[Guest]

what would happen if an infinitely divisible stick were cut in two, then half a minute later each half were again cut in two, then a quarter of a minute later each fourth cut in two, and so on ad infinitum. At the end of one minute what would be left? An infinite number of pieces? Would each piece have any length?

[Guest]

Doesn't it depend on the length of the stick in the first place? If the length of the stick is finite, then it seems to me that there will be an infinite number of pieces with zero length. If it is infinite, then there would be an infinite number of pieces of infinite length.

Having said that, my answer, which i think is logical, seems not to make sense to me!!!

[Guest]

I would say that nothing would happen, because with the logic as it is spelled out....you could never reach a minute....there would be infinite parts of a minute and you would never quite get there...and you would never quite stop cutting...and I have to say that that is an amazing device to be able to cut the stick that fast...

[Guest]

You'd eventually cut it into singular molecules and then maybe if you go further you divide it into atoms then separate protons and electrons and neutrons then the particles would go out of control and react in a big explosion frying you to dust...

[Guest]

You'd have an infinite amount of sticks with length 0, because (starting length of stick) / infinity = 0

[Guest]

You would never make the first cut, because you cannot find the center of infinity to begin with, and if you could, the resulting pieces would still be infinely long.

[Guest]

You would never make the first cut, because you cannot find the center of infinity to begin with, and if you could, the resulting pieces would still be infinely long.

just a clarification...if I'm reading it right...the stick isn't infinitely long, it's infinitely divisible...that would say to me that the stick has a finite length, which you can find the center of.

I may be wrong, but I don't think so

[bushindo]

Interesting. The cut parts aren't easily to comprehend, but the properties of the whole stick are well understood.

This is a lot simpler to imagine if we consider the stick to be the interval from 0 to 1. From this interval of real numbers we remove all numbers of the form a/b, where a and b are integers, and b = 2ⁱ, where i = 1, ... infinity. Essentially from the real numbers we remove a subset of the rational numbers. The cut parts are hard to image, because between any two points P and Q in the interval, it's easy to see that there is infinite number of cuts between them. However, the density of the interval is still the same because the non-rational numbers on the interval [0, 1] are infinitely more dense than rational numbers. If we imagine that such a divisible stick existed, and we infinitely cut them as described above. Assuming that the pieces are left in place as they are being cut, at the end the stick would look exactly the same as before the cutting process.

Edited April 22, 2010 by bushindo

[Quantum.Mechanic]

You would never make the first cut, because you cannot find the center of infinity to begin with, and if you could, the resulting pieces would still be infinely long.

Every point on an infinite line can be taken as the midpoint. Each half-line is equally long. Note that a half-line is "just as long" as a whole infinite line, even though a 1:2 mapping can be constructed. Infinity = 2*Infinity = 3*Infinity = ...

[Quantum.Mechanic]

I would say that nothing would happen, because with the logic as it is spelled out....you could never reach a minute....there would be infinite parts of a minute and you would never quite get there...and you would never quite stop cutting...and I have to say that that is an amazing device to be able to cut the stick that fast...

If the problem statement was instead:

"A train is moving along a track at a constant speed. In the first 1/2 minute it moves 1/2 km. In the next 1/4 minute it moves 1/4 km, etc. How long does it take to move 1km?"

Obviously the train will get to 1km in 1 minute.

The only difference with this and the OP is the point of dividing a material object infinitely, or measuring space and time with infinite accuracy. If we accept the assumption that the stick can be divided in the manner prescribed, the infinite sum leaves us with an infinite number of 0 length objects, whose sum is the original length of the stick.

[Guest]

If the problem statement was instead:

"A train is moving along a track at a constant speed. In the first 1/2 minute it moves 1/2 km. In the next 1/4 minute it moves 1/4 km, etc. How long does it take to move 1km?"

Obviously the train will get to 1km in 1 minute.

The only difference with this and the OP is the point of dividing a material object infinitely, or measuring space and time with infinite accuracy. If we accept the assumption that the stick can be divided in the manner prescribed, the infinite sum leaves us with an infinite number of 0 length objects, whose sum is the original length of the stick.

I have to disagree with you...if you think of time in a linear fashion, and one minute is considered infinite...when do you get to the end...because 59.999999999999...... seconds is not one minute. same thing with a train. if you are going one km, but your definition of infinity is 1 km, the train will never reach the end. I think of it this way...to be able to cut the stick at that speed, you would have to travel at near the speed of light or even faster if it were possible, and you would have to get faster and faster as time went on, and at that speed, time stops...so you would keep cutting and cutting forever.

Edited April 22, 2010 by ezdrocks

[moonthirsty]

It really depends upon who is doing the cutting.

