Can a Rational Point Always be Found Amongst a Set of Points in the XY-Plane?

In summary, the conversation discusses the possibility of finding a point in a set of points on the xy-plane such that its distance to every other point in the set is rational. The speaker suggests drawing a circle with a rational radius around any of the points as a possible solution. However, it is pointed out that this solution may not work for all points in the set. The speaker also mentions that they are unsure how to produce an answer and that the question may be more complex than it initially seems. The problem was originally given by a friend and has stumped many mathematicians at a conference.
  • #1
Dragonfall
1,030
4
Given n points n1,...,nk in the xy-plane, is it always possible to find a point p such that d(ni,p) is rational for 0<i<k+1?
 
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  • #2
What if one point is (x1,0), with x1 rational, and the other is (x2,0), with x2 irrational?
 
  • #3
Draw a line segment AB between (x1,0) and (x2,0) and a line L bisecting the line segment AB perpenticularly.
 
  • #4
um...either I'm misinterpreting the OP or the answer can be seen by drawing a circle radius p/q (where p/q is rational) around any of the points (with the point as the center).
 
  • #5
daveb said:
um...either I'm misinterpreting the OP or the answer can be seen by drawing a circle radius p/q (where p/q is rational) around any of the points (with the point as the center).

Perhaps you are misinterpreting. The question was whether, given a finite set of points, there exist a point p such that its distance to every point in the set is rational. Certainly every point on the circumference of your circle has rational distance (p/q) from the center, but what about the other points in the set?
 
  • #6
I've tried everything I know. I don't know how to produce an answer.
 
  • #7
Was this given to you as an assignment, or did you just think of it yourself? It may be a much deeper question than it appears.
 
  • #8
A friend sent this 'funny problem' that he got from a 'funny book'. I brought it to the Canadian undergraduate math conference last week and everyone was stumped.

EDIT: Oh and d is the Euclidean metric. No cheating.
 
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FAQ: Can a Rational Point Always be Found Amongst a Set of Points in the XY-Plane?

What is the problem with physics points?

The problem with physics points is that they are not a fundamental concept in physics. They were originally introduced as a convenient mathematical tool, but they do not have a physical meaning or representation in the real world. This can lead to confusion and limitations in understanding and predicting physical phenomena.

How do physics points differ from other concepts in physics?

Physics points differ from other concepts in physics because they do not have a physical manifestation. Concepts such as mass, energy, and force have tangible effects on the physical world, but physics points are simply mathematical constructs used to describe the position and movement of objects.

Can physics points be observed or measured?

No, physics points cannot be directly observed or measured. They are a theoretical concept used to simplify and model real-world situations, but they do not have a physical representation that can be observed or measured.

Are physics points still used in modern physics?

Yes, physics points are still used in some areas of modern physics, such as classical mechanics and Newtonian physics. However, they are not as widely used in more complex and advanced theories, such as quantum mechanics and general relativity.

What are the limitations of using physics points in scientific research?

One of the main limitations of using physics points in scientific research is that they do not accurately represent the complexities of the real world. As a result, they may not be able to fully explain or predict certain phenomena, and can lead to inaccuracies in scientific models and theories. It is important for scientists to recognize the limitations of physics points and use them appropriately in their research.

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