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

PREDETERMINE THE UNIT ROD - An alternative

For several years now I've been using a variation of Kato's method to help me predetermine a unit rod when solving solve problems of multiplication and division. Because I like it so much Kato's method is the one I most often use. But I love learning alternatives and here's a good one.
 

WHERE MULTIPLIERS ARE WHOLE NUMBERS

(Use multipliers and multiplicands of any size, follow these simple rules and products will always fall neatly on a unit rod.)

THE RULES

The Multiplier consists of:
One digit ===>  move the multiplicand one to the right of a unit rod
Two digits ==>  leave the multiplicand as is
Three digits => move the multiplicand one to the left of a unit rod

EXAMPLES
As an example, if we consider rod C to be the unit rod the number 85 would be set on rods BC as shown below.
A B C D E F
    .     .
0 8 5 0 0 0

Using the above set of rules, re-set the multiplicand before multiplying. This will allow the product (answer) to fall neatly on new unit rod F

Example 1: 85 x 5 - because this example has a one digit multiplier set 85 one rod to the right. Set 85 on rods CD
A B C D E F
    .     .
0 0 8 5 0 0
      + 2 5
    (-5)   
0 0 8 0 2 5
    + 4 0
  (-8)     
0 0 0 4 2 5

Example 2: 85 x 35 - because this example has a two digit multiplier leave 85 as is. Set 85 on BC
A B C D E F
    .     .
0 8 5 0 0 0
    + 1 5
      + 2 5
  (-5)     
0 8 0 1 7 5
  + 2 4 
    + 4 0
(-8)       
0 0 2 9 7 5

Example 3: 85 x 835 - because this example has a three digit multiplier set 85 one rod to the left. Set 85 on AB
    A B C D E F
        .     .
    8 5 0 0 0 0
      + 4 0
        + 1 5
          + 2 5
     (-5)      
    8 0 4 1 7 5
    + 6 4 
      + 2 4
        + 4 0
  (-8)         
    0 7 0 9 7 5
 

AS MULTIPLIERS GET LARGER

As multipliers get larger it's important to see how this works. Here are a few examples to get you started.
Note: As multipliers get larger products get larger too. Notice how many unit rods a multiplier occupies. In this respect product answers mirror the multiplier. In the examples below multipliers span two unit rods. Therefore the new unit rod will move two units to the right of the chosen unit rod.

The multiplier 4567 spans two unit rods
____._____._____.
A B C D E F G H I
0 0 4 5 6 7 0 0 0
and is considered to be one digit because 4 falls on rod C. Choose any unit rod and set your multiplicand one rod to the right.


The multiplier 34567 spans two unit rods
____._____._____.
A B C D E F G H I
0 3 4 5 6 7 0 0 0
and is considered to be two digits because 34 falls on rods BC. Choose any unit rod and set the multiplicand as is.

The multiplier 234567 spans two unit rods
____._____._____.
A B C D E F G H I
2 3 4 5 6 7 0 0 0
and is considered to be three digits because 234 falls on rods ABC. Choose any unit rod and set the multiplicand one rod to the left ..... and so on.
 

WHERE MULTIPLIERS ARE DECIMAL NUMERALS

The technique for multiplying pure decimal numerals differs only slightly to multiplying integers. But for me this was the revelation. In the case of decimals the way to set the multiplicand onto the soroban depends on the multiplier and it's multiple.

HOW TO DETERMINE A MULTIPLE

The dots (or unit rods) on a soroban help us to make sense of numbers. In order to be able to work with decimal numerals efficiently we need to understand the multiples of decimal numerals quickly. The soroban shows us how to do this.
EXAMPLES
Consider rod C to be the unit rod. Set the following numbers onto the soroban:
Example 0.0063
A B C D E F G H
    .     .    

          6 3
in this example if we didn't know that the number set onto the soroban was the decimal numeral 0.0063, we could just as easily take it to be 6.3  Because 6 occupies rod F (rod with a dot) it has a multiple of one.


Example 0.063
A B C D E F G H
    .     .    

        6 3
in this example if we didn't know that the number set onto the soroban was the decimal numeral 0.063 we could just as easily take it to be 63. Because 63 occupies rods EF it has a multiple of two.


Example Set 0.63
A B C D E F G H
    .     .    

      6 3
in this example if we didn't know that the number set onto the soroban was the decimal numeral 0.63 we could just as easily take the number to be 630. Because 630 takes up rods DEF it has a multiple of three.

RULES

The Multiplier Consists of:
One digit multiple ===> move the multiplicand one to the right of a unit rod
Two digit multiple ===> leave the multiplicand as is
Three digit multiple => move the multiplicand one to the left of a unit rod
 
Rod C is a unit rod, set 3 on rods C
A B C D E F G H
    .     .    

