Finding quadratic irrationals from their decimal form

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Discussion Overview

The discussion revolves around the challenge of finding quadratic irrationals from their decimal representations, particularly focusing on the use of continued fractions and the limitations of representing polynomial roots in exact forms. Participants explore the theoretical aspects of polynomial solutions and the nature of irrational numbers.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses a desire to convert decimal approximations of polynomial roots into exact forms, specifically quadratic irrationals, using continued fractions.
  • Another participant argues that there are infinitely many irrationals that could fit a given decimal, many of which are not in the form of surds, suggesting that the search for a unique representation may be futile.
  • A participant asserts that not all zeroes of polynomials can be represented as quadratic irrationals, emphasizing that while all quadratic irrationals can be expressed as surds, this does not apply to all polynomial roots.
  • There is mention of the theoretical possibility of representing all continued fractions as surds, which raises questions about the limitations of continued fractions in solving polynomial equations.
  • Some participants discuss the implications of inverting polynomial functions, particularly quintic equations, and the operations required for their solutions.
  • A later reply reiterates the difficulty of converting certain zeroes into the desired form and acknowledges the need for high precision in calculations when using continued fractions.

Areas of Agreement / Disagreement

Participants generally disagree on the feasibility of representing all polynomial roots as quadratic irrationals. While some acknowledge the potential of continued fractions, others highlight the limitations and the existence of multiple irrational solutions.

Contextual Notes

The discussion reveals limitations regarding the precision required for continued fractions and the theoretical constraints on polynomial solutions. There is also an acknowledgment of the complexity involved in converting decimal forms to exact representations.

rabbit boy
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Okay, this is something that's been bugging me for a while. A lot of the polynomials with higher powers can't be solved using algebraic methods and must be estimated. So my idea was to take these answers and find a way to convert them into an exact form, such as quadratic irrationals/surds.

I do not know if all zeroes of polynomials can be represented as quadratic irrationals. However, I did find out that you can use continued fractions to get a quadratic irrational solution from its decimal value to its \dfrac{a + b\sqrt{c}}{d} form.

The problem I'm having with continued fractions is that you have to keep going until you find where they repeat, for many quadratic irrationals there are dozens of numbers in the series, and I don't have enough accuracy in my calculations to find where it repeats.

And then of course there's guess and check, but I don't know a way of doing it efficiently yet.

So what should I do?
 
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No matter how many decimal places you got, there are in theory infinite number of irrationals that fit, and a lot of them not in the form of surds. So I think it will to futile to look for such a way.
 
wywong said:
No matter how many decimal places you got, there are in theory infinite number of irrationals that fit, and a lot of them not in the form of surds. So I think it will to futile to look for such a way.

I already know it's possible to find what the quadratic irrational is with continued fractions. I made a thread about it on another forum.

http://bbs.zoklet.net/showthread.php?t=11555
 
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But that was not your question. You asked about using continued fractions to get exact solutions to higher degree equations. "I do not know if all zeroes of polynomials can be represented as quadratic irrationals." No, they cannot. All quadratic irrationals can be represented as surds. Not all zeroes of polynomials can be. Further, it is at least theorectically possible to represent all continued fractions as surds so it is not possible to represent all solutions to polynomial equations as continued fractions either.
 
I once read that if we could invert the function x^5 + x = y to be x = f(y) then we could solve all 5 degree polynomials using algebraic operations together with the function f.
 
confinement said:
I once read that if we could invert the function x^5 + x = y to be x = f(y) then we could solve all 5 degree polynomials using algebraic operations together with the function f.

What you read is true, so what? All polynomials can be solved in any algebraically complete field such as the complex numbers. The question is what operations are needed to effect the solution. Field operations and root extractions suffice for at most quartic equations. A possible additional operation to allow solution of quintics in the bring radical you mention.
 
HallsofIvy said:
But that was not your question. You asked about using continued fractions to get exact solutions to higher degree equations. "I do not know if all zeroes of polynomials can be represented as quadratic irrationals." No, they cannot. All quadratic irrationals can be represented as surds. Not all zeroes of polynomials can be. Further, it is at least theorectically possible to represent all continued fractions as surds so it is not possible to represent all solutions to polynomial equations as continued fractions either.

My question is, for zeroes that can be represented in the form \dfrac{a + b\sqrt{c}}{d}, what is a good way to convert them from decimal form to that?

I already know how to do it with continued fractions, but sometimes it requires too many digits of precision. so I'm probably looking for a different way to do it.

I'm sorry for not being clear about that.

And thank you, you two, for answering whether all of them can be represented in that form. I was just a bit grumpy yesterday.
 
rabbit boy said:
My question is, for zeroes that can be represented in the form \dfrac{a + b\sqrt{c}}{d}, what is a good way to convert them from decimal form to that?
I think in general it would be hard.
 

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