Is it possible to use an algebraic formula, equation, concept, or principle to determine with perfect accuracy (or high precision, if not perfect) whether or not a number is rational?
An example number I have in mind is $\sqrt{937}$.
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1One can show that $\sqrt{p/q}\in\Bbb Q$ if and only if $p\in\Bbb Z$ and $q\in\Bbb Z\setminus{0}$ are perfect squares. – Stahl Oct 20 '13 at 07:17
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I obtained that number with $\sqrt{85 + 132 + 720}$ in an attempt to solve the perfect Euler brick problem. None are squares, but is there any chance the expression could still be rational? – kettlecrab Oct 20 '13 at 07:24
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@Stahl: (assuming $p$ and $q$ are coprime, that is) – Robert Israel Oct 20 '13 at 07:37
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1$p=8$ and $q=2$ are not perfect squares, but $\sqrt{p/q} \in \mathbb Q$. – Robert Israel Oct 20 '13 at 07:58
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Ah, of course. Silly forgetting the other direction! – Stahl Oct 20 '13 at 08:00
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I think you need to consult an expert on this, but I would boldly suggest the answer is unknown. The current method is to assume the number coming from a certain algebraic equation of degree $n$, then check the decimal part of the number and see if it become linearly dependent after taking the power a few times, etc. I think for a lot of constants like $\zeta(2n+1)$ or Euler's constant, we do not actually know whether they are rational or irrational, largely because the above method fails, which in fact only tells us the weaker result that the number should be algebraic by bounding its "height". For quadratics it is easy, but if the number is not a period or coming from a non-obvious limiting process, then the answer can be tricky and difficult.
Again, excuse me if I made any mistakes. A real expert in algebraic number theory would be better to answer your question.
Bombyx mori
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Whether a number is rational or irrational is not a matter of precision.
In the case of an algebraic number, you can find (by algebraic methods) the minimal polynomial
this satisfies over the rationals. If that has degree $> 1$, the number is irrational.
Robert Israel
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Could you give an example of how I might find a polynomial to satisfy a single number? I'm not quite sure I understand. – kettlecrab Oct 20 '13 at 07:46
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For example, $\sqrt{2} + \sqrt{3}$ is a root of $z^4 - 10 z^2 + 1$, which is irreducible over $\mathbb Q$. I found this using Maple. – Robert Israel Oct 20 '13 at 08:01
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Ah, I see. Too bad that $\sqrt{937} \not \in \mathbb{Q}$ (my theory was that if it was rational, multiplying the number by 10^(the number of decimal places) could get an answer for $\sqrt{lwh}$ which could then be done in the same way for length, width, and height to find the perfect cuboid. – kettlecrab Oct 20 '13 at 08:14