Square roots near 1

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  • 1. Square roots near 1 1.1 1.05 - Dave Coulson, 2012
  • 2. For square roots of numbers just slightly above or below 1,the approximation process is simple: 1 1 x 1 2 x 1.1 1.05 (Out by 0.1%) 0. 8 0.9 (Out by 0.4%)
  • 3. This comes about from putting a straight line against the square root curvenear the value x=1. y mx cWhere m d dx ( x 0.5 ) x 1 0.5and ( x, y) (1, 1)
  • 4. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y y m( X x)Where m d dx ( x 0.5 ) x 1 0.5and ( x, y) (1, 1)
  • 5. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 1 0.5( X 1)Where m d dx ( x 0.5 ) x 1 0.5and ( x, y) (1, 1)
  • 6. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 0.5 X 0.5 1Where m d dx ( x 0.5 ) x 1 0.5and ( x, y) (1, 1)
  • 7. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 0.5 X 0.5Where m d dx ( x 0.5 ) x 1 0.5and ( x, y) (1, 1)
  • 8. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 0.5 X 0.5Say X = 1 + a for some small value ‘a’.
  • 9. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 0.5(1 a) 0.5Say X = 1 + a for some small value ‘a’.
  • 10. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 0.5 0.5a 0.5Say X = 1 + a for some small value ‘a’.
  • 11. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  Y 1 0.5aSay X = 1 + a for some small value ‘a’.
  • 12. This comes about from putting a straight line against the square root curvenear the value x=1. y mx c  sqrt (1 a) 1 0.5aSay X = 1 + a for some small value ‘a’.
  • 13. A similar approach could be used to approximate square roots near otherfamiliar numbers. y mx c  sqrt (4 a) 2 1 4 a a N a N 2 N
  • 14. aIn general, N a N 2 N
  • 15. 3For example, 52 49 2 49 3 7 14 7.21 (Out by less than 0.02%)
  • 16. [END] dtcoulson@gmail.com