Thread: Does 0.9 repeated = 1?

Originally Posted by Universal Mind

I am not asking for proof, 8th grader. I am asking for explanation. How many times do I need to explain that to you? I know that the sums are cauchy. I am looking for explanation, not labels or proofs. Give it your best shot.

Can you explain what you mean by "explanation"? I don't think a single person here knows what you're asking for.

Originally Posted by drewmandan

Can you explain what you mean by "explanation"? I don't think a single person here knows what you're asking for.

I am asking for how the paradox I have discussed is resolved, not mere labels and proofs that 1 = 0.9... I am asking for how, not if or "but this branch of math says it does". So far, you are just addressing those beside the point issues and making the circular argument that the two figures equal each other and therefore equal each other. I know you understand the paradox. Resolve the paradox without mere labels and confirmations. Go as deep as you possibly can.

Originally Posted by Universal Mind

I am asking for how the paradox I have discussed is resolved, not mere labels and proofs that 1 = 0.9... I am asking for how, not if or "but this branch of math says it does". So far, you are just addressing those beside the point issues and making the circular argument that the two figures equal each other and therefore equal each other. I know you understand the paradox. Resolve the paradox without mere labels and confirmations. Go as deep as you possibly can.

But the analysis proof isn't circular. It uses epsilons and deltas and all that crap.

Originally Posted by drewmandan

But the analysis proof isn't circular. It uses epsilons and deltas and all that crap.

Again, proof is unnecessary. We agree that 1 = 0.9... That is not at issue with me. I want to know why. The converging geometric series label only says that it does happen, not how it can happen.

Just look at it this way. Explain why converging geometric series are real (not prove that they are real), then give the reason behind that explanation, then give the reason behind that explanation, etc. Go as far as you can possibly go.

Then again, if you don't know how to resolve the paradox, you can say that. I don't know how to resolve it either. In fact, I don't think anybody who has ever lived does.

Originally Posted by Universal Mind

Again, proof is unnecessary. We agree that 1 = 0.9... That is not at issue with me. I want to know why. The converging geometric series label only says that it does happen, not how it can happen.

Just look at it this way. Explain why converging geometric series are real (not prove that they are real), then give the reason behind that explanation, then give the reason behind that explanation, etc. Go as far as you can possibly go.

Then again, if you don't know how to resolve the paradox, you can say that. I don't know how to resolve it either. In fact, I don't think anybody who has ever lived does.

Define "reason".

Do the 9's go on forever, or do they end? If they go on forever, then how do they stop at 1? If they end, how many 9's are in there?

Wow, the fifth time you've asked the same question...

The 9s go on forever. No 9 is ever bumped up to a 10, and such an event does not need to occur, because 0.999~ is just another way of writing 1. There's nothing special about 1 which requires the 0.999~ version to be bumped up to it, just as there's no reason that some part of 1 needs to be bumped down to ~0.999, or the two digits in 2/4 needs to be bumped down to 1/2.

Originally Posted by Xei

Wow, the fifth time you've asked the same question...

The 9s go on forever. No 9 is ever bumped up to a 10, and such an event does not need to occur, because 0.999~ is just another way of writing 1. There's nothing special about 1 which requires the 0.999~ version to be bumped up to it, just as there's no reason that some part of 1 needs to be bumped down to ~0.999, or the two digits in 2/4 needs to be bumped down to 1/2.

In other words, they are the same number because they are the same number. Thank you for the "fifth" time you have posted that circular argument which answers nothing. Again, saying "just another way of writing 1" is worthless. It does not resolve the paradox.

The paradox is that the 9's go on forever even though they lead to exactly 1. That is what is so bizarre about the fact that the two figures have the same value. Saying that we are just talking about two different ways of writing the same number does not clear up the issue. My issue concerns how they are the same number, not whether they are the same number, what label can be used to justify saying the same number, or what branch of mathematics says they are the same number. My issue is about exactly how they can be the same number in light of the paradox I keep describing. Do I need to explain that again?

The 9s go on forever. Yes. Which is another way of writing 1 (as has been proved by a couple of methods). Yes. What is the problem?

Originally Posted by Xei

The 9s go on forever. Yes. Which is another way of writing 1 (as has been proved by a couple of methods). Yes. What is the problem?

The walrus was Paul.

Originally Posted by Universal Mind

The paradox is that the 9's go on forever even though they lead to exactly 1.

http://en.wikipedia.org/wiki/0.999#S...m_in_education

Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1

Well at least you got the 12 year olds on your side.

you're asking for an explanation why 0.999... = 1

Imagine that you've never heard of the paradox. Now lay a line of 0.999~ before your feet. It stretches out to infinity. You can walk along the line for as long as you want, but the line never ends. You're asking for an explanation. The truth is, the 'proof' is the explanation. That's the way maths work.

Just like the rounding of the universe in the fourth dimension (not counting time) can you imagine that? Of course not!

Originally Posted by drewmandan

Well at least you got the 12 year olds on your side.

Nice answer, especially considering the fact that I accept that the two figures are equal, as I keep saying and you keep ignoring. Now the paradox has been explained away by your brilliant articulation of something to the effect of, "Uh, uh, you're like a 12 year old." Thanks for answering nothing, again.

Originally Posted by ThreeLetterSyndrom

you're asking for an explanation why 0.999... = 1

Imagine that you've never heard of the paradox. Now lay a line of 0.999~ before your feet. It stretches out to infinity. You can walk along the line for as long as you want, but the line never ends. You're asking for an explanation.

