I remember me and my friend debating about this in grade 12 physics last year, and neither of us came to any solid conclusion.

Can anyone explain to me the solution to this?

For those unfamilliar with the idea: A sprinter runs a 100m race, but in order to complete that he must pass the 50m mark, and then the 75m mark, and then the 87.5m mark, and so on for every midpoint possible... till infinity. The official question is: how can one cross an infinite number of midpoints and yet still reach a finite distance?

Our best guesses were that the mathematical explanation would be that his midpoint crossings resemble a logarithmic function with a horizontal asymptote at 100m. I don't think anyone here needs an example, but I'm new and don't know the overall IQ of this forum :D ; http://img.sparknotes.com/figures/1/15debaf09555bfc7c688d9ee8ae574bc/asymptote_horizontal1.gif (not quite the right parameters, but the whole approaching the line bit is hopefully explained)

However we couldn't expand on that and eventually forgot about it.

The floor is open to someone older and wiser...

lets just say you take 1m strides, no matter how many midpoints you add in afterwards, it was still the 100 strides alone that got to the end, not getting to the 50% then the 75% etc

Pete mentioned it in another thread....Zeno's paradox I think he called it. "Dichotomy"

Of the 40 arguments attributed to Zeno by later writers, the four most famous are on the subject of motion:



The Dichotomy: There is no motion, because that which is moved must arrive at the middle before it arrives at the end, and so on ad infinitum.

The Achilles: The slower will never be overtaken by the quicker, for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must always be some distance ahead.

The Arrow: If everything is either at rest or moving when it occupies a space equal to itself, while the object moved is always in the instant, a moving arrow is unmoved.

The Stadium: Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time.
source:
http://www.mathpages.com/rr/s3-07/3-07.htm

Well, no matter how many marks there are, the sprinter is still going at the same pace. So, if there where an infinite number of marks there, it wouldn't matter, because the sprinter would just run right over them. So it would look like this:

0
|
/\
                        -            -      -   - ---- Finish
0 seconds

                                  0
                                  |
                                 /\
                        -            -      -   - ---- Finish
5 seconds

                                                           0
                                                           |
                                                          /\
                        -            -      -   - ---- Finish
10 seconds

- is a mark.

(The timing is probably off, but you get the general idea ;))

Our best guesses were that the mathematical explanation would be that his midpoint crossings resemble a logarithmic function with a horizontal asymptote at 100m.
http://img.sparknotes.com/figures/1/15debaf09555bfc7c688d9ee8ae574bc/asymptote_horizontal1.gif
What do the divisions on the horizontal axis represent?
What would it look like with time on the horizontal axis?

I remember me and my friend debating about this in grade 12 physics last year, and neither of us came to any solid conclusion.

Can anyone explain to me the solution to this?

For those unfamilliar with the idea: A sprinter runs a 100m race, but in order to complete that he must pass the 50m mark, and then the 75m mark, and then the 87.5m mark, and so on for every midpoint possible... till infinity. The official question is: how can one cross an infinite number of midpoints and yet still reach a finite distance?

Our best guesses were that the mathematical explanation would be that his midpoint crossings resemble a logarithmic function with a horizontal asymptote at 100m. I don't think anyone here needs an example, but I'm new and don't know the overall IQ of this forum :D ; The mistake in this "paradox" is the assumption that it must take an infinitely long time to pass through an infinite number of discreet distances.

Basic calculus shows us that in fact it can take a finite amount of time to pass through an infinite set of distances if the distances are constantly decreasing. That's what's happening here; it's true that you must pass through an infinite set of half-distances in order to reach the finish line, but the length of each half-distance is always decreasing. So, although you do indeed pass through an infinite number of half-distances, there's no reason to think that it will take you an infinite amount of time to do it. In fact, the sum of the times necessary to travel through your infinite set of half-distances will equal the total time necessary to run the race.

Basic calculus shows us that in fact it can take a finite amount of time to pass through an infinite set of distances if the distances are constantly decreasing.

Yeah. We knew it was obviously a finite finish time, but we didn't know how to express it mathematically. Because when I think of "infinite" I imagine everything else goes to infinity, such as time.

It wouldn't be a log if he ran at a steady pace, it'd be a straight line.
Here's an example:
http://www.analyticcycling.com/DiffEqMotionFunctions_PlotDistance.gif

At any time t, he'd have gone x meters. So at 87.5 meters, all you need to do is trace the graph to find the time he is at 87.5 meters. When you split distance up, you split time up to more and more precise pieces.

