How might Newton’s Method be used to find the square roots of numbers that are otherwise hard (eg, “the square root of 2”)?

ajaar

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How might Newton’s Method be used to find the square roots of numbers that are otherwise hard (eg, “the square root of 2”)?
 
We could develop a general formula using the function:

[MATH]f(x)=x^2-k[/MATH] where \(0<k\)

Then we get the recursion:

[MATH]x_{n+1}=x_{n}-\frac{x_n^2-k}{2x_n}=\frac{1}{2}\left(x_n+\frac{k}{x_n}\right)[/MATH]
 
We could develop a general formula using the function:

[MATH]f(x)=x^2-k[/MATH] where \(0<k\)

Then we get the recursion:

[MATH]x_{n+1}=x_{n}-\frac{x_n^2-k}{2x_n}=\frac{1}{2}\left(x_n+\frac{k}{x_n}\right)[/MATH]
MarkFL has shown you the method - which is also known as Piccard's iteration. In the days of yore, without calculator's, (in INDIA) we use to use this method to approximate the square-root in head (without pencil and paper). The process goes something like this:

You want to calculate square root of 30. Guess an answer - doesnot have to be accurate. Say I guess 5.

N divide the original number (30) by your guess (5).

The result is 6 - far away from you original guess (5).

Now take an average of this quotient (6) and your immediately previous guess → [(5+6)/2 =] 5.5.

So your next guess is 5.5. Approximately divide the original number by this new guess (5.5) → [ 30/5.5 ~] 5.46 - close enough to my original guess.

So I guess √30 ~ 5.46. All these can be done by just thinking - without pencil and paper and of course calculator or slide rule or abacus .......
 
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