If I am doing the cutting, then at the end of a minute I am going to have bloody stumps where my fingers once were. And a really dull knife. But other people might get better results.

[Guest]

Reminds me of the dichotomy of Zeno's arrow paradox.

An archer shoots an arrow at a target, but in order for the arrow to reach the target, it must first travel half the distance from the archer to the target. At this point, the arrow must travel half the remaining distance, bringing it 1/4 the original distance. This is continued on, ad infinitum, so that the arrow is never able to hit the target. Interesting to postulate.

Hint: This will not stop an arrow from hitting your friend, so don't try this at home

Fun note: I once passed a Zeno's Motel on the side of the highway but wondered if I would ever be able to get there

[moonthirsty]

Interesting. The cut parts aren't easily to comprehend, but the properties of the whole stick are well understood.

This is a lot simpler to imagine if we consider the stick to be the interval from 0 to 1. From this interval of real numbers we remove all numbers of the form a/b, where a and b are integers, and b = 2ⁱ, where i = 1, ... infinity. Essentially from the real numbers we remove a subset of the rational numbers. The cut parts are hard to image, because between any two points P and Q in the interval, it's easy to see that there is infinite number of cuts between them. However, the density of the interval is still the same because the non-rational numbers on the interval [0, 1] are infinitely more dense than rational numbers. If we imagine that such a divisible stick existed, and we infinitely cut them as described above. Assuming that the pieces are left in place as they are being cut, at the end the stick would look exactly the same as before the cutting process.

That's not "simpler to imagine", bushindo. But very cool answer, none-the-less.

The stick is already infinitely cut

So infinitely small

We can't tell at all.

[Guest]

This is a lot simpler to imagine if we consider the stick to be the interval from 0 to 1. From this interval of real numbers we remove all numbers of the form a/b, where a and b are integers, and b = 2i, where i = 1, ... infinity. Essentially from the real numbers we remove a subset of the rational numbers. The cut parts are hard to image, because between any two points P and Q in the interval, it's easy to see that there is infinite number of cuts between them. However, the density of the interval is still the same because the non-rational numbers on the interval [0, 1] are infinitely more dense than rational numbers. If we imagine that such a divisible stick existed, and we infinitely cut them as described above. Assuming that the pieces are left in place as they are being cut, at the end the stick would look exactly the same as before the cutting process.

This is the most reasonable answer. However, certain necessary assumptions that apply to the explanation above are unlikely to apply to cutting a stick. For instance, a cut of magnitude 0: ie, no length lost due to the cutting process. Also, leaving the stick together leaves an incorrect notion of the dimension of the pieces and while true leads to faulty conclusions. Take piece A with the center point of Ax. Given any thickness (y) that we presume A to retain after the cutting is complete (assuming the process completed in a finite timespan which is a rational assumption) would lead us to conclude that Ax-y/2<Ax<Ax+y/2 unless y=0. If y=0, A has no thickness and we have proven my point. Assuming y>0 (we know y cannot be less than 0) we know that there exist a rational number between Ax-y/2 and Ax+y/2, and specifically, that there exist a number of the form a/2^i where i is a positive integer, and by definition of the algorithm applied this number further subdivides A. This contradicts our assumption that the thickness of A is greater than 0 and the cutting process is complete. Obviously, A must have a thickness of 0. However, the question of dividing infinity is intriguing (which I realize was not the intent of the inquisitor, but is by far a more difficult question). While it is obvious that dividing an infinite bar in half is a simple matter of cutting it anywhere as Quantum.Mechanic explained, the process of dividing the half-line segment (necessary to divide the stick into fourths) in half is impossible. As an illustration consider [0,infinity) That set has no midpoint. Numerically it can be split by saying [0,1), [1,2), ...[n,n+1), ... (and an infinite number of other ways) but to define a midpoint and "cut it in half" is not possible. I would propose then that while this is a simple matter of limits for a stick of finite length, it is entirely impossible for an stick of infinite length.

[Guest]

I saw a lot of talk and guesses but no real math or references to math so...

This is really the harmonic series ∑(1/(2ⁿ)) (n goes from 1->inf), or ∑(1/2)*(1/2)^n-1 (n goes from 0->inf). The sum of this is (1/2)+(1/4)+(1/8)...,in this example each term is equal to that fraction of a minute (ex: the (1/2) term also indicates the passage 1/2 minute) and the denominator also happens to equal the number of pieces you have cut (note: the cut happens before the passage of time, but this is irrelevant). The sum of a harmonic series in general =a/(1-r), where a is the first term (1/2) and r is the part with the exponent (1/2) (from the eqn ∑(1/2)*(1/2)^n-1) so the sum works out to be (1/2)/(1-(1/2))=1, but remember that this is the sum of infinity terms. So we will get to 1 minute after infinity terms (or an infinite number of fractions of a minute) have been added up, and thus we will make an infinite number of cuts to pieces infinitely small. If you don't know calculus I know some of this may seem weird, adding an infinite number of terms and all, but trust me it works.