Yeah. We knew it was obviously a finite finish time, but we didn't know how to express it mathematically. Because when I think of "infinite" I imagine everything else goes to infinity, such as time.If it takes T units of time to run the complete race, then the time necessary to run the first half-distance would be T*1/2. The time for the next half-distance would be T*1/4. Then the next time would be T*1/8, and so on. The total time necessary to run the race would be the sum of the times of all the half-distances, which would be T/2 +T/4 + T/8...continuing out to T/infinity. You could factor that out to T * (1/2 + 1/4 + 1/8...). If you sum up 1/2 + 1/4 +1/8...out to 1/infinity, it all sums to 1, giving us T*1. Which proves that if you add of the time necessary to traverse the infinite number of ever-decreasing half lengths, you get the total time necessary to run the race.

The problem with the problem set up is that it describes a model which is confined to the time of the race. The model can never tell us about the time after the race, or even about the time at the very end of the race.
This doesn't mean that the time at the end of the race doesn't exist... just that the model of the problem can't tell us about it.

Let's say the runner is running at 10m/s.
She takes 5 seconds to reach 50m.
Another 2.5 seconds to reach 75m.
Another 1.25 seconds to reach 87.5m.
Another 0.625 seconds to reach 93.75m.
etc.

What happens when we graph the total distance against total time?

Here it is:
<img src="http://www.sciforums.com/attachment.php?attachmentid=3941&stc=1">

The only data points we have are in the ten seconds of the race. Our model of the problem simply doesn't tell us what happens at 10 seconds or after. However, it is clear from the graph that after 10 seconds, the runner reaches 100m and passes the realm of our flawed model.

But why do we need to worry about the time after the race? Should we write his biography, put his face on a cereal box?
No, because that's irrelevant to the question at hand, just as the graph of him passing the finish line.

Yeah, Achilles and the turtle. The turtle gets a 10 meter head start.

Same thing, The speed of Achilles is 10m/s and the speed of the turtle is 1m/s. As Achilles is where the turtle started, the turtle has gone 1 meter further. And when achilles reaches that mark, the turtle has gone 0.1 meters further. So Achilles never reaches the turtle.
But.. i drew this thingie that some guy who looks like a bumm once showed me.

http://sciforums.com/attachment.php?attachmentid=3947&stc=1

I beleive that paradox's tells us that something we assume about the universe or some assumption we have made is wrong. ie. a paradox means one has either facts or theories wrong.

In this case, there is the assumption that space can be divided infinitely and therefore movement can be made in ever tiner steps forever. However, Planck discovered in 1900 that such was not the case. One reaches a point where a particle is on one position or the other but not in between. The energy required was called a quantum. The amount of distance traveled is called Planck's constant. Science has observed that an electron can circle about a nucleus at a certain radius. One can apply small amounts of energy and the radius will be the same. It takes a certain level of energy to cause the electron to change its orbital radius. When that level of energy is applied, the electron leaps to a new higher orbit without ever orbiting at the interim levels.


Thus if one measures in the smallest units possible: Planck's constant, one can only do integer math. there can never be a "half unit"

I believe that paradoxs tells us that something we assume about the universe or some assumption we have made is wrong. In other words, a paradox means one has either facts or theories wrong.

In this case, there is the assumption that space can be divided infinitely and therefore movement can be made in ever tiner steps forever. However, Planck discovered in 1900 that such was not the case. One reaches a point where a particle is on one position or the other but not in between. The energy required was called a quantum. The amount of distance traveled is called Planck's constant. Science has observed that an electron can circle about a nucleus at a certain radius. One can apply small amounts of energy and the radius will be the same. Once one quantum of energy is applied, the electron leaps to a new higher orbit without ever orbiting at the interim levels.

Thus if one measures in the smallest units possible, Planck's constant, one can only do integer math. The universe has declared that there can never be a "half unit" step at this level.

We knew it was obviously a finite finish time, but we didn't know how to express it mathematically.

It is a geometric series. See http://mathworld.wolfram.com/GeometricSeries.html for a good explanation on how to set it up and solve it mathematically. In your case you are starting from k=1 and you have r=1/2 (look at formula 8 and 9)

-Dale

Fortget Plank. There is no paradox. The paradox appears because no statement is made regarding the runner's speed. Assuming the runner's velocity is constant, the time taken to finish an interval is 1/2 the time taken to finish the preceding interval. 1/2 + 1/4 + 1/8 + ... = 1. Simple limit theory. On the other hand, if one assumes the runner's speed is halved for interval, the runner indeed fails to reach the end line in finite time. 1/2 + 1/2 + 1/2 + 1/2 + ... = infinity.

To the math "teacher" in the previous thread on Zeno's Paradox: 0.9999... = 1.0. There is no number between the two; 0.9999... and 1.0 are different representations of the same mathematical quantity. In any base a>1, the number 0.(a-1)(a-1)... is 1.0. Let x = 0.(a-1)(a-1)... Then a*x = (a-1).(a-1)(a-1)... so (a-1)*x = a-1, or x = 1.

