Looking to perform multiplication on a Japanese abacus?
Congratulations! This means you are very serious about learning the soroban.
Multiplication on a Japanese abacus may seem intimidating at first, but it is actually quite approachable once you know and master the method.
Prerequisites
If you haven't already, we first recommend reading our previous explanatory articles on how to use the Japanese abacus.
Adding and Subtracting on a Soroban
It goes without saying, but performing multiplication will also require you to know your multiplication tables well!
Different Multiplication Techniques
Today we will show you how to perform multiplication on a soroban. There are several techniques for doing this.
However, today we have chosen to present the so-called modern method, which is the most efficient. There are older, more traditional multiplication techniques, but this is the one most commonly used. This is a technique that was also common in Japan around 1930 before being replaced by the standard method used today in Japan. This variant is still favored by a number of experts, including participants in abacus competitions, because it is slightly faster than the standard Japanese method.
Advantages
Besides being faster, the advantage of this technique is that determining the unit rod is very simple and decimal numbers are easy to handle. To determine the unit rod, simply look at the multiplier; count the digits before or after the decimal point, then shift the unit rod to the left or right accordingly.
This technique is therefore better suited for more complex calculations.
Disadvantages
The only real difficulty with this technique is that the operator must remember the digits of the multiplicand from one step to the next, as they are removed from the frame. (See the examples below for an explanation).
Determining the Unit Rod - Counting the Digits of the Multiplier
- When the digits of the multiplier are whole numbers or mixed decimal numbers, count only the whole number part before the decimal. For each whole digit, move the unit rod one rod to the right.
- When the digits of the multiplier are pure decimal numbers, count only the trailing zeros after the decimal point. For each trailing zero, move the unit rod one rod to the left (see next section for explanation).
- When a multiplier has neither whole digits nor trailing zeros, the unit rod does not shift.
Shifting the Unit Rod
1.04..... One whole digit - shift the unit rod 1 rod to the right.
47.009..... Two whole digits - shift the unit rod 2 rods to the right.
0.85..... No whole digits, no trailing zeros, the unit rod does not shift.
0.0189..... One trailing zero - shift the unit rod 1 rod to the left.
0.006..... Two trailing zeros - shift the unit rod 2 rods to the left
Example 1: Multiply 8 x18 x 6 = 48
Step 1: Rod F is the unit rod. Place the multiplicand 8 on rod F and the multiplier 6 to the left.
Step 2: Multiply 8 x 6 = 48, add the product 48 to rods F and G. Note the technique of this step; the 8 on rod F becomes the 4 from the product 48.
Determine the new unit rod: The multiplier has one whole digit, so shift one rod to the right from rod F. The new unit rod is rod G: this gives the answer 48.
Example 2: Multiply 78 x 7 = 546
Step 1: Rod F is the unit rod. Place the multiplicand 78 on rods E and F and the multiplier 7 to the left.
Step 2: Multiply 7 x 8 = 56, add 56 to rods F and G. This step changes the 8 on rod F to 5.
Step 3: Multiply 7 x 7 = 49; change the 7 on E to 4, add 9 to F.
Determine the new unit rod: The multiplier has one whole digit, so shift one rod to the right from rod F. The new unit rod is rod G, leaving the answer 546.
Example 3: Multiply 23 x 45 = 1035
Step 1: Rod I is the unit rod. Place the multiplicand 23 on rods H and I. Place the multiplicand 45 to the left.
Step 2: Multiply 3 x 4 =12, add 12 to rods I and J. For the next step, remember that the multiplicand was 3.
2a: Multiply 3 x 5 = 15, add 15 to rods J and K. This leaves 2 on H and the partial product 135 on rods I, J and K.
Step 3: Multiply 2 x 4 = 8, add 08 to rods H and I. For the next step, remember that the multiplicand was 2.
3a: Multiply 2 x 5 = 10, add 10 to rods I and J leaving 1035 on rods H, I, J and K.
Determine the new unit rod: The multiplier has two whole digits, so shift two rods to the right from rod I. The new unit rod is rod K, leaving the answer 1035.
Example 4: Multiply 0.0756 x 0.87 = 0.065772
Step 1: Rod F is the unit rod. Place the multiplicand 756 on rods H, I and J. Remembering that the multiplier is 0.87, place 87 to the left.
Step 2: Multiply 6 x 8= 48, add 48 to rods J and K. For the next step, remember that the multiplicand was 6.
2a: Multiply 6 x 7 = 42, add 42 to rods K and L. This leaves 75 on H and I and the partial product 522 on rods J, K and L.
Step 3: Multiply 5 x 8 = 40, add 40 to rods I and J. For the next step, remember that the multiplicand was 5.
3a: Multiply 5 x 7, add 35 to rods J and K. This leaves 7 on H and the partial product 4872 on rods I, J, K and L.
Step 4: Multiply 7 x 8 = 56, add 56 to H and I. For the next step, remember that the multiplicand was 7.
4a: Multiply 7 by 7, add the product 49 to rods I and J.
Determine the new unit rod: The multiplier has neither whole digits nor trailing zeros. The unit rod does not move. Rod F remains the unit rod, leaving the answer 0.065772.
The Learning Through Play blog demonstrates another multiplication technique, which may be easier for beginners; we recommend checking it out if you are having trouble with our explanation.
Alcula Virtual Soroban used for illustrations
We also recommend checking out our review of "The Soroban Method," the most comprehensive course for learning the soroban.