Zeno's Paradox

[The Sum of Awe]

Has anyone ever solved it? What is the solution? It's interesting indeed, I was just reminded of it today when reading a book, but I knew it before then, but not as good.

Basically, you can always cut movements in half, like half of a mile would be 6 feet, half of that would be 3 feet, and you can keep going on half and half forever, and you cannot move an infinite amount of times in a finite matter of time.

 

[Meow Mix]

Has anyone ever solved it? What is the solution? It's interesting indeed, I was just reminded of it today when reading a book, but I knew it before then, but not as good.

Basically, you can always cut movements in half, like half of a mile would be 6 feet, half of that would be 3 feet, and you can keep going on half and half forever, and you cannot move an infinite amount of times in a finite matter of time.

Yes, it's been solved for a long time. It's solved with limits and the calculus: an infinite sum with continually diminishing components converges to a finite number.

E.g., (I can't use math symbols on my work computer, so E = sigma)

(At infinity, where n = 1) E 1/2^n = {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...} = 1

In other words, it doesn't really take infinite time to cross the space because it only takes some finite number to cross the infinite divides.

 

[The Sum of Awe]

Yes, it's been solved for a long time. It's solved with limits and the calculus: an infinite sum with continually diminishing components converges to a finite number.

E.g., (I can't use math symbols on my work computer, so E = sigma)

(At infinity, where n = 1) E 1/2^n = {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...} = 1

I don't understand, I've only made it to pre-algebra

What does that mean basically?

 

[religion99]

Has anyone ever solved it? What is the solution? It's interesting indeed, I was just reminded of it today when reading a book, but I knew it before then, but not as good.

Basically, you can always cut movements in half, like half of a mile would be 6 feet, half of that would be 3 feet, and you can keep going on half and half forever, and you cannot move an infinite amount of times in a finite matter of time.

It is such a coincidence that I was reading about Zeno's paradox for last three months. I didn't dare talk about it here , because I thought nobody will understand it.

If you don't mind , can you share with us the book you were reading?

 

[The Sum of Awe]

It is such a coincidence that I was reading about Zeno's paradox for last three months. I didn't dare talk about it here , because I thought nobody will understand it.

If you don't mind , can you share with us the book you were reading?

Ah, it was just a bit of detail on it, not much information to pick up off of it if that's what you were looking for.

I forgot the name of the book, I'll bring it home tomorrow, but it is a book about many written philosophies in the history of mankind.

 

[Meow Mix]

I don't understand, I've only made it to pre-algebra

What does that mean basically?

Well, let's say I'm trying to cross a meterstick. To get to 1/2 the meterstick I have to pass 1/4. To pass the 1/4 I have to have already passed the 1/8. So on and so forth. Zeno says that it must take infinite time for me to do this because I'll never reach the 1/2, because I'll have to pass the 1/4 and the 1/8 and the 1/16 (etc., forever).

But the point is that it doesn't take infinite time to do that. It's just counterintuitive: the distance that I have to cross to get to the other side of the meterstick is {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...}, meaning I keep adding halves forever. You don't have to add them one at a time, though: it turns out that 1/2 + 1/4 + 1/8 + 1/16 (etc.) forever equals exactly 1. 1 is a finite number, so I can cross the meterstick in finite time, so there is no infinity to cross after all!

This is like the counterintuitive truth that 0.999... (where the 9's go on forever) = 1. It's the same concept: 0.9 + 0.09 + 0.009 = 0.999, right? Well, if you add 0.9 + 0.09 + 0.009 + 0.0009 + ... (keep adding 9's FOREVER) it turns out that it equals one exactly: an infinite sum of sequentially smaller numbers turns out to, in these special cases, equal a finite number. Thus to try to cross a distance measured by such infinite sums of numbers isn't to actually cross an infinity; it's to cross a finite distance just as makes sense to us in the everyday world

 

[The Sum of Awe]

Well, let's say I'm trying to cross a meterstick. To get to 1/2 the meterstick I have to pass 1/4. To pass the 1/4 I have to have already passed the 1/8. So on and so forth. Zeno says that it must take infinite time for me to do this because I'll never reach the 1/2, because I'll have to pass the 1/4 and the 1/8 and the 1/16 (etc., forever).

