Math Is Fun Forum

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Spot the error

To be able to regroup terms, you must first show what properity of the sequence?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Dross

Member

Registered: 2006-08-24

Posts: 325

Re: Spot the error

To be able to regroup terms, you must first show what properity of the sequence?

...convergence.

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Bad speling makes me [sic]

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Spot the error

Exactly.  Something I did not do.  It is entirely possible to show convergence as 1/(i^2+I) < 1/(i^2), and 1/(i^2) converges.  I just didn't do it.

Alright Dross, you're turn if you want it.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Dross

Member

Registered: 2006-08-24

Posts: 325

Re: Spot the error

I put it to you good people that a ladder, when pulled away from a wall by it's base at a constant speed, will fall infinitely fast.

Let a ladder of length L rest against a vertical wall. Let the horizontal distance from the base of the ladder to the base of the wall be given by x(t), and let the vertical distance between the base of the wall and the top of the ladder at time t be given by y(t).

Now, since the ladder, the distance x and the distance y form a right-angled triangle, we have by pythagoras that:

Now let the ladder be pulled away by it's base, such that the base moves away from the wall with a constant speed v. Differentiating the above with respect to t, we find that:

Now, as the ladder is pulled away at a constant speed, so x will approach L. Thus the numerator for the above expression for

will approach the non-zero term -Lv, whilst the denominator will approach zero. Thus

will approach infinity, becoming arbitrarily large as x approaches L.

Last edited by Dross (2006-09-21 11:41:10)

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Bad speling makes me [sic]

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Spot the error

what does the dot above y and x mean?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Spot the error

"Change in".

is distance, so

is change in distance, or speed.

As for proving you wrong, I don't suppose I'm allowed to just give a video of a ladder falling over as a counter-example?

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Dross

Member

Registered: 2006-08-24

Posts: 325

Re: Spot the error

what does the dot above y and x mean?

Sorry, I should have been more clear! Indeed Mathsyperson is right,

denotes change in

with respect to time, or

- the speed of

.

As for proving you wrong, I don't suppose I'm allowed to just give a video of a ladder falling over as a counter-example?

Of course you aren't - that would just take all the fun out of it!

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Bad speling makes me [sic]

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

Well, Dross’s problem is more to do with physics than mathematics, I’d say. The mistake is in assuming that the ladder always makes a right-angled triangle with the wall and floor. This is true when the ladder is stationary, but when you pull the base away from the wall, the top of the ladder is going to be pulled away as well. Then the Pythagorean relation between

and

no longer holds and hence the expression for

completely breaks down.

Dross

Member

Registered: 2006-08-24

Posts: 325

Re: Spot the error

Wow - I'd totally forgotten about this. I'm glad somebody got it!

I'd be interested in carrying on this thread, if you've got a false proof you can offer Jane?

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Bad speling makes me [sic]

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

Well then, here’s one I just made up. It should be quite easy, I hope.

Last edited by JaneFairfax (2007-03-01 12:41:08)

lightning

Real Member

Registered: 2007-02-26

Posts: 2,060

Re: Spot the error

the hence part?????

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

I’ll make it simpler. I’ll prove instead that

Now everyone should get it. This was precisely what my first fakse proof was all about; the differentiation was redundant, intended no more than to throw you off and make the error less obvious.

Last edited by JaneFairfax (2007-03-07 07:19:07)

lightning

Real Member

Registered: 2007-02-26

Posts: 2,060

Re: Spot the error

the -1=1 part ?????

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Devantè

Real Member

Registered: 2006-07-14

Posts: 6,400

Re: Spot the error

JaneFairfax, lightning hasn't learned about that yet.

Cool thread by the way.

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Re: Spot the error

I wouldn't know much about this, but I think it may have to do with the sin and cos functions being periodic and hence the x point at -1 is virtually the same as the x point at 1.

Nevermind.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

Nope, that’s not it. Look closely at the equations again.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Spot the error

It must have something to do with the square root having two possible values.

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It wanted to be normal.

lightning

Real Member

Registered: 2007-02-26

Posts: 2,060

Re: Spot the error

i know what a square root is just not a value or pie or x y stuff but in 3 months i'll be in first year of the high school

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

Exactly mathsyperson!

  is actually false.

The correct statement should be

.

Now your turn to post a false proof.

Last edited by JaneFairfax (2007-03-08 06:57:08)

lightning

Real Member

Registered: 2007-02-26

Posts: 2,060

Re: Spot the error

whats it gonna be math or math

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Spot the error

card(X) means the cardinality of a set X.

Prove that if card(X) ≤ card(Y) and card(Y) ≤ card(X), then card(X) = card(Y).

Proof: By definition, since card(X) ≤ card(Y), there exists a 1-1 function f:X->Y.  Similarly, there exists a 1-1 function g:Y->X.  But then every element x in X maps to a unique element y in Y.  Similarly every element y in Y maps to a unique element x in X.  Consequently X and Y must have the same number of elements and the functions f and g are 1-1 and onto.  Then by definition, card(X) = card(Y).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Spot the error

Oops, it looks like I missed Jane's confirmation. Which is a shame, because I have a really dandy false proof just waiting to be posted.

Unfortunately, it's going to have to wait because I can't find a flaw in Ricky's. The result of the proof is clearly correct, so it must just be that one of those steps is bending the rules of set theory.

As a random wild guess, I'll say that he needs to show that f = g[sup]-1[/sup], rather than just that two functions f and g exist.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Spot the error

That is correct Jane.  In general, if a mapping between two sets of the same cardinality is 1-1, then you are only guaranteed onto if their cardinality is finite.

Your turn Jane.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Spot the error

Woohoo, I got it right!

I don’t have a new false proof now. Since it was Mathsy’s turn last time, he can go next.