The rat, Sibyl, who lives beneath the trailer, wants to cross the room from one end to the other. Before she can cross the full distance, she must cross half the distance. Before she can cross the rest (half of the full distance) she must cross the next ¼. Before she can cover the last ¼, she must cover the next 1/8. And so on. Since covering any distance requires some time, it would take Sybil an infinite amount of time to cross all of the infinite distances. Therefore, Sibyl can never cross the room.
By the same argument, you could never cross the room either.
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9 comments:
Sir, this is a good problem. I'm going to think about it and try to come out with an explanation.
since Sybil is always crossing a fraction of the room, like in our ball bouncing sequence today or the o.99 sequence, it is like taking a percentage of the distance off everytime. The new distance he gets after crossing a section of the trailer becomes a whole and then another fraction is taken out of it. By this means, just like multiplying evey time by half, the distance will never meet exactly 0. For this reason, Sybil or any one who tries to use this method, will never actually cross the room completely, covering the whole distance. They would only get really close.
If we where asked to tell the limit of the sequence, we would state that:
DNE
Limtn= 0
n->oo
Susy, really good explanation. I guess it does make sense because, like you say, if you keep multiplying by half, we will always get close to the limit, but we would never get to it. It's a really interesting question sir.
SDC,
You say, "...anyone who tries to use this method." Sybil does not think to go half then a quarter etc.; she is just scampering along. But she still must cross those distances in that order whether she plans it that way or not.
Sybil's intentions notwithstanding, are you then concluding that she cannot cross the room?
Ahe cannot cross the room mathematically, but, we have to take into account her size... because, the distance from Sybill to the other end, will eventually become so small that she cannot move without reaching the other end.
I posted this on the 11b blog, but feel I should also post it here:
Well, a limit is, as you stated, a boundary. You can think of a boundary as a wall that the sequence would eventually either run into, or get really close to. The sybil question does have a limit, the limit is the size of the class. Even though sybil will never reach the end of the trailer, he will get infinitely close to the end. Think of it as a series with each term signifying the distance sybil has crossed.
(1/2), 1/4, 1/8, 1/16, 1/32, 1/64......etc
Now in order to find whats the limit of sybil's movement then we need to ad all the terms in the geometric series. This seems hard to do though because there are an infinte amount of terms. Lets look back at the sum of a geometric series formula:
S= t1 ((r^n-1)/(r-1))
In this case, T1 = 1/2 and r=1/2, what we need to figure out is what n is, and in this case its simple. Its infinity! (1/2)^infitiy = 0, try to think about why this is true, you may want to refer back to jaime's post (right above this one).Now we can solve for what the sum would be.
S=(1/2)((1/2)^infinity-1)/((1/2)-1))
S= (1/2)(0-1)/((-1/2)
S= 1
In our case the terms of the series signified the distance of the trailer: 1/2 meant 1/2 of the trailer, then 1/4 of the trailer etc, so seeing as out answer is 1, the limit of Sybil's movement is 1 trailer.
Rumidog,
yes, that is my initial statement. But, thinking about it again, obviously to cross another half of the room, you have to cross the next 1/4 of the distance. That way you cover half of the distance you have left to go. Mathematically, if we where to count how many sections he has to cover before reaching the whole trailer,(1 as anto says), It will take an infinite amount of time, we will not be able to finish counting and include all of them. But, in the "real" life, Sybil will eventually cross the room. It's like when I run half a basketball court, first I had to cross the smallest ever fraction of the court to get to the first 1/4 of it and then get to half-court. While crossing that half-court, and, afterwards, the whole court, I crossed an infinite amount of FRACTIONS of the court(all the fractions from 0-1 you can think of), to finally complete 1/1, or the whole court. But, I did cross it, as Sybil eventually will. That's unless he gets distracted with a bag of skittles in Mr.A's bag.**
Well if you take Sybil's problem to literally, then sybil can't cross the room because he won't be able to move. Since before he moves 1/2 h needs to move /4 and before he moves 1/4 he needs to move 1/8 etc etc. So you would need to cover an infinite amount of distances in order to move a required distance, this is impossible and therefore movement is an illusion. The same logic can also be applied to time...
What limits do is end all seemingly logical arguments like the one Mr.A gave to you. Limits prove that sybil does in fact cross the room. Although I understand how most of you feel, the explanation is just not good enough, and I also feel the same way.
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