Math Fun

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Re: Math Fun

Postby Abstract » Thu Jan 03, 2013 3:29 pm

tralix the answer to you problem is
AE FHIK LMNTVWXY Z


so is know one going to chance another guess at my problem?
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Re: Math Fun

Postby Tralix » Fri Jan 04, 2013 2:55 am

Abstract wrote:tralix the answer to you problem is
AE FHIK LMNTVWXY Z


so is know one going to chance another guess at my problem?



Correct. But even more importantly is anyone going to resolve Zenos paradox?

I thought I already solved your problem with the prime non prime thing, if that's not the case then what more to it is there, the primes denote where the non primes appear and the non primes are hence there only as the solution? Is that wrong?

The next number is 2 if we ignore the primes, then the next number is 4 and then 8 if the primes are in the right place? Your problem is annoying if that's not the solution because it has millions of solutions. Each prime and where it appears shows where the non primes will appear and what value they have :P
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Re: Math Fun

Postby Abstract » Fri Jan 04, 2013 4:35 am

no the next number is 7 now whats the next?
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Re: Math Fun

Postby Tralix » Fri Jan 04, 2013 4:46 am

Abstract wrote:no the next number is 7 now whats the next?


The next number being 7 only means that the numbers following will be non prime according to it and how many primes do or dont follow. I don't get what you are asking?

There's no answer I can give accept that if there are several primes then the numbers that follow are intimately related to them. 7,2,8 and so on.
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Re: Math Fun

Postby Abstract » Fri Jan 04, 2013 4:52 am

here is the answer then
ignoring the first 4 primes you take the prime number and sum its digits and sum those if multiple digits till you get to one digit and that gives you the number...
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Re: Math Fun

Postby Tralix » Fri Jan 04, 2013 4:54 am

Abstract wrote:here is the answer then
ignoring the first 4 primes you take the prime number and sum its digits and sum those if multiple digits till you get to one digit and that gives you the number...


Yeah I am rubbish at this shit. :lol:

I'm much better at lateral puzzles. :P

Nice puzzle though, my only saving grace is no one else got it either. I do officially suck, hats off to Abstract. :)
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Re: Math Fun

Postby Carleas » Fri Jan 11, 2013 12:07 am

Tralix:
TVWXZ


EDIT: totally missed the other two pages of this thread! :oops:
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Re: Math Fun

Postby Silhouette » Fri Jan 11, 2013 12:36 am

Carleas wrote:Tralix:
TVWXZ

Wrong :P Well not wrong, but you missed one.

Hint:
It's very unphilosophical of you to not ask "why".

Abstract, I may have got yours if it hadn't:
missed out the single digit primes - though I can see why you did.

Tralix, I don't really understand your question. You just want someone to solve Zeno's Paradox mathematically?
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Re: Math Fun

Postby Tralix » Tue Jan 15, 2013 5:15 am

Carleas wrote:Tralix:
TVWXZ


EDIT: totally missed the other two pages of this thread! :oops:


Right but forgive me if I don't send you a cookie for that one. :P
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Re: Math Fun

Postby Tralix » Tue Jan 15, 2013 5:18 am

Silhouette wrote:
Carleas wrote:Tralix:
TVWXZ

Wrong :P Well not wrong, but you missed one.

Hint:
It's very unphilosophical of you to not ask "why".

Abstract, I may have got yours if it hadn't:
missed out the single digit primes - though I can see why you did.

Tralix, I don't really understand your question. You just want someone to solve Zeno's Paradox mathematically?


Is it so hard you can't solve the arrow and turtle problem? Fire an arrow never hits its target.

I'll give you a clue calculus!

Trust me this is not a hard problem, people solved this even in Aristotle's time. :)
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Re: Math Fun

Postby Abstract » Wed Jan 16, 2013 8:10 pm

Tralix wrote:Solve Zeno's paradox about firing an arrow and never being able to hit a moving target by any means necessary and show that it is in fact not a paradox. You may use any form of maths as long as it is logical and constrained by current set, number theories.

