By Takashi Kojima
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Extra info for Advanced abacus: Japanese theory and practice
STEP 1: Set 6 190 on IJKL, with L as the unit rod, and set 12 on CD. See that there are four vacant rods between the two numbers (Fig. 136). STEP 2: Comparing the divisor 12 and the 61 in 6 190, set the quotient figure 5 on G. Now multiply the 1 on C by this 5, and subtract the product 5 from the 6 on I. This leaves 1 190 on IJKL. Next multiply the 2 on D by the same 5 on and subtract the product 10 from the 11 on IJ. This leaves 190 on JKL (Fig. 137). STEP 3: Comparing the divisor 12 and the 19 on JK, set the second quotient figure 1 on H.
METHOD A: Adding from higher to lower denominations: STEP 1: Set 21 yd. on AB, with B as the unit rod. Set 1 ft. on D and 8 in. on G (Fig. 143). Refer to Fig. 123 at the beginning of this chapter. STEP 2: Add 15 to the 21 on AB. This makes a total of 36 on AB. Now add 2 to the 1 on D. This makes a total of 3 on D. 47 Advanced Abacus Japanese Theory and Practice, by Takashi Kojima Next add 7 to the 8 on G. This makes a total of 15 on FG (Fig. 144). STEP 3: Add 29 to the 36 on AB. This makes a total of 65 on AB.
A+b)2 a2 + 2ab + b2 202 + (2 x 20 x 5) + 52 400 + 200 + 25 625 Both the abacus method and this written method are based on algebraic equations. In a simple problem (one involving a two-digit answer) both methods of extracting a square 53 Advanced Abacus Japanese Theory and Practice, by Takashi Kojima root can be described by the binomial equation (a+b)2 - a2 - 2ab - b2 = 0. In a slightly more complicated problem (one involving a three-digit answer) both can be described by the equation (a + b + c)2 - a2 - 2ab - b2 - 2ac - 2bc - c2 = 0.