Thread: Does 0.9 repeated = 1?

Originally Posted by Kushna Mufeed

OK, now that this thread is reverting back to the immature insult throwing contest that is was before, I'll request for the THIRD time that it's locked again.

No, don't destroy the thread because of one immature little shit and an understandable response to the ridiculous way he acts. Just try to get the member banned. He is probably a troll and might not even be old enough to be a member here.

Originally Posted by Universal Mind

You have a point about convergence, but your unnecessary extra wording is laughable.

It's called being precise.

Originally Posted by Universal Mind

Do you claim infinity is a number?

No, infinity is not a real number.

Originally Posted by Universal Mind

If not, then why would infinite smallness be a number?

Axiomatically, it's 0.

Originally Posted by Universal Mind

I like how you dodged my questions and points about your obvious personality disturbance. How convenient. You know I'm right about it, don't you?

I'm trying to avoid the mudslinging you're so fond of.

This conversation has kind of steered a little, hasn't it? It is no longer about whether math defines that .9~=1, but it's about whether math can define anything...

Better to say nine tenths repeated...maybe

No, not nine tenths repeated since it's more than nine-tenths. One tenth more than nine-tenths to be precise.

A few members received messages regarding their behavior in the thread. Ease up on the personal attacks, guys. If you really feel the need to lay into each other, take it to PM. I encourage everyone to continue the discussion in a civil manner, but if I see more flaming in this thread, I will close it for everyone.

Originally Posted by drewmandan

It's called being precise.

... and irrelevant.

Originally Posted by drewmandan

No, infinity is not a real number.

Infinity is not a number at all, period. It is an expression of limitlessness, not a specific amount.

http://mathforum.org/dr.math/faq/faq.large.numbers.html

http://scienceblogs.com/goodmath/200...t_a_number.php

http://nrich.maths.org/public/viewer.php?obj_id=2756

Originally Posted by drewmandan

Axiomatically, it's 0.

It is virtually 0, apparently. My mind is still open to the possibility that the two are equivalent, but you have not convinced me.

Originally Posted by drewmandan

I'm trying to avoid the mudslinging you're so fond of.

Yet you iniated it. Axiomatically?

Just talk about the issue. This is too interesting of a thread to have it ruined.

x=0.999...
10x=9.999...
10x-x=9.999...-0.999...
9x=9
x=1
0.999...=1

Yay!

9(1/10)/1-(1/10)=1

I did it again!

1=9/9=9*1/9=9*0.111...=0.999...=1

And once more, wow!

This conversation has kind of steered a little, hasn't it? It is no longer about whether math defines that .9~=1, but it's about whether math can define anything...

Well to be honest that was proved yonks ago now.

The resulting conversation is interesting I'd say... that's the whole point of threads anyway IMO.

Originally Posted by spockman

x=0.999...
10x=9.999...
10x-x=9.999...-0.999...
9x=9
x=1
0.999...=1

Okay, THAT is a Hell of a good point. Did you come up with that? Based on your point that obviously has solid merit, 0 = 0.000...1, but that makes infinite smallness a number even though infinity is not a number. And how does any 9 in the infinite set of 9's ever get followed by a digit that is not a 9 but is instead a bump up to 10? Apparently it somehow does, but I still don't understand how. Again, that is parallel to the dilemma posed by Zeno's paradox, which was never cleared up in my calculus class. My metaphysics professor said he could not fully resolve it. I have never come across anybody who has. Even Zeno never resolved it.

Reality is a paradox?

...people have posted that about fifty times already in this thread??

It's an extremely well known proof.

Originally Posted by Xei

...people have posted that about fifty times already in this thread??

It's an extremely well known proof.

No, it is not extremely well known. You and Drew are trying to make it sound like we are talking about the Pythagorean Theorem or the area of a rectangle or something. What we are talking about is very obscure and obviously not completely cleared up, considering the questions I have asked that have not been cleared up, such as the ones you avoided from my last post.

I have not read all 12 pages of this thread, so please name at least ten of the posts where that equation was used. I must have missed them. What is the name of the proof, and who came up with it? Can you tell me that without Googling it?