Clue: you don't need calculus, in fact all you need is periodicity over time, but it might be easier if you use it.

Could you explain the problem completley.... so i don't have to look it up... then maybe i can solve it...
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Re: Math Fun

Postby Tralix » Thu Jan 17, 2013 2:52 am

Abstract wrote:
Tralix wrote:Solve Zeno's paradox about firing an arrow and never being able to hit a moving target by any means necessary and show that it is in fact not a paradox. You may use any form of maths as long as it is logical and constrained by current set, number theories.

Clue: you don't need calculus, in fact all you need is periodicity over time, but it might be easier if you use it.

Could you explain the problem completley.... so i don't have to look it up... then maybe i can solve it...


It's not that hard, Xeno claims a man fires an arrow at a tortoise, but before the arrow arrives the tortoise has moved on

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.[11] – as recounted by Aristotle, Physics VI:9, 239b5

In the arrow paradox (also known as the fletcher's paradox), Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.[12] It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.[13]


1. Dichotomy paradox: Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled.


Also.

It's basically a question that tries to resolve the problem of motion and time, if the arrow is aimed at the tortoise and time and space are instants, then according to logic the arrow should never hit it's target, because as it travels the tortoise is in motion also, and the arrow never meets the point where the tortoise was.
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Re: Math Fun

Postby Abstract » Thu Jan 17, 2013 3:13 pm

zeno stuff

you have the amount of time given and then you divide that into an infinite amount of instances you can then claim that each instant is of value zero but to then return and determine the summation of instances you would be multiplying zero times infinity which is indeterminate... the best claim is then that in reality you are dividing the time given by infinite which is not zero finite/inf = .000000000000000000...0000000000000001 so the actual instants have a size. in otherwords infinitesimally small does not mean zero in size...
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Re: Math Fun

Postby Silhouette » Thu Jan 17, 2013 5:35 pm

Tralix wrote:
1. Dichotomy paradox: Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled.

One of the conditions of this dichotomy paradox necessarily entails that the arrow will never reach the tortoise:
in dealing solely with distances d/(2^n), and as the opening words explicitly state, one is only contemplating the infinite divisibility of distance travelled BEFORE the entire distance d is travelled - "BEFORE an object can travel a given distance d".
So if one is concerned about the arrow actually reaching the tortoise, one ought to consider the whole picture and deal also with once an object HAS travelled a given distance d, and even AFTER an object has travelled a distance d.

It doesn't seem to matter whether the tortoise moves at all, or even if it moves towards the arrow, because the arrow still has to pass through an infinite number of fractions of the distance BEFORE reaching the tortoise, whatever the tortoise is doing. And then on top of that, "the arrow/fletcher's paradox" says the arrow isn't even moving...

Are you looking for a solution to the paradox such that the arrow DOES reach the tortoise? Whether we use calculus at all is irrelevant if one of the assumptions is that we're only dealing with what happens BEFORE distance d is travelled.
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Re: Math Fun

Postby Tralix » Thu Jan 17, 2013 6:40 pm

Abstract wrote:zeno stuff

you have the amount of time given and then you divide that into an infinite amount of instances you can then claim that each instant is of value zero but to then return and determine the summation of instances you would be multiplying zero times infinity which is indeterminate... the best claim is then that in reality you are dividing the time given by infinite which is not zero finite/inf = .000000000000000000...0000000000000001 so the actual instants have a size. in otherwords infinitesimally small does not mean zero in size...


You could of also said use a Taylor Maclaurin series or an integral with natural logs such as the half life equation, but yes correct.
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Re: Math Fun

Postby Tralix » Thu Jan 17, 2013 6:41 pm

Silhouette wrote:
Tralix wrote:
1. Dichotomy paradox: Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled.

