.9999... equals 1?

So what is the conclusion to this thread? Has everyone agreed as to what 0.9r is equal to?

So what is the conclusion to this thread? Has everyone agreed as to what 0.9r is equal to?

.999... = 1. It was concluded that before the thread even started, just ignorant, stubborn people chose to disagree.


~D-Draw

Really, you are all missing it entirely. Your minds are so narrowed upon human mathematics that you don't see actual reality. Mathematics are numbers and paper, that represent reality approximately.

This is a mathematical question.  The mathematics shows that 1 is exactly equal to 0.9 recurring.  There are a number of mathematical proofs that have been presented.  A mathematical proof is a fact within the formal system of mathematics.  Your argument about reality is stupid.  How often, in "reality," do you have to deal with equations including the number 0.9 recurring?  The thing about mathematical proofs is that you can't deny them with reason, you have to mathematically prove that they are incorrect.  In this case, that is impossible.

We are not infinite creatures, therefore our mathematics are not infinite; though in some equations we use symbols to represent infinite quantities. These infinite quantities are not actually truly usable in exact math, they would result in no solution in any formula that used them. But, to humans, such a difference does not matter. An infinitely small difference is the same as no difference, to us. We have margins which anything that falls into them is "exact" to us, though technically just approximately exact.

This is completely false.  Parts of mathematics would not exist without the concept of infinity.  The problem is that you don't understand infinity, or maths.  What is your training in maths?  What level have you studied?  Maths is exact and it's not a case of there being differences so small that the don't matter, rather, if you understand the maths, you'll see that there really is absolutely no difference at all.

In mathematics, the 0.999... is expressed as 1 because the difference makes no difference. 0.0...1 has no effect on humanity, as it is infinitely small and an infinitely small difference always falls within our margin of error.

What are you talking about?  In mathematics 0.9r is expressed as 1 because there are mathematical proofs that demonstrate that they are the same number.  These are not approximate proofs, and they have nothing to do with the effect on humanity, but rather, are facts within the mathematical formal system.  Nobody decided that because the differences are so small it didn't matter, they demonstrated, clearly, that the numbers are absolutely equal.

This does not change reality, just our rational human use of it. Reality is infinite, but humans experience it finitely.

You keep saying this, but can you actually provide any evidence that this is actually the case?  I don't think what you're saying is a fact at all, and it is misleading to continually present it as a fact, unless you have some evidence to back it up.  What you're expressing is your evidence, based on the fact that you have had no serious mathematical training.

Our perceptions are not able to directly experience reality, but do so through filters (our minds and sensory organs) that only picks up finite portions of the infinite reality; this means that our subjective reality is finite, but does not change the actualities of reality (they are simply irrelevant to us, part of a different world entirely separate from us).

If what you're saying is correct, which you have provided no evidence to suggest is the case, how would we see things, hear things, smell things, taste things or touch things differently if we could see reality?  What is your evidence that we do not experience reality?  How is the world, in factual terms, different from how we perceive it?  If you're going to make claims about the world, please put forward evidence to show that you're not just putting forward baseless speculation.

So, to us, 0.0...1 = 0, because we are finite and such a number has no effect on anything human; it is irrelevant; but in reality, 0.0...1 does exist, and we can reason that it does exist (everything is made up of infinite infinitely small quantities).

Please give an example of 0.0...1 existing in reality.  You claim that it does, so you must have seen a place where it does exist.  Or are you just speculating and presenting your speculation as fact?

This means that 0.0...1 does not truly equal 0, just that our only use for it is as a 0. The same is for 0.999, because of the same reason; the difference is so small it makes no difference, so we take it as 1 (but in reality it is not 1, just as close as it can possibly get to being 1; which makes no difference to us).

Mathematical proofs don't work like that.  They work by having a given fact, that is already accepted as true, and then using the formal rules of mathematics to show that a new fact has to be true if the previous fact was true. if 1/3 = 0.3 recurring, you have to accept that 0.9 recurring = 1.  Approximations are not made in the various mathematical proofs, they are dealing with facts.

I've beaten this dead horse to bits. It really isn't so complex. Do you understand yet?

What you've done is present a number of statements as facts, without providing any evidence of that they are actually facts.  You've failed to put forward and rebuttal of the actual mathematical proofs that have been presented, or of the fact that a number of mathematical bodies have demonstrated that they support the mathematical proofs.  You've given no indication of any serious knowledge or training in mathematics and then you've acted like those of us who do understand maths can't understand what you're saying.  We completely understand, it's just that all the actual observational evidence demonstrates conclusively that you're wrong, and there has been no evidence to suggest that you're right.

Rather than putting forward your speculative opinions on a subject that you have not received formal education in, why don't you give up this debate and go and learn about what you're talking about.  Just because you fail to understand something, it doesn't mean that it is wrong.  In fact your understanding of a concept has nothing to do with the factuality of that statement.  This is why I believe that it is crucial that we formulate our opinions based on things that we know are facts, rather than speculating and then behaving as if our speculations are correct.  Dogmatic thinking, as far as I can see, is the source of all the evil in the world.

