Thread: A short question...

Just something that I couldn't think of an answer to.

Ok, say you start at 0 and are aiming for 10. The first second you take a 'step' of an amount (lets say 2), the second step you go half the previous amount (so 1), next step you half again (so .5) and so on. So you are always halving your previous step. So those three steps would take you from 2 to 3 and then to 3.5 etc.

As every step brings you closer to 10, if you continued this infinitely would you ever reach 10?

Also, does it make a difference what number you start with? I know that a number like 7 or 8 would bring you to 10 in the second step, but is there a rule like numbers over half the target number will eventually arrive there?

Thanks everyone.

I think it depends on your shoe size.

This is a classic problem. The first step is to note that 1 + 1/2 + 1/4 + 1/8 + ... = 2. This is a commonly known infinite series. If you start with 2 steps then you just factor out the first number, 2 + 1 + 1/2 + 1/4 + ... = 2*(1 + 1/2 + 1/4 + ...) = 2*2 = 4. Starting with 8 is the same thing 8 + 4 + 2 + 1 + 1/2 + ... = 8*(1 + 1/2 + 1/3 + ...) = 8*2 = 16.

If you want to reach exactly 10 then you have to start with 5 steps since 10 = 5*2 = 5*(1 + 1/2 + 1/4 + ...) = 5 + 5/2 + 5/4 + 5/8 + ...

Essentially you will always make it twice as far as your first step.

Originally Posted by TheObserver

This is a classic problem. The first step is to note that 1 + 1/2 + 1/4 + 1/8 + ... = 2. This is a commonly known infinite series. If you start with 2 steps then you just factor out the first number, 2 + 1 + 1/2 + 1/4 + ... = 2*(1 + 1/2 + 1/4 + ...) = 2*2 = 4. Starting with 8 is the same thing 8 + 4 + 2 + 1 + 1/2 + ... = 8*(1 + 1/2 + 1/3 + ...) = 8*2 = 16.

If you want to reach exactly 10 then you have to start with 5 steps since 10 = 5*2 = 5*(1 + 1/2 + 1/4 + ...) = 5 + 5/2 + 5/4 + 5/8 + ...

Essentially you will always make it twice as far as your first step.

Ah, thank you, glad to know this is an actual thing. Didn't know about infinite series. Still, seems like it is just more convenient to have this fact and that in a practical situation the infinite would never arrive. Speaking of which, are there any practical uses of this?

If you reach maths/physics at a university level you will be dealing with infinite series nearly every day of your life.

Isn't this similar to Zeno's dichotomy paradox. To reach some goal a distance away, you first must move halfway there, then ¾ of the way there, etc, etc. Each step, you're covering only half the distance to the goal, thus you can never reach it ... it would take an infinite number of steps.

Originally Posted by jrmonroe

Isn't this similar to Zeno's dichotomy paradox. To reach some goal a distance away, you first must move halfway there, then ¾ of the way there, etc, etc. Each step, you're covering only half the distance to the goal, thus you can never reach it ... it would take an infinite number of steps.

Yeah, that does sound similar. Thanks for the link - I knew it'd be a philosophical issue too.

Just something that I couldn't think of an answer to.

Ok, say you start at 0 and are aiming for 10. The first second you take a 'step' of an amount (lets say 2), the second step you go half the previous amount (so 1), next step you half again (so .5) and so on. So you are always halving your previous step. So those three steps would take you from 2 to 3 and then to 3.5 etc.

As every step brings you closer to 10, if you continued this infinitely would you ever reach 10?

Hmm.. I think this can be represented in an exponential function: f(x)=b^x
Where b represents the amount of the halving (0.5) and x represents the amount of steps taken (1,2,3,4,5...) so... f(x)=.5^x
When this equation is graphed it will show a graph that is approaching a line that it will never touch (also known as an asymptote.)
So, this being said if you were to halve every step you took you would be approaching ten but you would never actually touch it.

*note* I don't know if I got that equation right so if anyone see's any problems please let me know!!

I think your equation is similar, but a slightly more detailed equation is ∑f(x)=∑ab^x-1 . So when x=1, f(x) is 'a', which we can make it as a starting number (here, it will be 2). Also, just adding to what TheObserver said, the general equation to this kind of problem is this: a/(1-b). If a starting step was 2, and it become half every time, then a=2 and b=1/2. Therefore, if we plug them in, it will be 2/(1-1/2)=4.

There is a difference between getting arbitrarily close to 10 and actually getting to 10 though. The series in question gets arbitrarily close to 10 but never actually reaches 10. Its the whole potential vs actual infinity problem and supertasks which always gives me a headache.

Lets consider a different example - the thompson lamp. Suppose I show you a standard lamp with a switch and furthermore suppose said lamp is not glowing. After 30 seconds I turn on the switch, after a further 15 seconds I turn it off, after a further 7.5 seconds I again turn it back on. I continue this little game, alternating between switching the lamp on and off and halving the time between these actions. You get bored of my game and walk away, glancing back at me after a minute. Is the lamp glowing or not?