One of the conditions of this dichotomy paradox necessarily entails that the arrow will never reach the tortoise:
in dealing solely with distances d/(2^n), and as the opening words explicitly state, one is only contemplating the infinite divisibility of distance travelled BEFORE the entire distance d is travelled - "BEFORE an object can travel a given distance d".
So if one is concerned about the arrow actually reaching the tortoise, one ought to consider the whole picture and deal also with once an object HAS travelled a given distance d, and even AFTER an object has travelled a distance d.

It doesn't seem to matter whether the tortoise moves at all, or even if it moves towards the arrow, because the arrow still has to pass through an infinite number of fractions of the distance BEFORE reaching the tortoise, whatever the tortoise is doing. And then on top of that, "the arrow/fletcher's paradox" says the arrow isn't even moving...

Are you looking for a solution to the paradox such that the arrow DOES reach the tortoise? Whether we use calculus at all is irrelevant if one of the assumptions is that we're only dealing with what happens BEFORE distance d is travelled.


Ok just cutting and pasting what is written on the wiki is not an answer. ;)

It's technically correct though. ;)
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Re: Math Fun

Postby Silhouette » Thu Jan 17, 2013 6:49 pm

Tralix wrote:Ok just cutting and pasting what is written on the wiki is not an answer. ;)

It's technically correct though. ;)

Er... I didn't?

And... ok - it doesn't feel like I solved anything though, I just picked out and criticised the main assumption.

You also said Abstract was right for saying something else, so what was the answer you were looking for?
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Re: Math Fun

Postby Tralix » Thu Jan 17, 2013 7:01 pm

Silhouette wrote:
Tralix wrote:Ok just cutting and pasting what is written on the wiki is not an answer. ;)

It's technically correct though. ;)

Er... I didn't?

And... ok - it doesn't feel like I solved anything though, I just picked out and criticised the main assumption.

You also said Abstract was right for saying something else, so what was the answer you were looking for?


The answer both I and Abstract gave, I am not sure what you are trying to say here, that the laws of calculus are wrong or that Zeno is wrong or both.

You are technically correct in what you say, but that is not the answer.

It probably helps to visualise this in terms of a bouncing ball, we know eventually it comes to rest, common sense tells us this, and that time and distance are not exactly halving. Hence Zeno is basically making a logical error of assuming time and distance are exactly halving in his paradox.
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Re: Math Fun

Postby Silhouette » Thu Jan 17, 2013 8:16 pm

Tralix wrote:Zeno is basically making a logical error of assuming time and distance are exactly halving in his paradox.

Ok so that's the answer?

It doesn't seem very satisfying since the point d/2 IS d/2, simply by virtue of the conditions of the paradox: premises are not subject to logical error within the paradox itself. Likewise for t/2 if the paradox is being applied to time rather than distance. Of course in reality, distance d/2 is not covered in t/2 units of time, due to various things such as air resistance, imperfectly flat ground for the tortoise to walk along and thus variable velocity and/or speed, and even concepts of relativity affecting spacetime differently for each object since they are each travelling at relatively different speeds.

But this paradox is a theoretical one, not a real one - as any physical re-enactment of the paradox would easily solve it.

I'm not sure if we're tabbing anymore on this one, since you seem to have found a satisfying answer and you have even said what you were looking for untabbed? But here is another technically correct solution:

At a small enough level of magnification, objects are seen to never actually be touching. Two surfaces are always repelled by the tiny repulsive magnetic charges exerted at a sub-atomic level, which gives a particular distance that cannot be closed between two objects. At some point during the d/(2^n) series, the remaining distance that the arrow has left to travel will equal this threshold, beyond which it cannot travel further anyway - at which point it can be said to be touching as much as it ever would, without having fully travelled distance d.

However, if d is simply redefined to take into account this fact, the paradox is restored.
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Re: Math Fun

Postby James S Saint » Thu Jan 17, 2013 8:41 pm

The answer to ALL of the Zeno paradoxes is the understanding of simple calculus.
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Re: Math Fun

Postby Silhouette » Thu Jan 17, 2013 9:52 pm

I think you mean an acceptance of simple calculus.

