Notes On Newton’s Method

I was privileged to show Newton’s Method in integer arithmetic. Newton wanted to use fractions to see more decimal places, and the duration between his efforts and my good fortune, attests to the difficulty of maintaining objectivity or gaining new perspective.

Here I must document that my breakthrough hasn’t solved the problem completely! (Quelle surprise!)

The method I develop requires raising very large numbers to appreciably large powers. The actual integers become impractical very quickly.

Note: It is fairly practical to get up to six (6) iterations for small n on numerals 1-9, with Python 3. Use this to estimate efficiency- how many decimals are becoming significant?

I stipulate further that it is impractical to guarantee an exact margin of error, using my solution.

Having supplied an answer to some algebra problem requiring Newton’s Method for its solution, I want to close by suggesting this:

If we still want to see more decimal places, is there material advantage to be found in investigating logarithms in more detail? 

[Logarithms already solve the n-th root problem. Is there a way to use THEM for more decimal places? -area of ignorance for me personally-]

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