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. 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 54,756 (Answer is 234)
"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.
54,756 -> (05|47|56) : 5 is the 1st group number. The root digits is 3.
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Step 1: Set 54756. 1st group is 5.
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Step 2: Square number equal to or smaller than 5 is 4=2^2. 2 is the 1st root. Place 4 which is 2x of 1st root 2. This 4 is double root.
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Step 3: Subtract 2^2 from the 1st group 5. 5-4=1 : -a^2
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Step 4: Focus on 14 and divide it by double root 4.
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Step 5: Answer=3 and this is 2nd root. Set second root 3 on E.
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Step 6: Subtract double root 4 x 2nd root 3 from 14. 14-4x3=02.
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Step 7: Focus on 27.
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Step 8: Subtract (2nd root 3)^2 from 2nd group 27. 27-3^2=18
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Step 9: Add 2x2nd root 3 to double root. Set 2x3=6 on B.
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Step 10: Focus on double root 46 and 185.
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Step 11: Divide 185 by double root 46. Answer=4 reminder 1. Set them on F and I.
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Step 12: Subtract (3rd root 4)^2 from 3rd group 16. 16–4^2 =0: -b^2
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Step 13: Square root of 54756 is 234.
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Final state: Answer 234
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著者について:
イン・ミンシェン(Mingsheng Ying)HP: http://quantum-lab.org/mingsheng/
Mingsheng Ying (h-index: 34) is currently a Distinguished Professor at the University of Technology Sydney (UTS) and Research Director of the Center for Quantum Computation and Intelligent Systems, at UTS. He was the Cheung Kong Chair Professor, in the Department of Computer Science and the Scientific Director of the National Key Laboratory of Intelligent Technology and Systems at Tsinghua University. His research interests are quantum computation and quantum information, programming language theory and artificial intelligence. In 2008 he received The National Science and Technology Award for contributions in computer science from China.
He is an Associate Editor of Artificial Intelligence (Elsevier) and he has published more than 100 papers in top international journals and conferences such as ACM Transactions on Programming Languages and Systems, Artificial Intelligence, IEEE Transactions on Information Theory, IEEE Transactions on Software Engineering, Information and Computation, Journal of Computer and System Sciences, Physical Review Letters, POPL, CONCUR, IJCAI. He is also the author of the book Topology in Process Calculus - Approximate Correctness and Infinite Evolution of Concurrent Programs (Springer 2001).
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.
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Step 1: Set 237169. 1st group is 23.
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Step 2: Square number smaller than or equal to 23 is 16=4^2. 4 is the 1st root.
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Step 3: Subtract 4^2 from the 1st group 23. 23-16=07
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Step 4: Place 8 which is 2x of 1st root 4. This 8 is double root. Focus on 77.
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Step 5: Divide 77 by 8. Answer=9 and this is 2nd root.
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Step 6: Subtract 8x9=72 from 77. Place 05 on FG.
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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.
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Step 8: Focus on 2nd root 8 and 131 on FGH.
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Step 9: Subtract 8^2 from 131 (FGH) and set the answer 067 to FGH.
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Step 10: Focus on double-root 80 and 2nd root 8.
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Step 11: Add 2 x 2nd root 8 (=16) to double-root 80. Place 2x8+80=96 on AB.
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Step 12: Divide 676 on GHI by double-root 96.
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Step 13: Answer=7 and this is 3rd root. Place 3rd root 7 on F. Place remainder 676-96x7=004 on GHI.
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Step 14: Focus on 3rd root 7 and 3rd group 49 on IJ.
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Step 15: Subtract 7^2 from 3rd group 49. Place 49-49=00 on IJ.
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Step 16: Square root of 237169 is 487.
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Final state: Answer 487
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 9 as root in the middle of calculation. 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 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.
323761 -> (32|37|61) : 32 is the 1st group number. The root digits is 3.
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Step 1: Set 323761. 1st group is 32.。
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Step 2: Square number smaller than or equal to 32 is 25=5^2. 5 is the 1st root.
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Step 3: Subtract 5^2 from the 1st group 32. 32-25=07
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Step 4: Place 10 which is 2x of 1st root 5. This 10 is double root. Focus on 73.
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Step 5: Divide 73 by 10. Answer=6 and this is 2nd root. Subtract 10x6 from 73. 73-10x6=13
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Step 6: Place 2x double root 12 on BC. Focus on 37.
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Step 7: Subtract 6^2 from 37. 37-6x6=01。
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Step 8: Focus on 2nd root 8 and 131 on FGH.
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Step 9: Divide 1016 by 12. Answer=9. 9 is the 3rd root on G.
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Step 10: Set 1016-112x9=0008 on HIJK.
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Step 11: Focus on 3rd root 9 and 81 on KL.
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Step 12: Subtract 9^2 from 81 on KL. Set 81-81=00 on KL.
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Step 13: Square root of 323761 is 569.
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Final state: Answer 569
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. There is Zero in root. 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 164,836
(Answer is 406)
"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.
164836 -> (16|48|36) : 16 is the 1st group number. The root digits is 3.
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Step 1: Set 164836. 1st group is 16.
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Step 2: Square number smaller than or equal to 16 is 16=4^2. 4 is the 1st root.
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Step 3: Subtract 4^2 from the 1st group 16. 16-16=00
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Step 4: Place 8 which is 2x of 1st root 4. This 8 is double root. Focus on 48.
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Step 5: Cannot divide 04 by 8. Place 0 on E as 2nd root.
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Step 6: Divide 483 by double root 80.
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Step 7: Answer is 6 and this is 3rd root.
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Step 8: Place 483-80x6=003 on GHI.
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Step 9: Subtract square of 3rd root 6 from 36 on IJ.
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Step 10: Place 36-6x6=00 on IJ.
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Step 11: Square root of 164836 is 406.
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Final state: Answer 406