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Originally posted by mikelomNice one, we can even assume that the stone comes down 8/9ths of the remaining way each time, and it still definitely hits the ground!
I think you're right, but didn't explain it in a more simplistic way.
A picturing point of view is for repetitive halfs which could be endless, and yet we know there is an end.
Consider the analogy of throwing a stone into the air. We know to come down, it must come half way as high as it went. It must come half way again, and then half of that distanc ...[text shortened]... fore, this infinite half way argument is flawed, same as 0.9 recurring doesn't = 1.
-m. 😉
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Originally posted by FabianFnasYeah, it's Zeno all over again, and a tenth again, and a hundredth again, and a thou...
This 'paradox' is known from the ancient greek times. At that time, without knowledge of infinitesimality, it was really a paradox. Nowadays it isn't.
Those who don't know about infinity cannot understand why 0.99999... is exactly equal to 1.00000...
By the way, welcome back.
Richard
You're, of course, right, Joe. Sorry about that. n+1, not n.
A modern variant of the dilemma..
Ask a taxi driver how far is the distance to to four blocks east and three north is, and he'll say seven. What if he's allowed a route closer to how-the-bird flies? even if you zigzag 400 times east and 300 times north, a hundredth of a block at a time, he'll still say seven. Four million tiny increments east and three million north, and the route is still seven. Arranging those terms so that the taxi will stay as close to the direct possible route as possible has the route approach the hypotenuse of a triangle with other sides three and four.
And thus.. with n the number of turns.. as n approaches infinity, lim 7 -> 5. ..?
Obviously the reason is not the same, but it goes well with fixing the never-ending problem of having high-school students suggest that sqrt(a+b) = sqrt(a) + sqrt(b), just like sqrt(ab) = sqrt(a) sqrt(b).
Some have learned to avoid that from having them find the mistake in the "proof" that 7 = 5;
7 = 4 + 3 = sqrt (16) + sqrt (9) = sqrt (16 + 9) = sqrt(25) = 5.
Originally posted by talzamirSo which is the "true" distance. 7 or 5?
You're, of course, right, Joe. Sorry about that. n+1, not n.
A modern variant of the dilemma..
Ask a taxi driver how far is the distance to to four blocks east and three north is, and he'll say seven. What if he's allowed a route closer to how-the-bird flies? even if you zigzag 400 times east and 300 times north, a hundredth of a block at a time, he'l ...[text shortened]... he "proof" that 7 = 5;
7 = 4 + 3 = sqrt (16) + sqrt (9) = sqrt (16 + 9) = sqrt(25) = 5.
Originally posted by talzamirDoes the limit really apply...I mean its only a "limit" in the visual sense( not in a calculus sense), regardless of "n" in this scenario the distance remains constant, at 7.
You're, of course, right, Joe. Sorry about that. n+1, not n.
A modern variant of the dilemma..
Ask a taxi driver how far is the distance to to four blocks east and three north is, and he'll say seven. What if he's allowed a route closer to how-the-bird flies? even if you zigzag 400 times east and 300 times north, a hundredth of a block at a time, he'l ...[text shortened]... he "proof" that 7 = 5;
7 = 4 + 3 = sqrt (16) + sqrt (9) = sqrt (16 + 9) = sqrt(25) = 5.
Originally posted by joe shmoA "limit" that differs the visual sense from the calculus sense is always wrong. When it does it is an origin of a paradox.
Does the limit really apply...I mean its only a "limit" in the visual sense( not in a calculus sense), regardless of "n" in this scenario the distance remains constant, at 7.
The Zeno paradox is an example of this. Visually it seems right, but calculus shows what is really happening.
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Originally posted by FabianFnasI think it is extremely hard to prove pythagoras' theorem via calculus, which is the same as trying to prove that the lower limit of the taxicab's journey is 5 units (without invoking pythagoras).
A "limit" that differs the visual sense from the calculus sense is always wrong. When it does it is an origin of a paradox.
The Zeno paradox is an example of this. Visually it seems right, but calculus shows what is really happening.
The vastly easiest approach is to prove pythagoras using some other technique (e.g a dissection) and then use pythagoras to derive the limit.
Here is one attempt to prove pythagoras with calculus, good luck following it!
http://www.scribd.com/doc/30552/A-Calculus-Proof-of-the-Pythagorean-Theorem
Originally posted by iamatigerI think not. As I recall I was presented the proof of Theorem of Pythagoras in an understandable way in the school before the age of 16.
I think it is extremely hard to prove pythagoras' theorem via calculus
To understand that by only going west and north n times it is not possible that the distance between two points is different regardless of n. When you know the concept of infinity it is obvious. Else it is impossible to understand.
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Originally posted by FabianFnasBut was that a proof using calculus? That is the salient point. There are many relatively simple ways of proving Pythagoras. Using calculus is probably not one of them.
I think not. As I recall I was presented the proof of Theorem of Pythagoras in an understandable way in the school before the age of 16.
Richard
Originally posted by Shallow BlueOf course it was. It have been repeated many times thereafter, the last time in University calculus. Along with numerous other calculus proofs.
But was that a proof using calculus? That is the salient point. There are many relatively simple ways of proving Pythagoras. Using calculus is probably not one of them.
Richard
Why? Don't you believe in the pythagoras theorem?
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Originally posted by FabianFnasInteresting
Of course it was. It have been repeated many times thereafter, the last time in University calculus. Along with numerous other calculus proofs.
Why? Don't you believe in the pythagoras theorem?
Could you show us this proof or give a link to it?
Off the top of my head I can only think of using the arc length formula to prove pythagoras using calculus. This is one extremely bad flaw in that my proof for the arc length formula assumes pythagoras so it is a circular argument.
I am wondering though what is elegant and satisfying about proving pythagoras using a more complicated way? Mathematical proofs are at their best when they are concise and easy to understand.
