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Posted 01 December 2010 - 10:30 PM
Posted 01 December 2010 - 10:47 PM
Quote:
Original post by Eelco
How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.
Posted 01 December 2010 - 10:51 PM
Quote:
Original post by JoeCooper Quote:
Original post by Eelco
How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.
They're different because it doesn't reduce the perimeter, and the perimeter is what we're measuring. No reducing is done. The figure's perimeter isn't changed at all. You might as well not change the figure.
The area is, and if you measure that, you'll find it most certainly does give us Pi.
Posted 01 December 2010 - 11:20 PM
Posted 01 December 2010 - 11:37 PM
Posted 02 December 2010 - 03:39 AM
Quote:
Original post by JoeCooper
We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.
This approach is only being questioned because it looks like the reduction approach. Since the perimeter is not reduced, it is not the reduction approach, and is therefore one of an infinite number of totally random activities that also don't do anything relevant or useful, like watching Stargate. Your arguments are equally valid in support of me watching Stargate, counting the minutes, adding a break to refill my coffee, dividing it by 9 and calling it Pi; I can't tell that this is the wrong approach unless I have some idea of what I'm looking for.
Posted 02 December 2010 - 04:04 AM
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In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'
Quote:
failed to enumerate what the characteristics of such a procedure should be
Posted 02 December 2010 - 04:32 AM
Posted 02 December 2010 - 05:28 AM
Quote:
Original post by JoeCooper
Can you answer why my Stargate method for producing Pi doesn't yield Pi without falling back on the fact that it's not designed to find Pi?
Quote:Why would I?
Can you at least show why the perimeter-of-a-square method should work?
Quote:
Quote:
In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'
That's not my answer. My answer is that I can't use it for what Pi is used for. I can't take Pi=4 and get the object's area without, at some point, winding up with the 3.14etc. figure somewhere else in the equation, at which point we're only renaming things.
Quote:
It's a useless activity. You earlier enumerated the characteristics and only succeeded in suggesting that an intentionally broken procedure should work.
Posted 02 December 2010 - 05:36 AM
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Original post by JoeCooper Quote:
the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point
Maybe that's not the case. Again, if you just zoom in, it's stair-steps, it's not a circle. The shape on your screen isn't really a circle either, given that it's also painted onto such a grid.
Quote:
We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.
Quote:
Original post by BlueSalamander
"If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too."
Posted 02 December 2010 - 05:36 AM
Quote:
Original post by BlueSalamander
When you switch from a finite number of iterations to an infinite number of iterations, the square with cut corners becomes a perfect circle and the perimeter changes from 4 to pi without warning.
Posted 02 December 2010 - 06:18 AM
Quote:
Original post by Way WalkerThat is, why the procedure that yields 3.141... is correct and the one that yields 4 isn't. The only explanation you have is that you knew beforehand that 3.141... is the correct answer, but by what procedure did you come up with that number? How did you decide that that procedure yielded the correct answer?
Posted 02 December 2010 - 06:35 AM
Quote:
Original post by Hodgman
Is 1/infinity * infinity equal to 1?
Posted 02 December 2010 - 06:44 AM
Posted 02 December 2010 - 06:48 AM
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Original post by Fenrisulvur Quote:
Original post by Hodgman
Is 1/infinity * infinity equal to 1?
God no.
Posted 02 December 2010 - 06:55 AM
Quote:
Original post by Mantear
I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).
Posted 02 December 2010 - 08:05 AM
Quote:
Original post by Fenrisulvur Quote:
Original post by Hodgman
Is 1/infinity * infinity equal to 1?
God no.
Posted 02 December 2010 - 08:08 AM
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you speak it like a philosophy student, and (seem to) get frustrated when others remain unconvinced of the ideas you attempt to convey (kind-of like a philosophy student >_> )
Posted 02 December 2010 - 11:07 AM
Quote:I'm assuming that "1/inf" is the smallest value that is still greater than zero.
Original post by Fenrisulvur Quote:
Original post by Mantear
I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).
Eh, I maintain that the inline division operator is an obscure and ambiguous notation.
Anyway, that's indeterminate, and it's not clear just skimming Hodgman's post where he derived the form from.
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