ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual

  Predetermining the Unit Rod

Before beginning a problem soroban operators must first designate a unit rod. Rods to the right of the designated unit rod carry the decimal numbers.

The following is a method that allows for placing both multiplicand and dividend onto the soroban in such a way that the unit and decimal numbers in both product and quotient fall naturally on predetermined rods. In other words you always know where your unit rod is which in turn allows you to simplify problems. Take the example 0.000025 ÷ 0.05 = 0.0005  Because you've already predetermined the decimal place, what looks like a rather complicated decimal problem involving a lot of zeros can be simplified to a much more manageable 25 ÷ 5 = 5

This method is a variation on one taught to me by Edvaldo Siqueira of Rio de Janeiro, Brazil. The method was developed by Professor Fukutaro Kato, a Japanese soroban teacher living in Brazil in the 1960's and was published in the Professor's book, *SOROBAN pelo Método Moderno*. (*SOROBAN by the Modern Method*)

As some of the following examples will reveal, the method works particularly well and helps to simplify problems having complicated decimal numbers.

Counting Digits

Where digits are whole numbers or mixed decimal numbers, count only the whole number before the decimal. Consider the result positive.

Examples:
0.253....... count 0 digits.
2.703....... count +1 digit.
56.0092... count +2 digits.
459.38..... count +3 digits.
1500........ count +4 digits, and so on.
Where digits are pure decimal numbers, count only the zeros that immediately follow the decimal. Consider the result negative.
Examples:
0.40077..... count 0 digits.
0.02030..... count -1 digit.
0.0092....... count -2 digits.
0.00057..... count -3 digits, and so on.


Setting Problems On The Soroban

Multiplication:

Soroban showing rods A through K with rod  F  acting as the unit rod. Rods to the left of the unit rod are considered positive (+) while rods to the right are considered negative (-)

Formula for Setting the Multiplicand: Rod = # of digits in multiplicand PLUS # of digits in multiplier.

Example: 0.03 x 0.001 = 0.00003

For this example, the formula for this problem is: Rod = -1 + (-2) = -3.

Explanation:

a) The multiplicand has one zero after the decimal. Count -1

b) The multiplier has two zeros after the decimal. Count -2. The equation becomes -1 + (-2) = - 3

c) Count MINUS 3 from rod F. Set the multiplicand 3 on rod I and Multiply by 1. The product 03 naturally falls on rods JK. With rod F acting as the unit rod, the answer is 0.00003.

Further examples for multiplication

30 x 8............R = 2 + 1 = 3, set multiplicand 30 on rods CD
2 x 3.14..........R = 1 + 1 = 2, set multiplicand 2 on rod D
12 x 0.75.........R = 2 + 0 = 2, set multiplicand 12 on rods DE
0.97 x 0.1........R = 0 + 0 = 0, set multiplicand 97 on rods FG
0.5 x 0.004.......R = 0 + (-2) = -2, set multiplicand 5 on rod H

Division:

Soroban showing rods A through K with rod  F  acting as the unit rod. Rods to the left of the unit rod are considered positive (+) while rods to the right are considered negative (-)

Formula for Setting the Dividend: Rod = # of digits in dividend MINUS (# of digits in divisor + 2)

Example: 0.0032 ÷ 0.00016 = 20

For this example, the formula becomes: Rod = -2 - (-3 +2) = - 1.

Explanation:

a) The dividend has two zeros after the decimal. Count -2.

b) The divisor has three zeros after the decimal. Count MINUS (-3 + 2) = +1.*

Putting it all together the equation becomes -2 + 1 = -1.

c) Count MINUS 1 from rod F. Set the dividend 32 on rods GH and divide by 16. Following "Rule I" for placing the first quotient number, the answer 2 naturally falls on rods E. With rod F acting as the unit rod, the answer shows 20.

 For more on "Rule I", please see Quotient Rules.


Further examples for division

365 ÷ 0.5.........R = 3 - (0 + 2) = 1, set dividend 365 on rods EFG
0.02 ÷ 0.4........R = -1 - (0 + 2) = -3, set dividend 2 on rod I
0.09 ÷ 0.003......R = -1 - (-2 + 2) = -1, set dividend 9 on rod G
64 ÷ 32...........R = 2 - (2 + 2)= -2, set dividend 64 on rods HI
640 ÷ 32..........R = 3 - (2 + 2) = -1, set dividend 640 on rods GHI
0.004 ÷ 0.0002....R = -2 - (-3 + 2)= -1, set dividend 4 on rod G

* Two negatives multiplied together equal a positive. ex. - (-3 + 2) = +1
 




Ciphering the Divisor - Quick Glance Table


The formula for setting the dividend onto the soroban is  rod = # of digits in dividend MINUS (# of digits in divisor + 2)

I like to break the formula down into two sections
The Dividend: The first part of the formula is  rod = # of digits in dividend  and presents no real difficulty. As shown above in the counting digits section, take the dividend and count either the whole numbers before the decimal or the trailing zeros after the decimal. It's that simple.

The Divisor: The second part of the formula is  rod =  MINUS (# of digits in divisor + 2)  and takes a little more figuring. As an alternative the following quick glance table offers a visual reference to how the divisor portion of the formula works. Rather than trying to commit the table to memory, take the time to understand the sequence. See how it works. Once learned setting the dividend correctly onto the soroban becomes second nature and takes but a moment.
n = any number or numbers in the divisor      (#) = number of trailing zeros after the decimal 
0.00000n (5)
0.0000n (4)
0.000n (3)
0.00n (2)
0.0n (1)
0.n
n.0
nn.0
3 to the left
2 to the left
1 to the left
no count
1 to the right
2 to the right
3 to the right
4 to the right
Adjusting Rods to the Left and Right

 
In the following examples rod  F  will act as the unit rod
Example 1:  0.0012 ÷ 0.3 = 0.004

The dividend:  0.0012  is a pure decimal number with two trailing zeros after the decimal place. Count 2 rods to the right of the the unit rod.
The divisor:  0.3  has neither whole numbers nor trailing zeros. It corresponds to  0.n  in the table. Therefore count a further 2 rods to the right for a total of 4. Set 12  on rods JK. Divide 12 by 3 and the answer 0.004  falls naturally on rods EFGHI


Example 2:  0.12 ÷ 0.003 = 40

The dividend:  0.12  has no whole numbers nor has it trailing zeros. With nothing to count 12 stays on rods FG.

The divisor:  0.003  is a pure decimal number with two trailing zeros. It corresponds to  0.00n  in the table and has no count. Therefore the dividend 12 stays on rods FG. Divide 12 by 3 and the answer 40  falls naturally on rods EF.


Example 3:  12 ÷ 300 = 0.04
The dividend:  12  contains two whole numbers. Count 2 rods to the left of the unit rod.

The divisor:  300  has three whole numbers. Even though the table doesn't show an example with three whole numbers it's easy enough to see the pattern in the sequence. Count 5 rods to the right. Place 12 on rods IJ and divide by 3. The answer 0.04  falls naturally on rods EFGH.


In the above examples I illustrate the technique showing the process of counting left and right. Eventually it will be enough to glance at a problem and quickly predetermine the rod. Example 3 is a good case in point. With a little practice it's easy to see 
2 left _ 5 right = 3 right allowing us to go ahead and place the dividend beginning on Rod I.

 

Back to Multiplication
Back to Division

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Print Quick Glance Table (pdf)
 


 Two alternative methods 
Locating the Decimal 1
&
Locating the Decimal 2

Abacus: Mystery of the Bead
© 2004, 2005 by Totton Heffelfinger & Gary Flom