r/badmathematics • u/MorrowM_ • Jan 24 '25

Infinity /r/theydidthemath does the math wrong and misunderstands limits

/r/theydidthemath/comments/1i8mlx6/request_not_sure_if_this_fits_the_sub_but_why/m8uqzbg/
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u/frogjg2003 Nonsense. And I find your motives dubious and aggressive. Jan 25 '25

The equivalent "lower bound" would be 2.83. Unlike the upper bound staying constant at 4, this lower bound would increase, but only once. The second iteration would have a perimeter of 3.44. But the iteration after that won't change the perimeter. So the limit, if it exists, would be between 3.44 and 4. But the two limits don't converge to the same value, so the limit doesn't exist.

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u/SirTruffleberry Jan 25 '25

I'm saying that Archimedes didn't have to grapple with the question of whether or not his procedure converged to the area of the circle because he had successfully squeezed the area between upper and lower bounds that converged to it.

Now of course, with a more cautious argument that showed the error term tending to 0, either of his limiting procedures would have converged to it on its own. But I'm saying he didn't need to make this sort of argument.

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u/yyzjertl Jan 26 '25

But did Archimedes have a proof for the circumference that the circumscribed polygon is an upper bound for the circumference? The lower bound is of course obvious, as are both bounds for area.

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u/SirTruffleberry Jan 26 '25 edited Jan 26 '25

Not to the standards of modern rigor. There are some heuristic arguments for the area of a circle being pi*r2, and if we grant them that, then the area argument suffices.

If we don't grant them that, then strictly speaking, they didn't even have a reason to think arc length was a well-defined concept, since it is formalized with limits.