[Guest]

it's basic algebra: this is an exponential decay problem! y=b(0.5)^x and the the amount of a minute is left is the amount of the stick is left on either side

[Guest]

There would be infinitely amount of sticks, each one having no length (sounds wierd) >.>. If the length of the time each stcik is cut in half is halved each time, the stick would be cut infinitely amount of times before the minute ends

[Guest]

There would be 16 pieces, each of them the same length. It does not say infinite long stick. It says infinite divisible stick. After 30 secs it is at 4 pieces, at 45 secs it is at 8 pieces, at 60 secs ( 1 minute) there are 16 infinitely divisible pieces at the same length.

[Guest]

DSmithhurst that's wrong... it would be sixteen after 53.5 seconds. As the pieces are cut in too, so is the time.

[Guest]

just a clarification...if I'm reading it right...the stick isn't infinitely long, it's infinitely divisible...that would say to me that the stick has a finite length, which you can find the center of.

I may be wrong, but I don't think so

You're probably right, however as I see it, if it is infinitely divisible, therefore you can have an infinite number of pieces, and if you have and infinite number of pieces, that makes it infinitely long to begin with, so all the pieces are still infinitely long.

[Guest]

"what would happen if an infinitely divisible stick were cut in two, then half a minute later each half were again cut in two, then a quarter of a minute later each fourth cut in two, and so on ad infinitum. At the end of one minute what would be left? An infinite number of pieces? Would each piece have any length?"

This is one of those things where results that make sense separately seem to make a contradiction when brought together.

One result is how an interval of the number line may always be divided, giving two subintervals, which may also be divided, and so on. We also know that each subinterval must be smaller than the interval from which it was cut.

Those familiar with Zeno or calculus know that an infinite number of cuts will be made when one minute is reached.

Making something smaller and smaller an infinite number of times, makes one think that it must disappear.

But we all know that given a stick you cannot cut it until it disappears (mass and energy conservation).

So where does the disconnect come from?

First question: what does infinity really mean?

There are different ideas associated with infinity. Most of these are not useful to us. As humans we deal with finite objects, so in order to make use of the concept of infinity we must interpret in a language that appeals to us, a language of finite objects.

For this language to be useful, we must be able to easily manipulate objects, and easily apply the concepts of infinity to our manipulations.

The connection between the mathematical tool set constructed before calculus to the concepts of infinity is something called a limit.

Consider regular polygons, Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon, ... etc.

As the number of sides increases, the differences between one shape to another get smaller and smaller.

I tell you to pick a number called n, and I will show you the polygon with n sides and the polygon with n+1 sides.

Now I ask you to tell me what shape will you see when n is made so big that you can't even tell any difference at all between the n sided polygon and the n+1 sided polygon (the differences are so small that you cannot detect them at all).

It would look like a circle. The circle is the limit of the regular polygons as the number of sides approaches infinity.

What about the number y = 1/x?

The bigger you make x the smaller y gets. What will y look like when x is so big that you can't see any difference at all in y with further increase in x?

y will look like 0.

Do I ever say that y is equal to 0? No. Just that y starts looking more and more like 0 the bigger x gets.

There is a more formal definition for the limit than this, but this gets the point across.

So let's look at our stick now.

Let's say the number of cuts is n, so that the number of pieces is y = 2^n.

What is the limit of the size of a piece as n goes to infinity?

Well the size of the piece looks more and more like 0.

We cannot say it is 0 for any number of cuts though, it is still there.

This could have been the answer, but raven34 was very clever and talked about a situation where an infinity actually is reached. Seems impossible, but actually infinity is being reached by many things all around us all the time.

In any instance of motion, an object crosses an infinite number of points between its source and destination. Nobody thinks that's anything special. It's not special because the points exist in our minds and we can make as many as we want and go crazy and when we can't think of anymore points to add we can just say that there are more and more. We are allowed to do this because points in space don't cost anything to make, they are imaginary not physical. Another point to note is that time itself can be divided into an infinite number of points. The ratio of two infinite things can be finite, like the number of points traversed divided by the number of time points elapsed.

If y = 2x, it's limit is infinity as x goes to infinity.