This is a very interesting topic, a not completely resolved topic, and still a debatable topic on the whole. Most of us agree that there are certain forms of evidence that 1 = 0.999..., but the idea still has unfilled holes in it. Debate and discuss all you want, but please stop acting like the truth of the matter is household knowledge and that there are no issues to discuss. This is a very real issue, and as far as I can tell, nobody can clear up all of the very reasonable questions raised by it.

It's something I learned I think I learned at 11 or so in a book and then was taught it as part of the national curriculum at 13 or so. So yes, everybody who has been through very basic British schooling should have seen it. It's just a bit below Pythagoras' Theorem I'd say. Just read the thread if you want to see the proofs, there's about four links and a few posted proofs in the first two pages alone... including another one I posted involving the sum of a geometric series.

I am completely serious when I say I see no issue at all here. I didn't respond to your questions because they are literally without meaning. For example,

And how does any 9 in the infinite set of 9's ever get followed by a digit that is not a 9 but is instead a bump up to 10?

It isn't. It really is an infinite set of 9s. And this is an alternate way of writing 1. There's no need for it to be 'bumped up' because they're the same number; no more than 1 needs to be 'bumped down' to 0.999~. That's what an equals sign means. They're the same.

PS. Zeno was an idiot. He wasn't some sort of legendary mathematician, he just came up with a very easily solvable problem about a man and a tortoise and then rephrased it loads of times. Then he argued that he'd proved motion was impossible. He clearly wasn't that intelligent.

everybody who has been through very basic British schooling

Then maybe it's well known in Britian. I can't remember that proof being taught earlier than Jr. year of highschool.

Originally Posted by Xei

It's something I learned I think I learned at 11 or so in a book and then was taught it as part of the national curriculum at 13 or so. So yes, everybody who has been through very basic British schooling should have seen it. It's just a bit below Pythagoras' Theorem I'd say.

Will you forgive me if I don't believe you?

Originally Posted by Xei

Just read the thread if you want to see the proofs, there's about four links and a few posted proofs in the first two pages alone... including another one I posted involving the sum of a geometric series.

I still haven't seen that, but maybe it's somewhere in the 12 pages. This is a long thread.

Originally Posted by Xei

I am completely serious when I say I see no issue at all here. I didn't respond to your questions because they are literally without meaning. For example,

It isn't. It really is an infinite set of 9s. And this is an alternate way of writing 1. There's no need for it to be 'bumped up' because they're the same number; no more than 1 needs to be 'bumped down' to 0.999~. That's what an equals sign means. They're the same.

Having no meaning is not synonymous with having a premise you disagree with. 1 is reached from something that infinitely approaches it but never has a step of reaching it. That is a paradox, and it remains unresolved. It is perplexing, and I think you agree with me. There is evidence that 1 is reached, but questions remain unanswered.

Originally Posted by Xei

PS. Zeno was an idiot. He wasn't some sort of legendary mathematician, he just came up with a very easily solvable problem about a man and a tortoise and then rephrased it loads of times. Then he argued that he'd proved motion was impossible. He clearly wasn't that intelligent.

Zeno was an idiot? Not quite. Why is he studied in metaphysics classes at top notch universities? Why can't the professors resolve his paradox? If you can, please do.

I am going to bump up a Zeno thread. Please clear it all up for us.

Yeah, if you're going to call Zeno an idiot for thinking motion impossible, you might as well call any philosopher who thought time wasn't real an idiot too.

Will you forgive me if I don't believe you?

I don't care what you think.

I still haven't seen that, but maybe it's somewhere in the 12 pages. This is a long thread.

I said first two. Yeah? You'll find the first posted version on the second page by Roxxor. My sum proof is shortly afterwards.

Having no meaning is not synonymous with having a premise you disagree with. 1 is reached from something that infinitely approaches it but never has a step of reaching it. That is a paradox, and it remains unresolved. It is perplexing, and I think you agree with me. There is evidence that 1 is reached, but questions remain unanswered.

I don't agree with you slightly??

I can assure you that I have no problem with this at all.