Originally Posted by river_rat

There is a difference between getting arbitrarily close to 10 and actually getting to 10 though. The series in question gets arbitrarily close to 10 but never actually reaches 10. Its the whole potential vs actual infinity problem and supertasks which always gives me a headache.

Lets consider a different example - the thompson lamp. Suppose I show you a standard lamp with a switch and furthermore suppose said lamp is not glowing. After 30 seconds I turn on the switch, after a further 15 seconds I turn it off, after a further 7.5 seconds I again turn it back on. I continue this little game, alternating between switching the lamp on and off and halving the time between these actions. You get bored of my game and walk away, glancing back at me after a minute. Is the lamp glowing or not?

This is actually a physics question instead of a math question. The filament would never completely cool off and so it would glow continuously at a reduced intensity. Then when the frequency became high enough, the inductive and capacitive impedance of the wiring would become more dominant until the light went out completely.

Lol Harold14370 Someone always uses physics to try and ruin logical puzzles.

As an aside, physics tends to make these problems worse and not better as we have learnt that many things we should consider universal and obviously true in all realities are not. Before relativity for example, considering a self consistent universe where clocks ran at different rates would be seen as an impossibility. But let me not get started on the evils of the second class of supertasks!

Originally Posted by river_rat

There is a difference between getting arbitrarily close to 10 and actually getting to 10 though. The series in question gets arbitrarily close to 10 but never actually reaches 10. Its the whole potential vs actual infinity problem and supertasks which always gives me a headache.

Rather than arbitrarily, I think "asymptotically" approaching 10 is a better word. The thing is, the dichotomy paradox is in reference to a real situation, i.e. it describes the real world. Well, if the real world is truly discrete at the fundamental level, should it not be described in discrete terms? As the paradox seems to assume a continuous space.

And as for infinite series, I'm not sure as to how mathematicians interpret the result. I thought an infinite series is treated such that the "infinity" is fulfilled, as opposed to approaching and unreachable.

Like what halorealm said, I think it's better to say that the number is "asymptotically" approaching 10 rather than "arbitrarily." If you say arbitrarily, then you are saying it is different from that number. But the thing is, the difference gets too small at the end that just say it's that number assuming it's going for infinity. One thing that helped me understand this concept better was the fraction of third. 1/3 is .333333.... so if you multiply it by 3, it will be 3/3 or .99999.... Here, it never quite reaches that number, but we know that 3/3=1 so it is basically same as 1.

Originally Posted by halorealm

Rather than arbitrarily, I think "asymptotically" approaching 10 is a better word. The thing is, the dichotomy paradox is in reference to a real situation, i.e. it describes the real world. Well, if the real world is truly discrete at the fundamental level, should it not be described in discrete terms? As the paradox seems to assume a continuous space.

And as for infinite series, I'm not sure as to how mathematicians interpret the result. I thought an infinite series is treated such that the "infinity" is fulfilled, as opposed to approaching and unreachable.

Hi Halorealm

The actual definition of the series is as follows: We say that an infinite for series sums to if for any measure of closeness (say ) I can give you a number such that the partial sum is within of . That is why I say arbitrarily close since the challenge is an arbitrarily small positive real number.

We can immediately see the issue with using this concept as a solution to the paradox - we never actually sum an infinite number of numbers at any point in the definition. Everything is finite and thus you cannot point to the infinite series as justification for being able to do a super task. The dichotomy problem is worse. At least in this problem we have no last step but we have a first step we can take. In the dichotomy problem there is not even a first step so how do we even start the task?

On the "physics" of the paradox we have no idea about the small structure of the universe, general relativity assumes both time and space are continuous and works really well as a description of the universe, for example. Zeno's paradox's come in a set of four: covering both discrete and continuous space and discrete and continuous time and arriving at contradictions no matter which mixture of the two you choose for your model.

Originally Posted by river_rat

Hi Halorealm

The actual definition of the series is as follows: We say that an infinite for series sums to if for any measure of closeness (say ) I can give you a number such that the partial sum is within of . That is why I say arbitrarily close since the challenge is an arbitrarily small positive real number.

Hello river_rat

Excellent response. I'm not very familiar with infinite series. Is this measure somewhat like the correlation coefficient ? I'm assuming in the case you provided above, as , .

We can immediately see the issue with using this concept as a solution to the paradox - we never actually sum an infinite number of numbers at any point in the definition. Everything is finite and thus you cannot point to the infinite series as justification for being able to do a super task. The dichotomy problem is worse. At least in this problem we have no last step but we have a first step we can take. In the dichotomy problem there is not even a first step so how do we even start the task?

What do you mean that the dichotomy scenario has no "first step"? (lest I misunderstood you) Isn't it really a wordy representation of ?

Well, possibly not. You might be right that there is no first step, as the paradox is originally stated with the first step being the infinitesimal one. But, as far as I know, arrangement does not matter in this case. So, maybe the paradox could be reworded with the half step being the first, etc. without changing the concept (???).