does anyone know how to write -1.3107026413 as a simple polynomial using a square root?

Question

-1.3107026413 is only an approximation because the decimals don't stop there. so I'm looking for the irrational number in terms of a rational number +/- a square root term:
like .6 - sqrt(2)
or (1 + sqrt(5))/2 or something like that.

how does one do this type of algorithm?

Answer 1

The correct term is 'continued fractions'. You can represent quadratic surds by periodic continued fractions. Your number has the following continued fraction representation:

-2 + [0, 1, 2, 4, 1, 1, 2, 1, 3, 2, 1, 2, 32, 5, 4, 15, 4, 17, 4]

Unfortunatelly, I can not see a periodicity. This doesn't mean that it is not a root of a polynomial equation. It may be a cubic root, for example.

Answer 2

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Answer 3

Use partial fractions. That might help. Might take a little while though.

Correction: Continuous Fractions.

Answer 4

...too many variables to try to wade through...

Answer 5

Yes, I do.