As has been said by Drew many times, 0.999~ does not 'approach' 1 because numbers are static objects, not iterative processes or whatever you're thinking of. This whole 'step of reaching it' literally has no meaning. 0.999~ isn't a series of steps, it's a number, equal to 1. As you indicated in your last post, you seem to be looking for some point at which a 9 digit is 'bumped up' to a 10 in order for it to become 1.000~. This isn't necessary because .999~ is just another way of writing 1. Hence the equals sign.

1 isn't 'reached', and there's no such thing as evidence in maths. There's proof. Try and disprove it if you want (although that will be futile, as it has been proved and something can't simultaneously be true and untrue...).

Zeno was an idiot? Not quite. Why is he studied in metaphysics classes at top notch universities? Why can't the professors resolve his paradox? If you can, please do.

What?? So everybody philosopher studied in a 'top notch university' class is highly intelligent and correct (even though different philosophers have absolutely mutually exlcusive ideas about things)? Personally I think the entirity of thelogical arguments were created by absolute divvies; they still teach them at Harvard.

The professors probably can't resolve his paradox because they lack basic maths skills. Don't bother bumping something, this thread'll suffice.

Zeno's best known version of his only original (yet still stupid) thought is that of a man chasing a tortoise. As soon as he gets to where the tortoise was, it has moved on a little further, and so on until infinity. Hence the man never catches the tortoise.

Well first off it's clearly wrong because I've overtaken tortoises in races many times without issue.

And the solution to the paradox (hence removing its status as such); each step takes a smaller and smaller amount of time, tending to 0.

Let's say each time the man gets were the tortoise was, it has moved on another half the distance. Let's also say that the first time this happens takes 1 second. The next amount of time will be 1/2 a second as he has to go half as far. To sum the infinity of steps,

∞
Σ 1/2^n = 1 / (1-1/2) = 1 / 1/2 = 2
0

So it takes 2 seconds. Not infinite.

I'm guessing there's still something wrong. And I'm also guessing that you wont be able to state it without vague references to things which make no sense.

Yeah, if you're going to call Zeno an idiot for thinking motion impossible, you might as well call any philosopher who thought time wasn't real an idiot too.

How can I possibly comment if I don't know what their logical arguments were?

Originally Posted by Xei

I can assure you that I have no problem with this at all.

As has been said by Drew many times, 0.999~ does not 'approach' 1 because numbers are static objects, not iterative processes or whatever you're thinking of. This whole 'step of reaching it' literally has no meaning. 0.999~ isn't a series of steps, it's a number, equal to 1. As you indicated in your last post, you seem to be looking for some point at which a 9 digit is 'bumped up' to a 10 in order for it to become 1.000~. This isn't necessary because .999~ is just another way of writing 1. Hence the equals sign.

1 isn't 'reached', and there's no such thing as evidence in maths. There's proof. Try and disprove it if you want (although that will be futile, as it has been proved and something can't simultaneously be true and untrue...).

It is a matter of approach from the standpoint that the succession of 9's has components, and each component is closer to one than the previous one. I don't mean the number is in motion or anything. I just mean there is a 9 after the decimal, then another 9, then another 9, etc. A hyperbola "approaches" an asymptote, but that does not mean the hyperbola is in motion. The 9's continue forever, and at no point is there a digit that makes the number reach the status of 1 even though 1 is apparently (?) reached. That is what makes the scenario a paradox. As I said, there is evidence that 0.999... = 1, but that fact causes the scenario to have a paradoxical nature.

Originally Posted by Xei

What?? So everybody philosopher studied in a 'top notch university' class is highly intelligent and correct (even though different philosophers have absolutely mutually exlcusive ideas about things)? Personally I think the entirity of thelogical arguments were created by absolute divvies; they still teach them at Harvard.

No, not "correct". "Intelligent"? Yes. I agree with the philosophies of very few famous philosophers, but I would not call any of them "idiots". You remind me of that guy in The Princess Bride who said that Aristotle and Socrates were morons compared to him.

