Square root 323,761 using abacus (Half-multiplication table method 6)

[Set 323,761 on Mr. Square root]

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We will continue from where we ended in the last article, the actual solutions to calculate Square root using abacus. Today's example is Half-multiplication table method (Hankuku method), root is 3-digits case. We require 9 as root in the middle of calculation. Please check the Theory page for your reference. You can check the Index page of all articles.

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 323,761
(Answer is 569)

"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.

323,761 -> (32|37|61) : 32 is the 1st group number. The root digits is 3.


Step 1: Place 323761 on CDEFGH.


Step 2: The 1st group is 32.


Step 3: Square number ≦ 32 is 25=5^2. Place 5 on B as the 1st root.


Step 4: Subtract 5^2 from the 1st group 32. Place 32-5^2=07 on CD.


Step 5: Focus on 73761 on DEFGH.


Step 6: Divide 73761 by 2. Place 36880.5 on DEFGHI.


Step 7: Divide 36 on DE by the current root 5.


Step 8: 36/5=6 remainder 6. Place 6 on C as 2nd root.


Step 9: Place remainder 06 on DE.


Step 10: Focus on 68 on EF.


Step 11: Subtract 2nd root^2/2 from 68 on EF. Place 68-6^2=50 on EF.


Step 12: Focus on 508 on EFG.


Step 13: Divide 508 on EFG by the current root 56.


Step 14: 508/56=9 remainder 4. Place 9 on D as 3rd root.


Step 15: Place remainder 004 on EFG.


Step 16: Focus on 405 on GHI.


Step 17: Subtract 3rd root^2/2 from 405 on H. Place 000 on GHI.


Step 18: Square root of 323761 is 569.


Final state: Answer 569

Abacus state transition. (Click to Zoom)


It is interesting to compare with the Double-root method.


Next article is also about Half-multiplication table method, more difficult example.


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

Square root 323,761 using abacus (Double-root method 7)
http://blog.goo.ne.jp/ktonegaw/e/e57e2bc935af3a511814efb2458b18f4


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