Curvature of a circle approaches zero as radius goes to infinity

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Discussion Overview

The discussion revolves around the mathematical expression of the concept that the curvature of a circle approaches zero as its radius approaches infinity. Participants explore the implications of this idea, including the relationship between curvature and radius, and the mathematical limits involved.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the curvature of a circle decreases as the radius increases, noting that curvature is the reciprocal of the radius.
  • Another participant proposes using the limit expression lim 1/r as r approaches infinity to express the relationship mathematically.
  • A later reply confirms the limit expression, stating that lim_{r \rightarrow ∞} (1/r) = 0 is an appropriate formulation.
  • One participant raises a question about the implications of an infinite radius on the circumference of the circle and the value of pi, suggesting that it leads to an indeterminate form.
  • Another participant asserts that pi remains constant regardless of the radius and discusses the concept of indeterminate forms in relation to limits.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical expression involving limits, but there is a disagreement regarding the implications of an infinite radius on the value of pi and the nature of indeterminate forms.

Contextual Notes

Participants express uncertainty about how to handle the concept of infinity in mathematical expressions, particularly in relation to the circumference of a circle and the value of pi.

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Hello,

this isn't a homework problem, so I'm hoping it's okay to post here.

I would like to know the correct way to mathematically express the idea in my title. It is intuitively obvious that as the radius of a circle increases, it's curvature decreases.

I looked it up and found that the curvature of a circle is equal to the reciprocal of it's radius. Certain assumptions are often made when looking at lenses, i.e the wave fronts reaching the lens are parallel, or have 0 curvature - In other words, the object distance is infinitely far away.

But, 1/∞ ≠ 0

So how do I express it properly?

In words, I think it goes something like this - As the radius tends towards infinity, the curvature of the circle tends towards zero.
 
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Wouldn't you just use the lim 1/r expressions with r-> infinity to express it?
 
jedishrfu said:
Wouldn't you just use the lim 1/r expressions with r-> infinity to express it?

That would be my guess but I'm unsure of how to formulate that.

lim_{r \rightarrow ∞} \frac{1}{r} = 0

Like that?
 
If you imagine a circle with infinite radius, then its circumference is also infinite.
Then what would be the value of pi be? Infinite divided by infinite. Can you say what
it is?
I think the real projective line may be a picture of this kind of "circle":
http://en.wikipedia.org/wiki/Real_projective_line
 
7777777 said:
If you imagine a circle with infinite radius, then its circumference is also infinite.
Then what would be the value of pi be?
The same as always. ##\pi## is a constant (its value never changes).
7777777 said:
Infinite divided by infinite. Can you say what
it is?
No. There are several indeterminate forms, including [∞/∞], [0/0], [∞ - ∞], and a few others. These are indeterminate, because you can't determine a value for them.

They usually come up when we are evaluating limits of functions.
7777777 said:
I think the real projective line may be a picture of this kind of "circle":
http://en.wikipedia.org/wiki/Real_projective_line
 

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