You yourself rightly deny that infinity is defined. So for limits to converge towards a definite answer, this is like saying something finite can result from something infinite.
I forget the outcome of this old thread about whether 0.9(recurring) equalled 1.
A convergent series of 9/10+9/100+9/1000+.... would definitely tend towards 1, but for it to equal 1 requires an intuitive leap rather than a strict and rigorous approach taken forever and ever just to never quite get there.

This intuitive leap is necessary for the acceptance of calculus, which is more of a mindset adopted for things like Zeno's paradox to no longer seem paradoxical, than a solution for it.
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Re: Math Fun

Postby Tralix » Thu Jan 17, 2013 10:35 pm

Silhouette wrote:I think you mean an acceptance of simple calculus.

You yourself rightly deny that infinity is defined. So for limits to converge towards a definite answer, this is like saying something finite can result from something infinite.
I forget the outcome of this old thread about whether 0.9(recurring) equalled 1.
A convergent series of 9/10+9/100+9/1000+.... would definitely tend towards 1, but for it to equal 1 requires an intuitive leap rather than a strict and rigorous approach taken forever and ever just to never quite get there.

This intuitive leap is necessary for the acceptance of calculus, which is more of a mindset adopted for things like Zeno's paradox to no longer seem paradoxical, than a solution for it.


Only because you don't understand the underlying laws of calculus.

Limits are asymptotic they neither denote anything "real" nor are they per se fictions. They are what values can approach but never reach.

http://en.wikipedia.org/wiki/Taylor_series

The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.[1] Liu Hui independently employed a similar method a few centuries later.[2]

In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama.[3][4] Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor,[5] after whom the series are now named.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.


Image

Second-order Taylor series approximation (in gray) of a function f(x,y) = e^x\log{(1+y)} around origin.

Image
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Re: Math Fun

Postby Silhouette » Thu Jan 17, 2013 11:10 pm

Tralix wrote:Only because you don't understand the underlying laws of calculus.

Limits are asymptotic they neither denote anything "real" nor are they per se fictions. They are what values can approach but never reach.

*Don't accept the underlying laws of calculus.

Nothing you've said gets rid of this problem of infinity in calculus. Asymptotes are just an example of what I was talking about that you need to make an intuitive leap to reach. And then you say they're approached but never reached, which is just what I said about convergent series never quite getting there if you take a strict and rigorous approach to them forever and ever...

You can't repeat my argument in order to say it's wrong. Quoting the wiki-history of the Taylor series doesn't prove anything either.
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Re: Math Fun

Postby Tralix » Thu Jan 17, 2013 11:17 pm

Silhouette wrote:
Tralix wrote:Only because you don't understand the underlying laws of calculus.

Limits are asymptotic they neither denote anything "real" nor are they per se fictions. They are what values can approach but never reach.

*Don't accept the underlying laws of calculus.

Nothing you've said gets rid of this problem of infinity in calculus. Asymptotes are just an example of what I was talking about that you need to make an intuitive leap to reach. And then you say they're approached but never reached, which is just what I said about convergent series never quite getting there if you take a strict and rigorous approach to them forever and ever...

You can't repeat my argument in order to say it's wrong. Quoting the wiki-history of the Taylor series doesn't prove anything either.


It isn't a problem an axiom is not an issue unless you make it one by semantics and then everyone will just think you are a crank.

The problem is solved and successfully disputed according to all science and maths. The fact that you don't understand it, is possibly interesting but not a reason to really discuss your ignorance with you.
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Re: Math Fun

Postby Silhouette » Thu Jan 17, 2013 11:35 pm

Tralix wrote:It isn't a problem an axiom is not an issue unless you make it one by semantics and then everyone will just think you are a crank.

The problem is solved and successfully disputed according to all science and maths. The fact that you don't understand it, is possibly interesting but not a reason to really discuss your ignorance with you.

I am a crank.
This is not the thread to discuss "my" ignorance, no, whether with or without appeals to authority rather than actual arguments. I'll drop it on the condition that you come up with more lovely puzzles :D I apologise that I know of none.
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