A curve that does not meet rational points

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wrobel
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This is just to recall a nice fact:

Any two points ##A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1## can be connected with a ##C^\infty##-smooth curve that does not intersect ##\mathbb{Q}^n##.

The proof is surprisingly simple: see the attachment
 

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Sadly the pdf doesn't open on my phone.
 
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That's a really cool fact! It's always interesting to see how seemingly unrelated concepts like smooth curves and irrational numbers can be connected. Can you explain the proof a bit more? It looks like the attachment is just a picture.
 

1. What is a curve that does not meet rational points?

A curve that does not meet rational points is a mathematical concept in which a curve or line on a graph does not intersect with any points that have rational coordinates, meaning they cannot be expressed as a fraction of two integers.

2. How is a curve that does not meet rational points different from other curves?

A curve that does not meet rational points is different from other curves because it does not have any points with rational coordinates, while other curves may have some or all points with rational coordinates.

3. What are some examples of curves that do not meet rational points?

Some examples of curves that do not meet rational points include circles, ellipses, and parabolas with certain parameters. These curves can have points with irrational coordinates, but not rational ones.

4. What is the significance of a curve that does not meet rational points?

A curve that does not meet rational points has significance in mathematics and number theory, as it can help identify patterns and relationships between rational and irrational numbers. It also has applications in cryptography and coding theory.

5. Can a curve that does not meet rational points ever intersect with rational points?

No, a curve that does not meet rational points will never intersect with rational points. This is because its definition states that it does not have any points with rational coordinates, so it is impossible for it to intersect with any points that have rational coordinates.

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