But the point is that it doesn't take infinite time to do that. It's just counterintuitive: the distance that I have to cross to get to the other side of the meterstick is {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...}, meaning I keep adding halves forever. You don't have to add them one at a time, though: it turns out that 1/2 + 1/4 + 1/8 + 1/16 (etc.) forever equals exactly 1. 1 is a finite number, so I can cross the meterstick in finite time, so there is no infinity to cross after all!

This is like the counterintuitive truth that 0.999... (where the 9's go on forever) = 1. It's the same concept: 0.9 + 0.09 + 0.009 = 0.999, right? Well, if you add 0.9 + 0.09 + 0.009 + 0.0009 + ... (keep adding 9's FOREVER) it turns out that it equals one exactly: an infinite sum of sequentially smaller numbers turns out to, in these special cases, equal a finite number. Thus to try to cross a distance measured by such infinite sums of numbers isn't to actually cross an infinity; it's to cross a finite distance just as makes sense to us in the everyday world

Ah thank you.

 

[9-10ths_Penguin]

Has anyone ever solved it? What is the solution? It's interesting indeed, I was just reminded of it today when reading a book, but I knew it before then, but not as good.

Basically, you can always cut movements in half, like half of a mile would be 6 feet, half of that would be 3 feet, and you can keep going on half and half forever, and you cannot move an infinite amount of times in a finite matter of time.

Here's one way to solve it: if you want to go 6 feet, and you move in increments that are each half the length of the previous one, then just start out to go 12 feet and stop when you're halfway there.

The thing that always bugged me about Zeno's paradox is that effectively implies that the universe changes depending on how we look at it. Why would the impersonal laws of motion care whether you're intending to go six feet, six miles, or or six parsecs? Zeno's paradox assumes that movement is possible; the rest is just a matter of subjective point of view.

Another way of phrasing Zeno's "paradox" is to say that because we can express any arbitrary finite value as the sum of an infinite series, this has some implication for reality. Well, we don't have to express any value as an infinite series, do we?

 

[Acim]

It's just counterintuitive: the distance that I have to cross to get to the other side of the meterstick is {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...}, meaning I keep adding halves forever. You don't have to add them one at a time, though: it turns out that 1/2 + 1/4 + 1/8 + 1/16 (etc.) forever equals exactly 1. 1 is a finite number, so I can cross the meterstick in finite time, so there is no infinity to cross after all!

This is like the counterintuitive truth that 0.999... (where the 9's go on forever) = 1. It's the same concept: 0.9 + 0.09 + 0.009 = 0.999, right? Well, if you add 0.9 + 0.09 + 0.009 + 0.0009 + ... (keep adding 9's FOREVER) it turns out that it equals one exactly: an infinite sum of sequentially smaller numbers turns out to, in these special cases, equal a finite number.

I am not understanding why it turns out to equal 1 exactly.

I actually want to agree with what you're saying, partially based on how you are saying it and how I've thought about it before. But it feels like leap in logic given the math you've presented (here). It is seeming to me that it would forever be just (ever so slightly) short of 1 when I review the math you've provided.

Is there anything more you can explain? Or anything you think I'm running into (as error) that you can help me overcome?

As much as there may be debate to be had, I am seriously seeking agreement, not disagreement.

Thanks.

 

[Meow Mix]

I am not understanding why it turns out to equal 1 exactly.

I actually want to agree with what you're saying, partially based on how you are saying it and how I've thought about it before. But it feels like leap in logic given the math you've presented (here). It is seeming to me that it would forever be just (ever so slightly) short of 1 when I review the math you've provided.

Is there anything more you can explain? Or anything you think I'm running into (as error) that you can help me overcome?

As much as there may be debate to be had, I am seriously seeking agreement, not disagreement.

Thanks.

Whether or not 0.999... = 1 is one of the most common questions encountered by calculus tutors and professors by students -- you're not alone in your confusion over this.

The problem you're experiencing is the same as Zeno's paradox: when you're trying to conceive of 0.999... as a number, you're conceiving it as 0.9 + 0.09 + 0.009 + ... as if it's added piece by piece over time, rather than conceiving it as 0.999... with infinite 9's all at once.

Yes, you would always be short of 1 if you started at 0.9 and just added from there piece by piece: but that's not what the number 0.999... represents. It represents a number with infinite 9's after the decimal: they're already there, you don't have to add them piece by piece infinitely.

