Irrational circles about the orgin

In summary, the OP was wondering if a circle with an irrational radius could pass through a rational point. The answer is if the irrational radius is the square root of the sum of two rational squares then of course it can. However, if the radius is a non-algebraic number then the circle might not pass through a rational point.
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ramsey2879
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I recall a post previously where the Op was wondering if any circle about the orgin having an irrational radius could pass through a rational point. The answer then was if the irrational radius was the square root of the sum of two rational squares then of course.

Now I am wondering what if the radius was a non-algebraic number? Could that circle pass through a rational point?
 
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Hint: Suppose that r is transcendental and is the radius of a circle centered at the origin. Show that the existence of a rational point on this circle implies that r is algebraic.
 
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Petek said:
Hint: Suppose that r is transcendental and is the radius of a circle centered at the origin. Show that the existence of a rational point on this circle implies that r is algebraic.
I was thinking that way but then I was wondering if transcendental numbers actually exist since it seems strange to me that one could travel a full circle and not come upon a rational point. Sorry but I have other things that I am working on and don't have time to look up the theory of transcendental numbers for now.
 
  • #4
ramsey2879 said:
I was thinking that way but then I was wondering if transcendental numbers actually exist since it seems strange to me that one could travel a full circle and not come upon a rational point. Sorry but I have other things that I am working on and don't have time to look up the theory of transcendental numbers for now.

I'm honestly confused at this point. You used the term "non-algebraic number" in your original post. Non-algebraic numbers are exactly the same as transcendental numbers (if we're confining this discussion to real numbers). If I changed my hint to

Hint: Suppose that r is non-algebraic and is the radius of a circle centered at the origin. Show that the existence of a rational point on this circle implies that r is algebraic.

would that make more sense?
 
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Yes I can see that x^2 + y^2 = r^2 would make r algebraic, but my real question was how can one move completely around a circle without passing over a rational point. Didn't that call into question the existence of non-algebraic values as real numbers?
PS, I took a look at some of the information of non-algebraic (transcendental) numbers and guess that they do exist. Moreover I noted that there are far more algebraic irrational numbers and transcendental numbers, so I guess that it must be true that such a circle has no rational point lying thereon.
 
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FAQ: Irrational circles about the orgin

What is an irrational circle about the origin?

An irrational circle about the origin is a geometric shape formed by plotting points on a graph that satisfy the equation x2 + y2 = r2, where r is an irrational number. This results in a circle that cannot be perfectly measured or divided into equal parts.

How is an irrational circle different from a regular circle?

An irrational circle is different from a regular circle because the radius is an irrational number, meaning it cannot be expressed as a simple fraction. This results in a circle with an infinite number of points and a circumference that cannot be accurately measured.

Why are irrational circles important in mathematics?

Irrational circles have been studied and used in mathematics for centuries, as they represent a fundamental concept of irrational numbers. They are also used in various mathematical applications, such as in geometry and trigonometry, to demonstrate complex mathematical concepts and relationships.

Can irrational circles be seen in real life?

No, irrational circles cannot be seen in real life as they are a mathematical concept. However, they can be represented and visualized using graphing tools and can help to explain real-world phenomena, such as the curvature of the Earth's surface.

What are some examples of irrational circles?

Some examples of irrational circles include the unit circle, which has a radius of 1 and an irrational circumference of 2π, and the Apollonian gasket, which is a fractal formed by repeatedly inscribing circles within circles using irrational ratios.

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