# Thread: Problem on Mathematics - continued

1. ## Re: Problem on Mathematics - continued

Originally Posted by vince123123
I find this difficult to understand because we know in real life that the faster runner can always overtake the slower. In the example above, if Achilles covers 100 feet in the time the tortoise takes to cover 10 feet, he will cover another 100 feet when the tortise covers another 10 feet. This makes Achilles 200 feet from his starting point, or 100 feet from the tortoise's starting point, and the tortoise is only 20 feet from the tortoise's starting point.
Its called Zeno's paradox and here is an explanation of its solution: From http://www.mathacademy.com/pr/prime/articles/zeno_tort/

Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.
What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.
Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?" If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn't be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm.
Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno's Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words,
1= 1/2 + 1/4 + 1/8 + 1/16 ...

At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn't – in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn't so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)
Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.
And poor old Achilles would have won his race.

2. ## Re: Problem on Mathematics - continued

I don't see why you should say my mind is closed just because I don't understand something. In that case everyone's mind's are closed since no one understands everything. Are you saying that your mind is more open than my mind? I think that statement is wholly unnecessary.

As for your explanation, I believe the statement made was that the faster runner will NEVER overtake the slower runner, not how many times he has to catch up, or the time it takes for him to overtake.

To me, the question is a yes or no question: Can the faster runner overtake the slower runner.

Hence it doesn't seem to be the same as the bee question, which asks how many times the bee can zip in between. If the archilles question was phrased similar to the bee question, then obviously I can see the logic. I read the question phrased in the extract as a simple "can overtake or cannot overtake" question.

Originally Posted by night86mare
a more open mind would help.

instead of thinking about the tortoise covering another 10 feet, think about how achilles has to cover the distance between him and the tortoise.

since the tortoise is always moving, whenever achilles catches up to the "last marked point" of the tortoise, he has a distance to go, even though he is much faster.. this catch-up distance gets smaller and smaller.

let's say eventually at the distance of 0.1 cm between them. the time taken for achilles to catch up is so minutely small that in just the next nanosecond he might have "caught up" to the tortoise 1000000000000000 (large number) times. get it?

and yes, mathematically there will be a solution as to how fast he takes to catch up, and it is not hard using relatively velocity from basic physics, or just fundamental brute force mathematics.

but if the question asks you, how many times achilles has to "catch up" to the "last marked point" of the tortoise, the answer is infinite number of points.

same for the bee thing. the crux lies in the questions you CAN answer, and the questions to which there are no answers. this is after all, a rather theoretical abstract concept which might have no practical bearing, but it isn't all that unimaginable.

3. ## Re: Problem on Mathematics - continued

I find it weird that these mathematicians (I'm assuming they are) think of crossing a room as crossing half, and half of half, and half of half of half. Why not just cross it as a whole? If you keep halving something, obviously you'll not reach the end since there's an infinite number of halves.

Its called Zeno's paradox and here is an explanation of its solution: From http://www.mathacademy.com/pr/prime/articles/zeno_tort/

Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.
What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.
Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?" If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn't be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm.
Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno's Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words,
1= 1/2 + 1/4 + 1/8 + 1/16 ...

At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn't – in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn't so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)
Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.
And poor old Achilles would have won his race.

4. ## Re: Problem on Mathematics - continued

Originally Posted by vince123123

Hence it doesn't seem to be the same as the bee question, which asks how many times the bee can zip in between. If the archilles question was phrased similar to the bee question, then obviously I can see the logic. I read the question phrased in the extract as a simple "can overtake or cannot overtake" question.
it is actually the same essence, that summation to 1 is based on a lot of halves.

i think maddyks has explained it pretty well. if you cannot see it then perhaps you should leave it alone, there seems to be a lack of affinity to maths (and perhaps logic, logic the subject, not logic the general meaning) here. you have to really imagine a hypothetical situation instead of just relating things to real life. this is theory, and it will not really be applicable in real life either.

but fact is fact, 1/2 + 1/4 + 1/8 + 1/16... etc will eventually sum up to give you 1 when the series is infinite. but it cannot and will not end. the same logic applies.

5. ## Re: Problem on Mathematics - continued

woah................. so much disturbance going around here. dun later thread kana closed sia.

want to try an easy explaination of why conditions for this question cannot be met and that there isn't an answer for this question (or meaningful answer in this sense), and hopefully can bring the question to a nice happy closure.

ok, for this question, it is asking for the number of times the bee meets with two cars travelling towards each other till they meet. now, let's reverse the question: two cars and a bee are at the same point. once the time starts, the cars start moving away and the bees "bounces" between them like a pinball. both scenarios are symmetrical and answer does not change as it is time-symmetrical.

now, here's the problem: at t=0, all three things are at s=0. however, the bee's velocity is faster than the cars. immediately when they start, the bee already ceases to be in the center which is why conditions of the problem cannot be met. to make this a solvable question, a non-zero time must be given to the cars to move away first before the bee starts to move(or for the non-reversed question, the answer exists if the cars stop at a non-zero distance away from each other).

hope this explaination might make it clearer. (though the mathematical proof always looks nicer)

6. ## Re: Problem on Mathematics - continued

Neo says : " There is no bee. "

7. ## Re: Problem on Mathematics - continued

Sure, since you can't see my point either, you should leave it alone too since its a lack of affinity to real life situations

It was my question that was directed at Jenova and was not directed at you to begin with anyway.

