ABACUS: MYSTERY OF THE BEAD

The Bead Unbaffled - An Abacus Manual## Soroban and Shifting the Unit Rod - Multiplication & Division

I've always been fascinated by decimal numbers. In earlier tutorials I've discussed at length how I deal with decimal numbers when solving problems of multiplication and division. Here's yet another method that's so simple I find I'm using it almost exclusively. While this method is similar to

predetermining the unit rod, the biggest difference is the decimal place is determined after completing the problem.Although this is a soroban technique and I'm using Kojima's methods for both

multiplicationanddivisionthese examples could also be solved using a 2/5 bead suan pan.

Using these techniques to find the correct decimal answer is remarkably straightforward. Solve the problem and simply count either the whole numbers or trailing zeros in the the multiplier or divisor. Then shift the unit rod left or right accordingly. It's that easy. Here's how it works.

## In Multiplication : Where the multiplier has....

1.03....... one whole number - unit rod shifts 1 rod to the right.

45.003... two whole numbers - unit rod shifts 2 rods to the right.

0.75....... no whole numbers, no trailing zeros, unit rod does *not* shift.

0.0125... one trailing zero - unit rod shifts 1 rod to the left.

0.0036... two trailing zeros - unit rod shifts 2 rods to the left...etc.## In Division : Where the divisor has....

1.003..... one whole number - unit rod counts 1 rod to the left.

45.03..... two whole numbers - unit rod counts 2 rods to the left.

0.75....... no whole numbers, no trailing zeros, unit rod does *not* shift.

0.0125... one trailing zero - unit rod counts 1 rod to the right.

0.0036... two trailing zeros - unit rod counts 2 rods to the right...etc.

Where to begin the count? Two rules......

1 In multiplication alwaysstart the countone rod to the rightof the unit rod.

2 In division always start the countone rod to the leftof the unit rod.## Examples where rod F is the unit rod;

In multiplication always start the count beginning one rod to the right of the unit rod (in this case rod G)

## In division always start the count beginning one rod to the left of the unit rod (in this case rod E)

## Multiplication And Shifting the Unit Rod

Example 1:67 x 0.023 = 1.541Choose rod

Fto be the unit rod and set the problem remembering that the multiplier is 0.023 on rods AB. (Fig. 1)(Fig. 1)

Step 1:Using Kojima'smultiplication techniquessolve the problem leaving 1541.

Step 2 and the answer:Choose the new unit rod. The multiplier has one trailing zero. Starting on rodG(one rod to the right of the unit rod) count one to the left. RodFremains the unit rod, therefore the answer to 67 x 0.023 = 1.541 (Fig. 2)(Fig.2)

Example 2:0.067 x 2.3 = 0.1541Choose rod

Cto be the unit rod and set the problem remembering that the multiplier is 2.3 on rods AB.(Fig. 3)

Step 1:Using Kojima'smultiplication techniquessolve the problem leaving 1541.

Step 2 and the answer:Choose the new unit rod. The multiplier has one whole number. Starting on rodD(one rod to the right of the unit rod) count one rod to the right. The new unit rod is rodEtherefore the answer to 0.067 * 2.3 = 0.1541 (Fig. 4)(Fig.4)

Example 3:67,000 x 0.23 = 15410Choose rod

Ito be the unit rod and set the problem remembering that the multiplier is 0.23 on rods AB. (Fig. 5)(Fig. 5)

Step 1:Using Kojima'smultiplication techniquessolve the problem leaving 1541.

Step 2 and the answer:Choose the new unit rod. The multiplier has neither whole numbers nor trailing zeros. Starting on rodJ(one rod to the right of the unit rod) count neither right nor left. The new unit rod is rodJtherefore the answer to 0.23 x 67,000 = 15410 (Fig. 6)(Fig. 6)

## Division And Shifting the Unit Rod

Example 1:1541 ÷ 23 = 67Choose rod

Ito be the unit rod and set the problem remembering that the divisor is 23 on rods AB. (Fig. 7)(Fig. 7)

Step 1:Using Kojima'sdivision techniquessolve the problem leaving 67.

Step 2 and the answer:Choose the new unit rod. The divisor has two whole numbers. Starting on rodH(one rod to the left of the unit rod) count two to the left. The new unit rod is rodFtherefore the answer to 1541 ÷ 23 = 67 (Fig. 8)(Fig. 8)

15.41 ÷ 0.0023 = 6700

Example 2:Choose rod

Fto be the unit rod and set the problem remembering that the divisor is 0.0023 on rods AB. (Fig. 9)(Fig. 9)

Step 1:Using Kojima'sdivision techniquessolve the problem leaving 67.

Step 2 and the answer:Choose the new unit rod. The divisor has two trailing zeros. Starting on rodE(one rod to the left of the unit rod) count two to the right. The new unit rod is rodGtherefore the answer to 15.41 ÷ 0.0023 = 6700 (Fig. 10)(Fig. 10)

Example 3:1.541 ÷ 0.23 = 6.7Choose rod

Fto be the unit rod and set the problem remembering that the divisor is 0.23 on rods AB. (Fig. 11)(Fig. 11)

Step 1:Using Kojima'sdivision techniquessolve the problem leaving 67.

Step 2 and the answer:Choose the new unit rod. The divisor has neither whole numbers nor trailing zeros. Starting on rodE(one rod to the left of the unit rod) count neither right nor left. The new unit rod is rodEtherefore the answer to 1.541 ÷ 0.23 = 6.7 (Fig. 12)(Fig. 12)

Print Shifting the Unit Rod

Two alternative methods

Locating the Decimal1

&

Locating the Decimal2## ▪

Abacus: Mystery of the Bead

▪Advanced Abacus Techniques## © September, 2013

Totton Heffelfinger Toronto Ontario Canada

totton[at]idirect[dot]com