Proof That Pi = 2: Intuitively Wrong?

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The discussion centers around a flawed proof suggesting that pi equals 2 through a series of semicircles and partitions. Participants argue that the proof misinterprets the convergence of arc lengths, emphasizing that the limit of a sequence of paths does not necessarily equal the length of the limit path. The conversation highlights the distinction between piecewise linear approximations and the actual lengths of curves, particularly in the context of limits and continuity. A key point raised is that even as paths become infinitely refined, their total lengths can remain distinct from the lengths of their straight-line counterparts. Ultimately, the discussion underscores the importance of rigor in mathematical proofs and the subtleties of limits in geometry.
  • #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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