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I am not a student of mathematics but dabble with its concept every now and then.

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- Thread starter Ahmed Abdullah
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In summary, the conversation discusses the need for a denser point arrangement than rational numbers in order to ensure continuity in Euclidean geometry. The concept of earlier infinity is mentioned, which points to the continuum hypothesis. The use of algebraic numbers as points in geometry is also discussed, as well as the limitation of computable numbers in representing certain geometric values. The concept of non-computable numbers is introduced, with an example being Chaitin's constant.

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I am not a student of mathematics but dabble with its concept every now and then.

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What do you mean by "earlier infinity?"Ahmed Abdullah said:

I am not a student of mathematics but dabble with its concept every now and then.

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Mark44 said:What do you mean by "earlier infinity?"

Now I understand i was looking for infinity that is bigger than infinity associated with rational number but smaller than that of real number. I know that it is undecidable.

But i was originally wondering what kind of infinity is necessary for continuity in the sense of euclidean geometry.

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However, what happens when I want to know the length of the diagonal of a square of side length 1? The answer is [itex] \sqrt{2}[/itex]. What about the circumference of a circle of radius 1? The answer is [itex]2\pi[/itex].

You quickly move from rational numbers to irrational as you start looking at polygons, and then quickly to transcendental when you start looking at curves.

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Thank you for your response. I understand your point about algebraic number. If points of the plane is defined by pair of algebraic number, then the intersection of any two lines will also be a point (I get it). But i don't get your last response about computable number. I'll really appreciate if you elaborate the point.jbriggs444 said:

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The value of a trigonometric function or inverse trigonometric function evaluated at a computable number is, if defined at all, computable. The length of a circular arc with a computable radius between two computable endpoints is, therefore, computable.

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What are some examples of non-computable numbers?jbriggs444 said:

The value of a trigonometric function or inverse trigonometric function evaluated at a computable number is, if defined at all, computable. The length of a circular arc with a computable radius between two computable endpoints is, therefore, computable.

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One can nonetheless write down a definition for a non-computable number. For instance, http://en.wikipedia.org/wiki/Chaitin's_constant

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