Here is the question: -
There is a circle with centre (pi,e). The radius of teh circle could be anything. How many rational points (points whose both coordinates are rational) exist on the circle?
I have been unable to come p with any line of thought as of now. Please contribute!
Are you asking “given any radius then is there a point on the circle centered at and radius is there a point on that circle that has rational coordinates?”
On the other hand, you were shown that given any point with rational coordinates say there is always a circle centered at containing : .
Please try to clarify the wording of this question.
What exactly are you asking? What does “to come p with any line of thought as of now” mean?
The first reply to the question makes the argument that we already know there's one rational point, reducing the entire procedure to asking is there any other such point?
Here lies my problem. How do we know there is atleast one rational point on the given circle?
The distance between the two points defines the RADIUS and thus the circle.
The center of the circle is
A random point (call it P) has coordinates
Is it possible for you to calculate the distance between the center of the circle and point P?
If you can calculate the distance, you have found the radius of the circle and thus proof that the point is on the circle.
An 8th grade student was able to do this:
Does that answer your question?
If you intended to ask: with the given irrational coordinates for the center of the circle, and some rational points on the circle, is it possible to have a RATIONAL RADIUS?
The answer is no.
And you can prove that along the same line of reasoning that the square root of 2 is not rational.