Originally Posted by night86mare
it is actually the same essence, that summation to 1 is based on a lot of halves.

i think maddyks has explained it pretty well. if you cannot see it then perhaps you should leave it alone, there seems to be a lack of affinity to maths (and perhaps logic, logic the subject, not logic the general meaning) here. you have to really imagine a hypothetical situation instead of just relating things to real life. this is theory, and it will not really be applicable in real life either.

but fact is fact, 1/2 + 1/4 + 1/8 + 1/16... etc will eventually sum up to give you 1 when the series is infinite. but it cannot and will not end. the same logic applies.

8. ## Re: Problem on Mathematics - continued

Originally Posted by vince123123
Sure, since you can't see my point either, you should leave it alone too since its a lack of affinity to real life situations

It was my question that was directed at Jenova and was not directed at you to begin with anyway.
haha and nvr the twain shall meet..
cheers guys..

9. ## Re: Problem on Mathematics - continued

OK, I'll take a simplistic formal stab at it...

d(t) := 20t = total distance travelled by bee, irrespective of direction.
b(t) := 1-10t = distance remaining between the two cars.
t is time in hours

How many fractions of b(t) can d(t) take?

What is t when b(t) is 0?

0 = 1-10t
t = 1/10

What is the ratio d(t)/b(t)?

lim(t->1/10)[(d(t)/b(t)]
= lim(t->1/10)[(20t/(1-10t)]
= lim(t->1/10)[20t]/lim(t->1/10)[1-10t]
= 2/(lim(t-<1/10)[1-10t])
= infinite, as denominator approaches zero as t approaches 1/10.

...which is what most people already know here, but teacher, do I get the A+?

Next exercise... prove that the above ratio of d/b will still be infinite irregardless of bee size as long as it's under 1km... (otherwise, time to call in Ultraman).

10. ## Re: Problem on Mathematics - continued

Originally Posted by vince123123
It was my question that was directed at Jenova and was not directed at you to begin with anyway.
oh. this is an open forums. but then why so sensitive because cannot understand?

peace, peace, one day you will know. maths and theory is very easy one.. just work at it hee hee hee! just like taking photo.. if you are too overly defensive when people try to explain the basics to you, very hard to learn one. this is just heartfelt advice which is well-intended!

cheers!

11. ## Re: Problem on Mathematics - continued

Originally Posted by ahbian
Neo says : " There is no bee. "
shouldn't it be..

neo freezes time.

neo squishes bee.

neo squishes two cars.

no more problem

12. ## Re: Problem on Mathematics - continued

Originally Posted by satay16
now, here's the problem: at t=0, all three things are at s=0. however, the bee's velocity is faster than the cars. immediately when they start, the bee already ceases to be in the center which is why conditions of the problem cannot be met. to make this a solvable question, a non-zero time must be given to the cars to move away first before the bee starts to move(or for the non-reversed question, the answer exists if the cars stop at a non-zero distance away from each other).
bro, you forget that if the cars meet, die die the bee also gone case

the trouble is, in the small moments before the car meet, the bee has a lot of small distances to cover and keeps flying to and fro.. so your explanation also not very accurate. to illustrate (though not very accurate - the graph should get progressively squished as proceed to the right)..

after a certain point cannot draw anymore, cos distance too small, but the bee is still there bouncing.

13. ## Re: Problem on Mathematics - continued

Errr.. can I go macro the bee first

14. ## Re: Problem on Mathematics - continued

I think a similar problem can be posed as the question: "What is the size of a dot?"

15. ## Re: Problem on Mathematics - continued

Originally Posted by vince123123
Sure, since you can't see my point either, you should leave it alone too since its a lack of affinity to real life situations

It was my question that was directed at Jenova and was not directed at you to begin with anyway.
Hi Vince:

There will be a few open minded people who will tell you if you have one leg on ice and the other leg in fire, the average feeling you experience will be nice.
Like wise, if you replace the bee with the opened minded person, he can run infintely between the two cars and will never get crushed.

16. ## Re: Problem on Mathematics - continued

oh i see my legendary stalker has arrived.

quantity > quality

17. ## Re: Problem on Mathematics - continued

Originally Posted by night86mare
oh. this is an open forums.
is this not an open forums?

18. ## Re: Problem on Mathematics - continued

Originally Posted by Silence Sky
is this not an open forums?
well i find it very worrying when people tend to obsess too much about me.

other than personal problems, your last few posts are generally flamebaiting ones, addressed indirectly to me, but made vaguely.

nothing i can do if the mods don't do anything. it is after all, reinforcement to you of the mentality that it is ok to do so because you can get away with it.

but don't push your luck too far. after all, loki thought he was very smart too.

19. ## Re: Problem on Mathematics - continued

So you show us a harmonic graph to explain your case.
You say the distance will become too small for you to plot as time tends towards infinity.
Will you trancate the harmonic at certain point in time? or you recon the bee will continue bouncing even when the distance between the two cars is smaller than the size of the bee?

Originally Posted by night86mare

after a certain point cannot draw anymore, cos distance too small, but the bee is still there bouncing.

20. ## Re: Problem on Mathematics - continued

Hi vince123123, wht I am trying to say is that most likely we've attempted the questions wrongly like how Achiles attempted to solve his prob.. Becoz my ans follows a geometric series like how his did and is an infinite series.

I apologize for wasting ur time attempting my little homwoek

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