Originally Posted by Xei

Zeno's best known version of his only original (yet still stupid) thought is that of a man chasing a tortoise. As soon as he gets to where the tortoise was, it has moved on a little further, and so on until infinity. Hence the man never catches the tortoise.

Well first off it's clearly wrong because I've overtaken tortoises in races many times without issue.

And the solution to the paradox (hence removing its status as such); each step takes a smaller and smaller amount of time, tending to 0.

Let's say each time the man gets were the tortoise was, it has moved on another half the distance. Let's also say that the first time this happens takes 1 second. The next amount of time will be 1/2 a second as he has to go half as far. To sum the infinity of steps,

∞
Σ 1/2^n = 1 / (1-1/2) = 1 / 1/2 = 2
0

So it takes 2 seconds. Not infinite.

I'm guessing there's still something wrong. And I'm also guessing that you wont be able to state it without vague references to things which make no sense.

How can I possibly comment if I don't know what their logical arguments were?

First of all, Zeno was not arguing that the man in reality cannot possibly pass the tortoise or that motion really is impossible. He presented a "paradox", which means he presented what appears to be a contradiction in a matter that does exist in reality.

The man is moving toward the tortoise as the tortoise is moving. The man is at point A, and the Tortoise is at point B. The man must get to point B before he gets to the tortoise, but when the man is at point B, the tortoise has moved ahead to point C. Then the man must get to point C before getting to the tortoise, but by then, the tortoise is at point D. This goes on infinitely, so how does the man ever reach the previous tortoise point AND the tortoise SIMULTANEOUSLY when the tortoise is constantly moving? Obviously he does, but that does not mean you can explain how it happens. How does that step ever happen? Tell me. Thanks.

I can't believe you don't find this stuff interesting.

The 9's continue forever, and at no point is there a digit that makes the number reach the status of 1 even though 1 is apparently (?) reached.

Oh for goodness sakes this is the third time you've asked exactly the same thing. The 9s do continue forever. There are two ways of writing a number, one with an infinite series of 9s and one with an infinite series of 0s.

No, not "correct". "Intelligent"? Yes. I agree with the philosophies of very few famous philosophers, but I would not call any of them "idiots". You remind me of that guy in The Princess Bride who said that Aristotle and Socrates were morons compared to him.

There are many people vastly more intelligent than me, but Zeno was probably not one of them. His paradox is supremely easy to solve.

The man is moving toward the tortoise as the tortoise is moving. The man is at point A, and the Tortoise is at point B. The man must get to point B before he gets to the tortoise, but when the man is at point B, the tortoise has moved ahead to point C. Then the man must get to point C before getting to the tortoise, but by then, the tortoise is at point D. This goes on infinitely, so how does the man ever reach the previous tortoise point AND the tortoise SIMULTANEOUSLY when the tortoise is constantly moving? Obviously he does, but that does not mean you can explain how it happens. How does that step ever happen? Tell me. Thanks.

Because a it's a converging geometric series.

There are an infinite series of steps, so you can assign no one step the status of the 'last' step.

Originally Posted by Xei

Oh for goodness sakes this is the third time you've asked exactly the same thing. The 9s do continue forever. There are two ways of writing a number, one with an infinite series of 9s and one with an infinite series of 0s.

The third time? Then this is the third time you have not answered my question. I know the 9's continue forever. That is exactly my point. They continue forever. In continuing forever and ever and ever, they are still just 9's, each of which represents "not quite 1". Do you see the paradox?

Originally Posted by Xei

There are many people vastly more intelligent than me, but Zeno was probably not one of them. His paradox is supremely easy to solve.

Then why has nobody ever done it?

Originally Posted by Xei

Because a it's a converging geometric series.

That gives a broad explanation of the phenomenon on the whole, but it does not answer my specific question. We are looking at converging geometric series under a microscope as it applies to motion. There is a last step, and it is when the man reaches the tortoise. Right? Well, that point is also one of the previous tortoise points. Right? The paradox is that the man can never reach the tortoise without reaching the previous tortoise point first, and the tortoise stays in motion. Yet, the man reaches the tortoise, so the tortoise is in a position that is ALSO a previous tortoise point. Apparently the distance between the tortoise and that last previous tortoise point is infinitely small.

