Square root using abacus: Theory (Double-root method)

Square root & Cube root

[Japanese]

This is beginning of articles about how to solve Square root. Today is the theory of Double-root method. (Baikon 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) 2x(1st root), find the 2nd root

Divide remainder by 2x(1st root) and get the 2nd root. This is why we call the calculation as "Double-root method".



Step 3) Subtract square of the 2nd root




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. Use next formula for the 3rd root of Square root calculation.



Step 1) Find 3rd root

Divide by Double root: 2(a+b) and find 3rd root.



Step 2) Subtract square of 3rd root




Case 3) Root is 4-digits

1st root=a, 2nd root=b, 3rd root=c, 4th root=d, then S (Square) is given by following formula.



Use next formula for the Square root calculation.



Step 1) Divide by Double root: 2(a+b+c) and find 4th root.

Step 2) Subtract square of 4th root




Next article shows how to solve Square Root by Double-root 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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