There are many, many, many proofs professors have come up with to convince their students that 0.999... = 1 even before teaching them exactly why they're the same number (just expressed differently); i.e. by teaching them the calculus. If this interests you, then I encourage learning calculus because it can be interesting and fun contrary to popular opinion! But, here is a neat way to think about it even without the calculus:

1/3 = 0.333...

This is easier to accept, right? 1/3 is just another way of expressing 0.333... because they're the same number, just expressed differently. They both represent the number you have when one is divided by three.

Well, 1/3 * 3 = 1, right? If I have a third, and I multiply it by three, I'll end up with one whole.

So, what is 0.333... * 3?

You'll get 0.999...

But remember that 1/3 = 0.333..., and that 1/3 * 3 = 1.

That means that 0.333... * 3 = 0.999..., which must also = 1!

 

[Acim]

I've read what you've written and appreciate the explanation.

There's how I understand what you're saying which I'll get at (quickly) and then there's the debate that is philosophical. I desire to go there, but will admit there is relative inequality at work (that I think goes two ways, you to me, and me to you). IOW, we may both show up as having convictions that are talking over the head of the other, but is something we 'completely' believe / understand.

How I understand what you're saying is: what we commonly refer to as "1" is an approximation and is equal to .999....

That may not be exactly how you are explaining it, and yet is what I'm compelled to stand by, until something 'more logical' presents itself.

From this point on in this post, I am looking for philosophical debate on what you wrote. Which stems from the approximation understanding and goes into more detail.

Whether or not 0.999... = 1 is one of the most common questions encountered by calculus tutors and professors by students -- you're not alone in your confusion over this.

The problem you're experiencing is the same as Zeno's paradox: when you're trying to conceive of 0.999... as a number, you're conceiving it as 0.9 + 0.09 + 0.009 + ... as if it's added piece by piece over time, rather than conceiving it as 0.999... with infinite 9's all at once.

Be that as it may, it strikes me as illogical to conceive of it as .999 (with infinite 9's) and not see that as approximation of 1. Such that, if I saw 'something' that was .999 (with say 100 nines), I would think I would not have any noticeable or discernible difference between that and the full version. And arguably the full version could always be the .999 variation, and I may never know. Like if you say to me, here is 1 tomato. That may be the .999 version you are giving me, and I would rather easily take that to be "1 tomato."

Yes, you would always be short of 1 if you started at 0.9 and just added from there piece by piece: but that's not what the number 0.999... represents. It represents a number with infinite 9's after the decimal: they're already there, you don't have to add them piece by piece infinitely.

I really think I get this but I also acknowledge that from your perspective, you are convinced I'm not getting this fully. And is where there could be gap of debate that I'm not sure how we'd get around this.

There are many, many, many proofs professors have come up with to convince their students that 0.999... = 1 even before teaching them exactly why they're the same number (just expressed differently); i.e. by teaching them the calculus. If this interests you, then I encourage learning calculus because it can be interesting and fun contrary to popular opinion!

This just strikes me as teaching someone that the approximation is the same thing as the 'full thing' by more or less convincing the student that what you always interpreted as 'full thing' was (and always will be) an approximation.

But, here is a neat way to think about it even without the calculus:

1/3 = 0.333...

This is easier to accept, right? 1/3 is just another way of expressing 0.333... because they're the same number, just expressed differently. They both represent the number you have when one is divided by three.

I disagree they are the same number.

1/3 = 1 part of 3. Such that I can see that 1 + 1 + 1 = 3.

.333 = an approximate relation to 1 that goes on infinitely, and I would say technically (logically) is not equal to 1/3. I realize for some it is equal.

There is part of me that wishes to say if .999... equals 1, that arguably .333... could be equal to 1. Not sure how I would prove this other than to take the proof for .999 = 1, and then manipulate that to fit what I'm getting at. For I am saying .999 equaling 1 strikes me as manipulating (pure) logic.

Well, 1/3 * 3 = 1, right? If I have a third, and I multiply it by three, I'll end up with one whole.

Agreed, though I think of it as equaling 3, to be honest. And that 3 is a whole number.

This is also where I want to make the point that all whole numbers are equal to 1 (in essence) even while in some method of me proving that, I'm not sure how I would do this. Perhaps someone else has already done that, and I'm off the hook for having to try and prove it. I understand how we traditionally say (and you have said) 1/3 * 3 = 1, though with logic that is this thread, I would admit that truly understanding 1/3rd is challenging if relating it to 1.