It doesn't stop at 1?

Originally Posted by ThreeLetterSyndrom

You're asking for an explanation. The truth is, the 'proof' is the explanation. That's the way maths work.

No, the proofs are not the explanations for the specific issue I have raised. The proofs we have seen prove that 0.999... = 1, not how a number with an infinite number of digits greater than 0 equals a whole number. All we have gotten are comments like, "Math says it does," "Here's another name for the phenomenon," and, "You're like a 12 year old." Everybody is stumped. Few people admit it.

Originally Posted by ThreeLetterSyndrom

Just like the rounding of the universe in the fourth dimension (not counting time) can you imagine that? Of course not!

Time is the fourth dimension.

Hmm. Let me give this one a shot. It's not an assamtote, since an assamtote keeps getting closer to it's targets in smaller and smaller increments,
(1/4+1/8+1/16,) so it would stand that an assamtote only approaches 1 and never is 1. .999~, however, doesn't have to get closer and closer to it's target since all the infinite nines are layed out right there instantly, inherently in the number, no progession required.

But having an infinite amount of nines can't be in itself a number unless it's equivalent to the next step up, since it's, I don't know, pushing the envelope? I don't think I'm expressing my thoughts ont he subject very well. Uhh... It's as if it's a puzzle with an infinite number of peices. You can't see all the peices, it seems as if the peices if put on one at a time could never complete the picture, but all the peices are together.

I recognize that there are major holes in my statement and that it probably doesn't make perfect sense, but it's the best that I can do. There, at least I gave it a shot. In full honesty, I don't really understand the HOW either.

Originally Posted by Universal Mind

It doesn't stop at 1?

Frankly, I can't comprehend the level of failure of your math education that could prompt such a question.

Originally Posted by drewmandan

Frankly, I can't comprehend the level of failure of your math education that could prompt such a question.

What you can't comprehend is that the question is meant to produce explanations of the full picture and not stump somebody into saying infinity actually stops. You can't explain away the paradox, and you know it because I have made it clear to you, and you are acting out your frustration by acting like a spoiled little 12 year old.

Originally Posted by spockman

Hmm. Let me give this one a shot. It's not an assamtote, since an assamtote keeps getting closer to it's targets in smaller and smaller increments,
(1/4+1/8+1/16,) so it would stand that an assamtote only approaches 1 and never is 1. .999~, however, doesn't have to get closer and closer to it's target since all the infinite nines are layed out right there instantly, inherently in the number, no progession required.

But having an infinite amount of nines can't be in itself a number unless it's equivalent to the next step up, since it's, I don't know, pushing the envelope? I don't think I'm expressing my thoughts ont he subject very well. Uhh... It's as if it's a puzzle with an infinite number of peices. You can't see all the peices, it seems as if the peices if put on one at a time could never complete the picture, but all the peices are together.

I recognize that there are major holes in my statement and that it probably doesn't make perfect sense, but it's the best that I can do. There, at least I gave it a shot. In full honesty, I don't really understand the HOW either.

Thank you! Not only does your post involve the best attempt at an explanation I have seen in this thread yet. It is also involves the second admission of being stumped. And you are the first person to recognize the issue I have raised. Plus, you didn't act like a childish little asshole who got grounded on Halloween, like some people I won't name. Cheers to you.

Originally Posted by Universal Mind

What you can't comprehend is that the question is meant to produce explanations of the full picture and not stump somebody into saying infinity actually stops. You can't explain away the paradox, and you know it because I have made it clear to you, and you are acting out your frustration by acting like a spoiled little 12 year old.

But no one said infinity "stops".

Originally Posted by Universal Mind

It doesn't stop at 1?

As far as I understand, it doesn't have to stop. In the same way that 0.333~ is the limit of the series "keep adding 3s to 0.333" which approaches 1/3, 0.999~ is the limit of the series "keep adding 9s to 0.999"... The series itself goes on forever.
So, uh, I hope that makes sense .

Originally Posted by drewmandan

But no one said infinity "stops".

It doesn't. Hence the paradox.

Originally Posted by Universal Mind

It doesn't. Hence the paradox.

1 is actually 1.00000... with an infinite number of zeros. Is that also a paradox?

Originally Posted by drewmandan

1 is actually 1.00000... with an infinite number of zeros. Is that also a paradox?

No. It is not the case that each 0 adds to the amount represented by the previous digits. The 0's do not approach a new value forever.

Originally Posted by Universal Mind

No. It is not the case that each 0 adds to the amount represented by the previous digits. The 0's do not approach a new value forever.

But 1 is the sum of the series 2 - 0.9 - 0.09 - ...

Originally Posted by drewmandan

But 1 is the sum of the series 2 - 0.9 - 0.09 - ...

You mean the difference. Yes, it is. What are you saying?

Originally Posted by Universal Mind

You mean the difference. Yes, it is. What are you saying?

That the 0's in 1.0~ are just as funky as the 9's in 0.9~.

Originally Posted by drewmandan

That the 0's in 1.0~ are just as funky as the 9's in 0.9~.

No. Again, the 0's do not approach a number forever.

0.999.... : Infinite approach toward 1, yet the number equals 1.

1.0: A whole number with digits that do not approach anything.

The two figures equal each other. That is a paradox.