By Takashi Kojima
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Additional info for Advanced abacus: Japanese theory and practice
Subtracting 19 from 20, add 1 to the 15 on EF. This gives you 16 on EF and 99 on BC (Fig. 159). STEP 4: As you cannot subtract 10 from the 8 on I, borrow 1 from the 16 on EF, and subtracting 10 from 12, add the result 2 to the 8 on I. This gives you 10 on HI and 15 on EF (Fig. 160). STEP 5: Now add 26 to the 99 on BC, and you get 125 on ABC. Add 2 to the 15 on EF, leaving 17 on EF. Add 7 to the 10 on HI yielding 17 on HI. Finally subtract 12 from this 17 and carry 1 to the 17 on EF. This gives you 5 on I and 18 on EF.
Set the second trial quotient figure 5 on B (Fig. 183). NOTE: The 5 on B corresponds with the b of the expression. STEP 5: Multiply the 4 on A by the 5 on B, and subtract the product 20 from the 22 on CD. This leaves 25 on DE (Fig. 184). STEP 6: Square the 5 on B, and subtract the product 25 from the 25 on DE, and the board is cleared of the dividend (Fig. 185). STEP 7: Finally halve the 4 on A into its original digit 2. Our square root is 25 on AB (Fig. 186). METHOD C: STEP 1: Set 625 on CDE, with E as the unit rod (Fig.
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. To understand this compare the examples below, which show the standard written method (left) and the abacus method (right).