Quadratic approach to square roots         - Dave Coulson, 2012
This is another iterative technique that doesn’t really help you unless you havea calculator, in which case you can get th...
(12 k ) 2    15012 2    2(k )(12 ) k 2   150       144 24k k 2 150            24k k 2      6                   k2     6   ...
(12 k ) 2    150This number is quite small and can be ignored.                       k2     6                   k         ...
(12 k ) 2   150            6       k          0.25            24
(12 k ) 2     150Therefore    (12 .25 ) 2   150                            6                       k          0.25       ...
(12 k ) 2     150Therefore    (12 .25 ) 2   150      (Out by 0.02%)                            6                       k ...
We could trade accuracy for speed bychoosing a number further from the target.                           (10 k ) 2   150
We could trade accuracy for speed bychoosing a number further from the target.                            (10 k ) 2    150...
(10 k ) 2   150This at least gets me into theneighbourhood of the correctanswer in a few easy steps.                      ...
In general,       (10 k ) 2   N              100 20k k 2     N                              N 100                         ...
In general,                (10 k ) 2   N                     100 20k k 2       N                                       N 1...
In general,                (10 k ) 2   N                     100 20k k 2       N                                       N 1...
Amazingly, this approximation produces errorsno greater than 6% across the range 100-200,making it a good “ballpark-figure...
The method turns out to be exactly the same asa method I’ve described in other lessons.                                  1...
[END]        dtcoulson@gmail.com
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Square roots by quadratic method

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Square roots by quadratic method

  1. 1. Quadratic approach to square roots - Dave Coulson, 2012
  2. 2. This is another iterative technique that doesn’t really help you unless you havea calculator, in which case you can get the square root at the click of a button. (12 k ) 2 150K is an adjustment made to a square root whichis in the neighbourhood of the target value.
  3. 3. (12 k ) 2 15012 2 2(k )(12 ) k 2 150 144 24k k 2 150 24k k 2 6 k2 6 k 24 24
  4. 4. (12 k ) 2 150This number is quite small and can be ignored. k2 6 k 24 24
  5. 5. (12 k ) 2 150 6 k 0.25 24
  6. 6. (12 k ) 2 150Therefore  (12 .25 ) 2 150 6 k 0.25 24
  7. 7. (12 k ) 2 150Therefore  (12 .25 ) 2 150 (Out by 0.02%) 6 k 0.25 24
  8. 8. We could trade accuracy for speed bychoosing a number further from the target. (10 k ) 2 150
  9. 9. We could trade accuracy for speed bychoosing a number further from the target. (10 k ) 2 150 10 2 2(k )(10 ) k 2 150 100 20k k 2 150 20k k 2 50 50 k 2.5 20
  10. 10. (10 k ) 2 150This at least gets me into theneighbourhood of the correctanswer in a few easy steps. 50 k 2.5 20
  11. 11. In general, (10 k ) 2 N 100 20k k 2 N N 100 k 20
  12. 12. In general, (10 k ) 2 N 100 20k k 2 N N 100 k 20 111 100Which means, for example 111 10 20
  13. 13. In general, (10 k ) 2 N 100 20k k 2 N N 100 k 20 111 100Which means, for example 111 10 20 1 2A  100 A 10 10
  14. 14. Amazingly, this approximation produces errorsno greater than 6% across the range 100-200,making it a good “ballpark-figure” method. 1 2 A  100 A 10 10
  15. 15. The method turns out to be exactly the same asa method I’ve described in other lessons. 1 1 A A2 2A 100 A 10 1 10 1 10 100 100 10 1 2 A  100 A 10 10
  16. 16. [END] dtcoulson@gmail.com

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