    3

Now follow the rules

Use the above rules to re-set the multiplicand before multiplying. This will allow the product (answer) to fall neatly on unit rod C.
Example: 3 x 0.0063 - one digit multiple, move the multiplicand one to the right of a unit rod. Set 3 on rod D
A B C D E F G
    .     .  

      3        
      + 1 8
        + 0 9
    (-3)     
        1 8 9     Answer = 0.0189

Example: 3 x 0.063 - two digit multiple, leave the multiplicand as is. Set 3 on rod C
A B C D E F G
    .     .  

    3          
    + 1 8
      + 0 9
  (-3)       
      1 8 9       Answer = 0.189

Example: 3 x 0.63 - three digit multiple, move the multiplicand one to the left of a unit rod. Set 3 on rod B
A B C D E F G
    .     .  

  3            
  + 1 8
    + 0 9
(-3)         
    1 8 9         Answer = 1.89


AS PURE DECIMAL MULTIPLIERS GET LARGER

C is the unit rod. Set the following examples;
The multiplier 0.000004 is considered as one digit multiple because 4 falls on unit rod I. Choose any unit rod and set your multiplicand one rod to the right.
____._____._____.__
A B C D E F G H I J
                4
The multiplier 0.00004 is considered as two digit multiple because 40 falls on rods HI. Choose any unit rod and set your multiplicand as is.
____._____._____.__
A B C D E F G H I J
              4
The multiplier 0.0004 is considered as a three digit multiple because 400 falls on rods GHI. Choose any unit rod and set your multiplicand one rod to the left..... and so on
____._____._____.__
A B C D E F G H I J
            4
 

DIVIDING WHOLE NUMBERS

(Use divisors and dividends of any size, follow these simple rules and quotients (answers) will always fall neatly on a unit rod.)

THE RULES

The divisor consists of:
One digit ===>  move the dividend one to the left of a unit rod
Two digits ==>  leave the dividend as is
Three digits => move the dividend one to the right of a unit rod
EXAMPLES
Use the above rules to re-set the dividend before dividing. This will allow the quotient to fall neatly on the new unit rod C
Example 1: 425 5 - consider rod F to be the unit rod. This example has a one digit divisor therefore set 425 one to the left on rods CDE
Set the quotient
A B C D E F G
    .     .  
0 0 4 2 5 0 0
 (8)
   -4 0      
0 8 0 2 5 0 0
   (5)
     -2 5    
0 8 5 0 0 0 0
 
Example 2: 2975 35 - consider rod F to be the unit rod. This example has a two digit divisor therefore set 2975 as is on rods CDEF
A B C D E F G
    .     .  
0 0 2 9 7 5 0
 (8)
   -2 4
     -4 0    
0 8 0 1 7 5 0
   (5)
     -1 5
       -2 5  
0 8 5 0 0 0 0
 
Example 3: 70975 835 - consider rod F to be the unit rod. This example has a three digit divisor therefore set 70975 one to the right on rods CDEFG
A B C D E F G
    .     .  
0 0 7 0 9 7 5
 (8)
   -6 4
     -2 4
       -4 0   
0 8 0 4 1 7 5
   (5)
     -4 0
       -1 5
         -2 5
0 8 5 0 0 0 0
 

WHERE DIVISORS ARE DECIMAL NUMBERS

HOW TO DETERMINE A MULTIPLE

 For division it's exactly the same as for multiplication (see Determine the Multiple)

THE RULES

The divisor consists of:
One digit multiple ===>  move the dividend one to the left of a unit rod
Two digits multiple ==>  leave the dividend as is
Three digits multiple => move the dividend one to the right of a unit rod
Example: 25 0.005 - consider rod F to be the unit rod. This example has a one digit multiple therefore set 25 one to the left on rods DE
A B C D E F G
    .     .  
0 0 0 2 5 0 0
   (5)
     -2 5    
0 0 5 0 0 0 0
 
Example: 25 0.05 - consider rod F to be the unit rod. This example has a two digit multiple therefore set 25 as is on rods EF
A B C D E F G
    .     .  
0 0 0 0 2 5 0
     (5)
        -2 5 
0 0 0 5 0 0 0
 
Example: 25 0.5 - consider rod F to be the unit rod. This example has a three digit multiple therefore set 25 one rod to the right on rods FG
A B C D E F G
    .     .  
0 0 0 0 0 2 5
       (5)
         -2 5
0 0 0 0 5 0 0

- Totton Heffelfinger (Feb, 2008)
 
References:
Yabuki, Shinichi R.
Sigma Educational Supply co.
Modern abacus: An effective Mathematical Tool

Abacus: Mystery of the Bead
Advanced Abacus Techniques
© February, 2008
Totton Heffelfinger   Toronto Ontario  Canada
Email
totton[at]idirect[dot]com