other system, please stick to it. I am not advocating the one here exposed is better to any other. All I can say that, at least for me, I found it easier to grasp than the other one taught by Kojima, which I never learned :)
Rio de Janeiro, Brazil
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