So, what is 0.333... * 3?

You'll get 0.999...

But remember that 1/3 = 0.333..., and that 1/3 * 3 = 1.

That means that 0.333... * 3 = 0.999..., which must also = 1!

This again assumes that we all agree that .333.... equals 1/3rd. I do not believe it does, and think with basic math that .333... * 3 equals (as you said) .999.... which is an approximation of 1, but via pure logic is not 1, and is less than 1.

/debate.

 

[Meow Mix]

Acim,

The heart of the issue is that you think of 0.999... as an approximation of 1, and you think of 0.333... as an approximation of 1/3.

Maybe this stems from the fact that normally when we use decimals we're trying to describe numbers between whole numbers: 1.1 is slightly above 1 (but approximate to it), for instance.

This is just a convention, though. There are many ways to write the exact same number. I can write the number four as "4," as "(2 + 2)," as "(5 - 1)", as "8/2," as "4.000...," as "11" (in binary), and yes, even as "3.999...". All of these numbers are the exact same number -- not approximations of some number, but exactly the same one.

Likewise, 1/3 is exactly the same number as (0.333...). The latter isn't just an approximation of 1/3, it's just another way of writing that same number: much the same way that I can say "three" or I can say "tres" (in Spanish) and mean exactly the same thing.

Decimals are only approximations when we're taking a measurement: if I have a meterstick and I measure something as being 5.5 centimeters, it's actually going to be ~5.50 centimeters (with the last zero representing the approximation that it's about exactly 5.5 cm.

0.3 is approximately 1/3, and 0.33 is closer to approximating 1/3, and 0.33333333333333333333333 is a very close approximation to 1/3, but 0.333... is exactly the same number as 1/3. It has to be an infinite series to be exact, though: if I have 0.333 (to the googleplexth place with 3's, but not infinite) then that is indeed only an approximation -- but with an infinity of 3's after the decimal, that number is exactly 1/3: not because the math works out that way (though it does) but simply because it's just another way of writing the same number.

In fact, 1/3 is the same as 0.333... is the same as 0.010101... (in binary). Different ways of writing the same number, not an approximation.

 

[Acim]

Acim,

The heart of the issue is that you think of 0.999... as an approximation of 1, and you think of 0.333... as an approximation of 1/3.

It isn't just that I think it, I am saying philosophically, it is this way.

Maybe this stems from the fact that normally when we use decimals we're trying to describe numbers between whole numbers: 1.1 is slightly above 1 (but approximate to it), for instance.

This is just a convention, though.

It is partially (perhaps mostly) the convention of it, but also the actuality of knowing 1. Again, it is like if you give me "1 tomato." That is also, to large degree, convention. While there is case to be made (at some level I would think) that it is 'actually 1.' But how would I / we know that? How would I know that what you gave me, isn't in fact .994 of a tomato? Likewise, and yet a tad more illogical, how I would I know that what was given me wasn't 1.003 of a tomato? I may never know, and the accuracy won't matter for sake of convention. I'm happy to have 1 tomato, and from this relative understanding I may quantify 5 tomatoes or any number relative to what I understand as 1.

There are many ways to write the exact same number. I can write the number four as "4," as "(2 + 2)," as "(5 - 1)", as "8/2," as "4.000...," as "11" (in binary), and yes, even as "3.999...". All of these numbers are the exact same number -- not approximations of some number, but exactly the same one.

I cannot go along with 'exact same' on all of them, and for sure not on the 3.999 one. That, being what I think this thread is about, still shows up to me as less than 4.

The equating of concepts to 4, I understand because of language that math is. But it becomes impure language the more the concepts are seen (or made out to be) exact same when the appearance is 'quite different.' Seems far more accurate to say, relatively the same, with stipulation that language (or convention) is presuming them to be same. Again, this to me is the philosophy part, and for much of what you're saying, I'm not really seeking deep debate, but on the 3.999 point, I am more interested in the debate, since it still strikes me as illogical to say that is exact same number as 4. Or as illogical as saying that 4 = 1, since we are talking about one whole number.