I know that converging geometric series is the explanation, but I am asking you to look further into the situation. The man always reaches the previous tortoise point before he can reach the tortoise, and the tortoise is always moving away from the last tortoise point. Get it? Tortoise and previous tortoise point end up converging, just like an object falls and hits the ground even though it goes half way to the ground, then half that distance, then half that distance, infinitely (another version of Zeno's Paradox). But I have never come across a detailed enough explanation that satisfactorily answers how any of this happens.

The third time? Then this is the third time you have not answered my question. I know the 9's continue forever. That is exactly my point. They continue forever. In continuing forever and ever and ever, they are still just 9's, each of which represents "not quite 1". Do you see the paradox?

Ughhh I don't believe it. They don't represent not quite one. They represent infinite 9s. Which is a way of writing 1.

Then why has nobody ever done it?

They have, they solved it back then and it's been thoroughly done several times since. Including by me just then.

That gives a broad explanation of the phenomenon on the whole, but it does not answer my specific question. We are looking at converging geometric series under a microscope as it applies to motion. There is a last step, and it is when the man reaches the tortoise. Right? Well, that point is also one of the previous tortoise points. Right? The paradox is that the man can never reach the tortoise without reaching the previous tortoise point first, and the tortoise stays in motion. Yet, the man reaches the tortoise, so the tortoise is in a position that is ALSO a previous tortoise point. Apparently the distance between the tortoise and that last previous tortoise point is infinitely small.

So you're saying the paradox is that there must come a step when the man moves to the tortoise's previous position and also to the tortoise's new position?

Well, for a start that's not how Zeno stated it. But okay.

I'll rerespond to the percieved problem; there are an infinite number of steps. Hence there is no last step to which you can assign this property.

This is analogous to the fact that there is no last 9 in 0.999~.

Originally Posted by Xei

So you're saying the paradox is that there must come a step when the man moves to the tortoise's previous position and also to the tortoise's new position?

Well, for a start that's not how Zeno stated it. But okay.

I'll rerespond to the percieved problem; there are an infinite number of steps. Hence there is no last step to which you can assign this property.

This is analogous to the fact that there is no last 9 in 0.999~.

Yes, that's the issue. As I said in the other thread, your explanation is about as good as one can be, and I agree that covering infinity is what makes the phenomenon possible. I don't agree with your point that there is no last step, however. The last step is when the man catches up with the tortoise. It seems to be the case that the last previous tortoise point would be an infinitely small distance before the tortoise, and if infinite smallness is 0, then the tortoise and the previous tortoise point are the same point. Like in the other problem we have been talking about, Zeno's Paradox seems to prove that 0 and infinite smallness are the same. But of course that raises issues like the ones I have discussed.

If you think Zeno's Paradox (and the like) is such a non-issue, why do you think it is studied in metaphysics classes, and why are the professors not taught the explanation even though they teach the paradox?

There's plenty of non-issues studied in philosophy. And like I said, some philosophers simply don't have the sufficient mathematical skills to understand these subtle concepts.

Originally Posted by Xei

There's plenty of non-issues studied in philosophy. And like I said, some philosophers simply don't have the sufficient mathematical skills to understand these subtle concepts.

I'm talking about modern day professors. A lot of silly issues are covered in philosophy classes, but Zeno's Paradox seems to be one of the more respected and baffling issues covered even though the professors have plenty of access to the explanations given. They might not understand the math well enough, but it seems that they would have come across a mathematician they respect who gives an explanation they can at least trust to some extent to have merit.

I have another idea to bring up about this stuff. It is the idea that I first ran into in this department. Let's say that instead of the man and the tortoise being in constant motion, the man and the tortoise stop for a moment every time the man reaches one of the previous tortoise points. Would he ever reach the tortoise? Also, if the falling object were to somehow stop for a moment every time it moves half way to the ground from a previous stopping point, would it ever reach the ground? The answer seems to be that they both would reach their ends after an infinite number of moves. Could that ever happen? The other scenarios involve the same principle, except the moments last 0 time units. And obviously they reach their ends in that situation.