What is the limit of y/x as x goes to infinity? If x is 10 billion, y will be 20 billion,

the ratio will always be 2. If x is the number of points in time elapsed over some scale, and y is the number of points in space elapsed in some scale, then y/x is the speed of the object and the ratio of the scale of space over the scale of time.

Physically cutting a stick does not exist in our minds though, it is very real.

Each cut costs something. Energy, entropy, time, etc...

You cannot physically do what raven34 asks because cutting a stick is a physical process, not an imaginary one.

So what? We approximate reality by imaginary stuff all the time. When you throw a ball up in the air, you say that gravity accelerates it downwards at 9.8 m/s2 so it's trajectory is v0*t - gt^2/2. Do you think this equation is actually true? It is not. We don't consider the overlap of the quantum mechanical wavefunctions of the ball with its surroundings or the gravity of jupiter's effect on the ball. Those things are real, but too subtle and complicated to worry about.

Can we approximate the physical process of cutting the stick an infinite number of times by an imaginary mathematical scenario where infinity is allowed?

You can try, but you won't get anything useful, because this is a case, where the approximating equation will not give a result resembling reality for an infinite number of cuts even though it does resemble reality for a small number of cuts.

Effects that your equation did not take into account will get magnified as your pieces get smaller and smaller. They will start resembling some other equation, and you will be waaaay off. Things like quantum mechanics will come into effect.

What if we ignore all that and pretend the stick is infinitely divisible like raven34 said (she covered many bases)?

Well since we are pretending now, we have no guarantee that our answer will resemble reality at all. Whatever answer we get, if it disagrees with our common sense and physical intuition, then we know we shouldn't be confused because our math no longer describes physical reality. There is no contradiction.

So what is the mathematical answer?

Well, since raven34 said we cut it an infinite number of times, she tells us that for our imaginary game infinity is reachable, and we will in fact have an infinite number of pieces.

They cannot have 0 length though. It is obvious that we are intended to assume that the stick is not subject to vanishing, and that the only change in state is to cut the stick, which we say never removes any matter in the stick. Thus the sum of all the cut pieces must be equal to the original sticks length which is finite. The lengths therefore cannot be zero.

The mathematical answer is to fight fire with fire. raven34 introduced to the problem this strange concept of infinity as being something unbounded in the greatness of its magnitude.

There is also a counterpart, a strange concept of something unbounded in the smallness of its magnitude, yet non zero.

This is called an infinitesimal. The lengths of the pieces will be infinitesimal, which is allowed if infinity is allowed because one can describe infinitesimals in any language that can be used to describe infinity.

We cannot take the limit because raven did not say

the limit as the number of cuts go to infinity.

She said the number of cuts IS infinity.

i.e.

not number of pieces = limit as n goes to infinity of 2^n

but rather number of pieces = 2^(infinity)

If the original length was L, the size of piece would be

size = L/(2^n)

In the limit as n goes to infinity, this size has limit 0.

If n equals infinity however, the size equals an infinitesimal.

That my friends is the answer, and I hope I have dispelled any contradictory or paradoxical feelings associated with the problem.

Edited April 23, 2010 by mmiguel1

[Guest]

I'm glad I'm not a carpenter! I had no idea how much math was involved with cutting wood.

[Quantum.Mechanic]

I have to disagree with you...if you think of time in a linear fashion, and one minute is considered infinite...when do you get to the end...because 59.999999999999...... seconds is not one minute. same thing with a train. if you are going one km, but your definition of infinity is 1 km, the train will never reach the end. I think of it this way...to be able to cut the stick at that speed, you would have to travel at near the speed of light or even faster if it were possible, and you would have to get faster and faster as time went on, and at that speed, time stops...so you would keep cutting and cutting forever.

But we know that 1 minute does eventually end. And 1km has an end. It's very close to Zeno's Paradox about Achilles and the Tortoise.

[Guest]

But we know that 1 minute does eventually end. And 1km has an end. It's very close to Zeno's Paradox about Achilles and the Tortoise.

I agree with you on that, and in a sense that this were actually possible, I would again agree. But since the parameters of the puzzle are NOT actually possible, you can't look for a POSSIBLE solution...you have to look at it theoretically, which is why I say that you would never reach one minute...there would always be a minuscule portion of a minute remaining.

I'm not the sharpest tool in the shed, but I can still cut!

[Guest]

Well you have a paradox don't you. If you follow the calculus route, the # of cuts approaches infinity as time approaches 1 min. You didn't say there was anything special about time here, so the time t = 1 minute will occur. So one might say, well you have made an infinite # of cuts. But it is not possible to make an infinite # of cuts, just as it is not possible to count to infinity. So I conclude that it is not possible to have the situation that you propose.

BasicPoke