How can you estimaite [tex]\sqrt 40.[/tex]

9514 1404 393

Answer:

  √40 ≈ 6 4/13 ≈ 6.31

Step-by-step explanation:

The estimate by linear approximation starts with the recognition that ...

  40 = 36 +4 = 6² +4

and the difference from 36 to the next square, 7² = 49 is 49-36 = 13.

The linear approximation to √40 is ...

  √40 ≈ 6 4/13 ≈ 6.31

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There are various methods for improving this approximation. Perhaps the fastest improvement is obtained using the "Babylonian method." The next iteration is the average of this value and 40 divided by this value.

  (82/13 +40/(82/13))/2 = 6 173/533 ≈ 6.3246 . . . . accurate to 4 decimal places

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Additional comment

When you write the value in terms of a square and a remainder:

  n = s² +r

then the above linear approximation of √n can be written as ...

  √n ≈ s +r/(2s+1)

Using the idea that a square root can always be represented by a continued fraction, we can continue the fraction one more iteration to give a slightly better approximation:

  √n ≈ s +r(2s+1)/(2s(2s+1) +r)

For the case at hand, this gives √40 ≈ 6+52/160 = 6 13/40 = 6.325, accurate to 3 decimal places. Additional iterations using the continued fraction idea can be had using ...

  n'' = s + r/(s +n') . . . . . where n' is an approximation of the root and n'' is a better approximation. This does not converge as fast as the Babylonian method, but may be easier to calculate on some calculators.

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