Square root using abacus: Theory (Half-multiplication table method)

Square root & Cube root

[Japanese]

This is beginning of articles about how to solve Square root second season. Today is the theory of Half-multiplication table method. (Hankuku method)

Square root methods: Double-root method, Double-root alternative method, half-multiplication table method, half-multiplication table alternative method, multiplication-subtraction method, constant number method, etc.


"1st group number" is the left most numbers in the 2-digits groups of the given number for square root calculation. Number of groups is the number of digits of the Square root.

1,764 -> (17|64): 17 is the 1st group number. The root digits 2.
60,516 -> (6|05|16) : 6 is the 1st group number. The root digits is 3.

Square root of 60,516 is 246. We call 1st root=2, 2nd root=4, 3rd root=6.


Case 1) Root is 2-digits

1st root=a, 2nd root=b, then S (Square) is given by following formula.



Use next formula for the Square root calculation.





Step 1) Find the 1st root and subtract its square

Find the 1st root and subtract its square from 1st group number.



Step 2) Half-remainder

Half-remailder is the divide the remainder by 2 after subtraction of the 1st-root. It means the division by 2 from the right-most of the remainder.





Step 3) Calculate 2nd root

Calculate the 2nd root by division by the 1st root.





Step 4) Subtract Half-square

Subtract Half-square of the 2nd root. Half-square is the half of the multiplication table. If the multiplication is 5x5=25, the Half-square is 5*5=25/2. (We use "*" instead of "x" for Half-multiplication in this article.)






Case 2) Root is 3-digits

1st root=a, 2nd root=b, 3rd root=c, then S (Square) is given by following formula.



Use next formula for the Square root calculation.



Follow same steps as root is 2 digits case until finding the 2nd root. But we use next formula for the 3rd root of Square root calculation because remainder is divided by 2.





Step 1) Calculate 3rd root

Calculate 3rd root by division by current root.





Step 2) Subtract Half-square

Subtract Half-square of the 3rd root.




Next article shows how to solve Square Root by Half-multiplication table method using "Mr. Square Root" abacus.


Related articles:

How to solve Cube root of 1729.03 using abacus? (Feynman v.s. Abacus man)
http://blog.goo.ne.jp/ktonegaw/e/cff5d6e7ecaa07230b9cc7af10b23aed

Index: Square root and Cube root using Abacus
http://blog.goo.ne.jp/ktonegaw/e/f62fb31b6a3a0417ec5d33591249451b


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