Likewise, 1/3 is exactly the same number as (0.333...). The latter isn't just an approximation of 1/3, it's just another way of writing that same number: much the same way that I can say "three" or I can say "tres" (in Spanish) and mean exactly the same thing.

And I'm still saying that 1/3rd is not the same as .333... logically. In terms of convention that is not apparently concerned with accuracy, it is similar enough that I would rarely question it (aloud) because the times it may come up for me, are impractical to question it. In this thread, I am saying it is not the same, logically speaking. You telling me it is the same, is akin to me telling you that Universe is equal to God, just because it is.

Decimals are only approximations when we're taking a measurement: if I have a meterstick and I measure something as being 5.5 centimeters, it's actually going to be ~5.50 centimeters (with the last zero representing the approximation that it's about exactly 5.5 cm.

What I think you're not getting with what I'm saying is that 1 is a measurement, and the decimals are working in relation to that (approximation). If 1 is a known number (which I actually believe it is, but not sure how I can 'prove' that), then .333... multiplied by (whole number) 3 will not equal 1, but rather an approximation of 1, logically speaking.

0.3 is approximately 1/3, and 0.33 is closer to approximating 1/3, and 0.33333333333333333333333 is a very close approximation to 1/3, but 0.333... is exactly the same number as 1/3. It has to be an infinite series to be exact, though: if I have 0.333 (to the googleplexth place with 3's, but not infinite) then that is indeed only an approximation -- but with an infinity of 3's after the decimal, that number is exactly 1/3: not because the math works out that way (though it does) but simply because it's just another way of writing the same number.

Yet technically it is not a number that is written, and to whatever level it is being explained so far, it is utilizing a leap in logic. It is assuming student agrees that "oh okay, I see, .333... is equal to 1/3rd, now I can see the math." Minus that assumption, and it doesn't logically add up.

In fact, 1/3 is the same as 0.333... is the same as 0.010101... (in binary). Different ways of writing the same number, not an approximation.

Sorry, just not seeing it how you are saying it. And seeing it instead as, "God equals Universe. Different ways of writing the same concept, not anything other than exact same concept."

 

[Meow Mix]

Well, there is not really a way for me to argue this without simply teaching the calculus. It's not a big enough deal that I should press the matter, but I do encourage doing some study on limits, infinite series, and infinite sums.

It's counter-intuitive, but it is logically and philosophically correct that 0.999... = 1, and that it isn't just an approximation. Trying to explain why algebraically can't do it much justice; but this is exactly why it took until Newton and Leibniz to develop the calculus.

0.999... does intuitively look like it will always be less than 1, but this intuition is wrong.

If this is a topic that really does interest you then I can get into the basics of limits, infinite series and infinite sums enough to explain it; but it's really just easier to either do some research or take a calculus class. That has such an arrogant tone to it: "Go read a book," but I think you know I don't mean it that way.

EDIT: I noticed that wikipedia has an article on 0.999... that explains succinctly why it is exactly 1, and not an approximation of 1. Maybe start with that?

 

[Acim]

If this is a topic that really does interest you then I can get into the basics of limits, infinite series and infinite sums enough to explain it; but it's really just easier to either do some research or take a calculus class. That has such an arrogant tone to it: "Go read a book," but I think you know I don't mean it that way.

EDIT: I noticed that wikipedia has an article on 0.999... that explains succinctly why it is exactly 1, and not an approximation of 1. Maybe start with that?

I'll go with Wikipedia for now. Thanks for the link.

Before responding to you (initially), I consulted with someone with advanced knowledge of math (and calculus).

 

[Acim]

First algebraic proof (fractions and long division), I'm saying fails. For reasons I've already cited.

Will continue on.

 

[Acim]

Second algebraic proof (Digital manipulation), strikes me as leap in logic. As I noted before, I could get this to say other things. I.E.

x = .333...
10x = 3.333....
10x - x = 3.333... - .333....
3x = 3
x = 1
(therefore 1 = .333...)

 

[Acim]

Will pick up the rest on another day....

 

[Meow Mix]

I'll go with Wikipedia for now. Thanks for the link.

Before responding to you (initially), I consulted with someone with advanced knowledge of math (and calculus).

Not to toot my own horn, but I am a physics grad student. I may not have a bachelors in math but a B.S. in physics is about the same.

 

[Acim]

I trust your wisdom and enjoy the dialogue.

Just wish we could do without the B.S.