Square root 237,169 using abacus (Double-root method 6)

[Set 237,169 on Mr. Square root]

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We will continue from where we ended in the last article, the article shows actual solutions to calculate Square root using abacus. Today's example is simple - basic Double-root method, root is 3-digits case. We require root reduction in the steps. Please check the Theory page for your reference.

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.


Abacus steps to solve Square root of 237,169 (Answer is 487)

"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.

237169 -> (23|71|69) : 23 is the 1st group number. The root digits is 3.


Step 1: Set 237169. 1st group is 23.


Step 2: Square number smaller than or equal to 23 is 16=4^2. 4 is the 1st root.


Step 3: Subtract 4^2 from the 1st group 23. 23-16=07


Step 4: Place 8 which is 2x of 1st root 4. This 8 is double root. Focus on 77.


Step 5: Divide 77 by 8. Answer=9 and this is 2nd root.


Step 6: Subtract 8x9=72 from 77. Place 05 on FG.


Step 7: You cannot subtract 9^2 (=81) from the 51 on GH, so the 2nd root 9 is over-root. Subtract 1 from 2nd root 9 then replace the 2nd root as 8, give back the double-root 8. Place 05+08=13 on FG.


Step 8: Focus on 2nd root 8 and 131 on FGH.


Step 9: Subtract 8^2 from 131 (FGH) and set the answer 067 to FGH.


Step 10: Focus on double-root 80 and 2nd root 8.


Step 11: Add 2 x 2nd root 8 (=16) to double-root 80. Place 2x8+80=96 on AB.


Step 12: Divide 676 on GHI by double-root 96.


Step 13: Answer=7 and this is 3rd root. Place 3rd root 7 on F. Place remainder 676-96x7=004 on GHI.


Step 14: Focus on 3rd root 7 and 3rd group 49 on IJ.


Step 15: Subtract 7^2 from 3rd group 49. Place 49-49=00 on IJ.


Step 16: Square root of 237169 is 487.


Final state: Answer 487

Abacus state transition. (Click to Zoom)



Next article is also about Double-root method.


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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