Chaotic measurements

[ughaibu]

Does chaos theory have any interesting implications for the nature of irrational numbers?

[Qfwfq]

No, why would it?

[ughaibu]

Presumably there are degrees of irrationality, I wondered if chaos theory had an idea about thresholds of irrationality interacting with levels of predictability.

[Qfwfq]

Rational numbers are the ratio of integers, irrational numbers aren't. Transcendental numbers are not the solution of any algebraic equation.

 

Can you give any definition of degrees of irrationality, in between the above two definitions?

[ughaibu]

Qfwfq: The number two has an irrational square root, and that root too has a root irrational to the original two. So we have numbers that can be rationalised by being raised to the power of two or three, etc, up to infinity, this is the kind of degree I had in mind.

 

A different matter, but: could you give a thumbnail description of hyper-real numbers, please. I had the impression that they're periods between rational infinitesimals, but I suspect I'm wrong about that.

[Qfwfq]

I'm not sure what you mean by having

a root irrational to the original two

but I see what you're thinking regarding powers. I don't agree though, why is that a degree of irrationality? It is defined only for a small subset of irrational numbers. Consider something as simple as the sum of two roots, even of the same index, e. g. the square root of 2 plus the square root of 3. Can you find a power of this sum which gives a rational number?

[ughaibu]

Qfwfq: Thanks for the reply. Sorry about "a root irrational to the original two", I was trying to avoid confusion with the "too", basically I was saying that two has an irrational cube root (I assume??). Posts 3 and 5 are just an explanation of the thinking that caused me to post the original question, not an attempt to suggest an answer to the question or to steer any possible answers in a particular direction. I assume your sum of roots would be an example of what I meant by a number becoming rational to the power of infinity. Unfortunately I dont know enough about these things to suggest other ways of defining degrees.

[Qfwfq]

I see what you mean.

basically I was saying that two has an irrational cube root (I assume??).

Yes, of course. All roots of two are irrational and so are most roots of most rationals, it's enough to factorize a rational to see whether it has any rational roots.

 

I assume your sum of roots would be an example of what I meant by a number becoming rational to the power of infinity.

You could choose to call it that but I think this degree is finite for a very small subset of them, even considering numbers that aren't transcendental, because the set of rational roots is countable wheras I don't think the set of irrational roots is.

 

I did however find something up your line :), looking the matter up in Mathworld. A rational is called an algebric number of degree n if it is a root of some rational polynomial of the same degree and not of any lower degree one. I hadn't thought of this, it effectively generalizes your criterion of raising to a power.

[ughaibu]

Okay, thanks for the link.