In particular, 1/2 + 1/4 + 1/8 + ... = 0.11111... (base 2) = 1.0.

Zeno's paradox has a flaw. It assumes that the infinite summation of finite amounts of time is infinite. Basically... 1/2 + 1/4 + 1/8 + 1/16 + ... = infinity

The mistake in this "paradox" is the assumption that it must take an infinitely long time to pass through an infinite number of discreet distances.

Basic calculus shows us that in fact it can take a finite amount of time to pass through an infinite set of distances if the distances are constantly decreasing. That's what's happening here; it's true that you must pass through an infinite set of half-distances in order to reach the finish line, but the length of each half-distance is always decreasing. So, although you do indeed pass through an infinite number of half-distances, there's no reason to think that it will take you an infinite amount of time to do it. In fact, the sum of the times necessary to travel through your infinite set of half-distances will equal the total time necessary to run the race.

The resolutin is the fact that one does not move in half distance increments over time. If one infact took half increment steps he would indeed require an infinite amount jof time to reach the end.

MacM, one is perfectly free to regard the motion of an object between any two points as a series of ever smaller half way steps. This does not in any way imply that it will take an infinite amount of time to execute this infinite number of halfway steps. One can, using basic kinematics, calculate the time it would take to run each step. The total time is then simply the sum of all these step times. As others have indicated, the infinite sum can be calculated using the methods of elementary calculus and produces the well known result that the total time is the distance divided by the velocity.

The paradox assumes that both location and momentum are knowable to an infinite degree of precision. Once reduced to an appropriate scale, the runner finishing the race is a matter of probability.

Say the race is 100 metres. The distances covered (in metres) are:

50, 25, 12.5, 6.25, 3.125, ...

Suppose the runner runs at constant speed of 10 metres per second. Then, the times taken to cover the above distances are (in seconds):

5, 2.5, 1.25, 0.625, 0.3125, ...

Add up the total time taken:

5 + 2.5 + 1.25 + 0.625 + 0.3125 + ... = 10 seconds.

In other words, the entire distance is covered in a finite time of 10 seconds.

Therefore, it is incorrect to say that the entire distance cannot be covered.

James R., exactly what I said.. only I didn't do all the math :)

The mathematical probability that we should exist now, in this particular instance in time, is 1 in infinity, and yet - here we are. What about the bouncing ball:
If you drop a ball from 1 meter with a bouncing efficiency of 50% then:
The first bounce will be 0.5 meter high, the second 0.25m, then 0.125m ...
Theoretically the ball will never stop bouncing.

Hi root,
Two distinct questions:

What happens to the time beween bounces?

How long does it take for a bounce to be indistinguishable from random vibrations?

hi root,

I'm curious, how did you arrive at this conclusion: "The mathematical probability that we should exist now, in this particular instance in time, is 1 in infinity, and yet - here we are."

hi root,

I'm curious, how did you arrive at this conclusion: "The mathematical probability that we should exist now, in this particular instance in time, is 1 in infinity, and yet - here we are."

How many fractions of seconds in all time, I would say infinate.
Why, what would you say is the probability?

Hi root,
Two distinct questions:

What happens to the time beween bounces?

How long does it take for a bounce to be indistinguishable from random vibrations?

Don't take this too serious, it's meant as a joke.

The mathematical probability that we should exist now, in this particular instance in time, is 1 in infinity, and yet - here we are.
I believe that the probability of *you* existing at the time of your post may have been 1 in infinity until I observed your post. At which point the probability became 100%.

The probability of *me* existing now has always been 100% for as long as I can recall.

I believe that is the quantum-mechanical view anyway ;)

-Dale

I believe that the probability of *you* existing at the time of your post may have been 1 in infinity until I observed your post. At which point the probability became 100%.

The probability of *me* existing now has always been 100% for as long as I can recall.

I believe that is the quantum-mechanical view anyway ;)

-Dale

Look at it this way:
Depict time as a line, infinitely long, stretching from infinite past to infinite future, every mm. represents one human lifetime. Color one mm. of the line in red to represent your lifetime. The instant ,*NOW*, could theoretically be located at any point of the line, if it happen to be any point outside your red mm. then you would not exist. So, what a coincidence that NOW is at a time that you actually exists!

(I know that not everybody will agree that time is infinite).

Look at it this way:
Depict time as a line, infinitely long, stretching from infinite past to infinite future, every mm. represents one human lifetime.
Interesting! Alternatively, since I know the probablity of me existing has been 100% for as long as I can recall, perhaps it is time itself that is the improbable thing here. Maybe the probablity of time itself existing was infinitesimal, at least until I started making observations. Now, of course, the probability is 100%. Solipsism meets quantum mechanics?

