A Method taught by Takashi Kojima in his book
- "Advances Abacus, Theory and Practice"
 

 

Multiplication by Complementary Numbers

As Kojima explains this is a quick and efficient method that can be used when a multiplier is a little smaller than 100, 1000 etc. Example 96 or 994. The idea is to multiply the *complement* of the multiplier with respect to 100, 1000 etc. and subtract its product form the multiplicand. This is best explained using an example.

Example 1:  57.83 x .96 = 55.5168

The answer can be found by the following simplified multiplication.
  57.83 x 0.96
=57.83 x (1 - 0.04)
=57.83 - (57.83 x 0.04)
=55.5168

Step 1: Set the multiplicand 57.83 on DEFG and set 4, the complement of 96 with respect to 100, on A. (Fig.1)  

Fig.1

NOTE: The unit number 7, is set on unit rod E.

 

Step 2: Multiply the 4 on A by the 3 on G and subtract the product 12  from HI. This leaves 57.8288 on DEFGHI. (Fig.2)

Fig.2

 

Step 3: Multiplying the 4 on A by the 8 on F, subtract its product 32 from GH. This leaves 57.7968 on DEFGHI. (Fig.3)

Fig.3

 

Step 4: Multiply the 4 on A by the 7 on E and subtract the product 28 from FG, leaving 57.5168 on DEFGHI. (Fig.4)

Fig.4

 

Step 5: Lastly multiply the 4 on A by the 5 on D. Subtract the product 20 from EF, leaving 55.5168 (Fig.5) the answer.

Fig.5

The answer 55.5168 on rods DEFGHI

 

Totton Heffelfinger
Toronto Ontario Canada
May 2004