Proof That Pi = 2: Intuitively Wrong?

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SUMMARY

The forum discussion centers on the flawed proof that suggests Pi equals 2 through a series of semicircles and straight-line approximations. The argument is dissected, revealing that the convergence of arc lengths does not equate to the actual length of the limit path. Key contributors clarify that while the approximation process may seem valid, it ultimately leads to incorrect conclusions regarding the lengths involved, particularly in the context of limits and piecewise differentiable curves.

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  • Understanding of limits in calculus
  • Familiarity with piecewise differentiable functions
  • Knowledge of arc length calculations
  • Basic principles of geometry, particularly regarding triangles and semicircles
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  • Study the concept of limits in calculus, focusing on convergence and divergence
  • Explore the properties of piecewise differentiable functions and their implications on arc lengths
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  • #31
Of course that's not true, and I never said that.
 
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  • #32
adriank said:
Of course that's not true, and I never said that.

I edited my post.
 
  • #33
Right, and it's precisely because \gamma_n'(t) doesn't converge that the lengths don't converge to the length of the limit. But every length involved exists.

This is a classic example where some behaviour that is constant during some limiting process is not preserved once you take the actual limit. (In other words, the "length of a path function" is not "continuous".)
 
  • #34
I was simply arguing that the lengths do not converge at all, not just to that particular value.
 
  • #35
The derivatives don't converge, but the lengths certainly do, since they're the constant \sqrt2! They don't converge to 1, that's all.
 

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