I must say, this thread has diverged quite far from the original topic. Of course that is what is making it so fun now.

-Dale

Say the race is 100 metres. The distances covered (in metres) are:

50, 25, 12.5, 6.25, 3.125, ...

Suppose the runner runs at constant speed of 10 metres per second. Then, the times taken to cover the above distances are (in seconds):

5, 2.5, 1.25, 0.625, 0.3125, ...

Add up the total time taken:

5 + 2.5 + 1.25 + 0.625 + 0.3125 + ... = 10 seconds.

In other words, the entire distance is covered in a finite time of 10 seconds.

Therefore, it is incorrect to say that the entire distance cannot be covered.

In other words consider the sum of the terms from 1 to infinity of the series 1/n^2. Graphically each term in the series can be looked at as a box of width one and height 1/n^2. If we superimpose the function f(x)=1/x^2 on this graph we will see that the area underneath it (from 1 to infinity) is larger than the sum each individual box in our series (from n=1 to infinity). We know that the sum of the boxes must be less than 1 + ∫(1-infinity)1/x^2*dx

Since that number exists, there is a limit for the summation, or the summation converges. A better picture and description is found http://www.math.purdue.edu/~rcp/MA301Ch7.pdf on page 86.

If one infact took half increment steps he would indeed require an infinite amount jof time to reach the end.
No, you are incorrect about this. It would not take an infinite amount of time to pass through an infinite number of half-distances. It is possible to sum an infinite series of numbers (in this case, the time needed to run each half-distance) and come to a finite result. See my previous post.

Look at it this way:
Depict time as a line, infinitely long, stretching from infinite past to infinite future, every mm. represents one human lifetime. Color one mm. of the line in red to represent your lifetime. The instant ,*NOW*, could theoretically be located at any point of the line, if it happen to be any point outside your red mm. then you would not exist. So, what a coincidence that NOW is at a time that you actually exists!

(I know that not everybody will agree that time is infinite). This is often refered to as bad probability analysis. You are starting with something you know to happen and then you are trying to attribute a probability for it happening. Probably the most prolific example of bad probability analysis is the Bible Codes in which random phrases are searched for encoded in the Bible and when found, attributed some probability that has largely been agreed upon to be bad probablility theory. The reasons are many, but one of the basic reasons is that not every probability is independent.

For a simplified example: If I have 5 digits which can have the value of 0 or 1, what is the probability of them all being 1? If all the digits were truely independent, then it would be 1 in 32. But if all digits were dependent on the first digit, as for an example rule: all equal to the first digit, then the probability would be 1 in 2.

Interesting! Alternatively, since I know the probablity of me existing has been 100% for as long as I can recall, perhaps it is time itself that is the improbable thing here. Maybe the probablity of time itself existing was infinitesimal, at least until I started making observations. Now, of course, the probability is 100%. Solipsism meets quantum mechanics?

I must say, this thread has diverged quite far from the original topic. Of course that is what is making it so fun now.

-Dale

Two more examples of these strange infinity things:
According to Einstein's theory when you approach the event horizon of a Black Hole then time starts slowing down. So if you could experience falling into a Black Hole it would take for ever. The closer you get to the event horizon the more time slows down until it comes to a virtual standstill.
Another one:
It is often said that at the moment before you die you see your whole life flash past in your mind. So you re-experience your whole life until you get to the point where you are about to die and the same thing happens again. You keep on re-experiencing your life and never get to the point where you actually die.

Imagine: falling into a Black Hole while you are about to die!

Hi root,

According to Einstein's theory when you approach the event horizon of a Black Hole then time starts slowing down. So if you could experience falling into a Black Hole it would take for ever.

I don't think this is true. While it is true that the usual Schwarzschild coordinate time diverges at r = 2M, it does in fact take a finite amount of proper time for an observer to fall through the horizon. Proper time is of course what you, the infalling observer, would experience.

Hi root,



I don't think this is true. While it is true that the usual Schwarzschild coordinate time diverges at r = 2M, it does in fact take a finite amount of proper time for an observer to fall through the horizon. Proper time is of course what you, the infalling observer, would experience.

OK, I will agree to that, but you will never get to the centre though as time slows down as gravity increases.

There is a very simple solution to this whole paradox. The question was: How can one cross an infinite number of midpoints and yet reach a finite distance?

Well look at it this way. A point has no dimentions, but a linear distance (displacement?) has one dimention. So how can you measure a dimention according to something with no dimentions? You cannot do something like that.

Well to make things simpler let say I ask you all: How many points are there in a metre? There can be no answer to that question (technically infinity is not an answer).