Skeptic, I already conceded.

But I would still like an answer to this:

What is 2 x 0.9r? Is it 2?

Yes!  And 54 x 0.9 recurring = 54.

The reason a calculator will say something different is because you're not calculating 0.9r, but actually just 0.999999999 or however many digits you entered.  These are significantly different numbers.

Because we're dealing with an infinite number, work it out starting from the first digit, instead of the convention of the last.

2x 0.9 = 1.8
2x 0.99 = 1.98
2x 0.999= 1.998
...
2x 0.9999999999 = 1.9999999998

So we can see that as 0.9 approaches an infinite number of 9s, the answer also approaches an infinite number of 9s.  While technically our answer is 1.9r8, we have already proven that 0.9r = 1.  Therefore 1.9r must equal 2, regardless of what digits are left at the end.

Likewise, as 0.9 approaches an infinite number of 9s, we can see that that number times 54 approaches 53.9r46.  Since we know that 0.9r = 1, 0.9r46 must also equal 1.  Were we dealing with finite decimals, that would not be the case, but we're not.

The reason a calculator will say something different is because you're not calculating 0.9r, but actually just 0.999999999 or however many digits you entered.  These are significantly different numbers.

I just used the windows calculator and divided 1 by 3, which would give me .3r. But when I multiplied the result by 3 again, I got 1. Why? The calculator would not round .9r up to 1, so what happened there? I should have gotten .9r as an answer...

well I guess 0.9r=1 as long as 0.0r1=0, I'm just not clear how a positive number alone can equal anything else but a positive number.

But in that case 0.9r(equalling 1)+0.0r1(equalling 0)=1. then 0.9r(meaning1)+0.0r1=1. But since we know that 0.9r+0.0r1 really equals 1, and not 1.0r1, we find a problem, specifically that 1=1.0r1. Which again is all solved if 0.1r1=0, but it would have to extend to every instance of the amount, and not just exactly 0.0r1. This means 1.0r1•10^infinite would be just 10^infinite. So numbers get dropped somewhere. And I didn't think 1/3=0.3r anyway, isn't it 0.3r+0.3r+0.3r4=1?

well I guess 0.9r=1 as long as 0.0r1=0, I'm just not clear how a positive number alone can equal anything else but a positive number.

But in that case 0.9r(equalling 1)+0.0r1(equalling 0)=1. then 0.9r(meaning1)+0.0r1=1. But since we know that 0.9r+0.0r1 really equals 1, and not 1.0r1, we find a problem, specifically that 1=1.0r1. Which again is all solved if 0.1r1=0, but it would have to extend to every instance of the amount, and not just exactly 0.0r1. This means 1.0r1•10^infinite would be just 10^infinite. So numbers get dropped somewhere. And I didn't think 1/3=0.3r anyway, isn't it 0.3r+0.3r+0.3r4=1?

The only problem is that you can't have 1.0r1 in the first place.  Having another number after the repeating number implies that the repeating number comes to an end at some point for that non-repeating number to come in.

This is why the question cant be answered. Because 1-0.9r must equal a positive number. But this positive number can never be found.



Because if 0.9r=1, it also means 0.89r=0.9. And 0.79r=0.8. And 0.49r=0.5. So 0.49r•2 must equal one, right?

But it doesnt. 0.49r•2=0.9r8. Which doesnt equal 1.

so that is saying 0.5•2=/=1, when it does.

maybe it would be better explained if you divide 1.1r by 1, and you get 1, but if you divide 1.1r by 0.9r, you get something like 1.1r21 I think. It's really late here and I'm pretty tired.

double-post, since it really doesnt have much to do with my first, I think were toob usy thinking of the number 1 as definign a point and not defining an infinitely small field. This means 0.9r, 1.0r, and 1.0r1 all equal one, while 0.9r8, 0.9r, and 1.0r all equal 0.9r, 1.0r1, 1.0r, and 1.0r2 are equal, 0.0r, 0.0r1 and -0.0r1 are all equal, etc. Our definition of the poitn of a number is the center of one defining field, which is also where two defining fields overlap.

If you think that .9r does not equal 1, and you can't come up with a mathematical proof to show it, why do you keep arguing?  There is no such thing as 0r1 in mathematics!  It simply shows that you don't understand the concept of infinity. 

If you think that .9r does not equal 1, and you can't come up with a mathematical proof to show it, why do you keep arguing?  There is no such thing as 0r1 in mathematics!  It simply shows that you don't understand the concept of infinity. 

What I love about numbers is that you can make up any number you like and it exists. 0.0r1 certainly could exist. Good luck distinguishing it from 0 though.

There is no such thing as 0r1

On the theoretical number line which 0.9r is a point, 0.0r1 would be the very next point after zero. Also, on that same theoretical number line, 0.9r would be the number preceding one.

Quote from: RoundisWrong on March 10, 2007, 11:58:06 PM

There is no such thing as 0r1

On the theoretical number line which 0.9r is a point, 0.0r1 would be the very next point after zero. Also, on that same theoretical number line, 0.9r would be the number preceding one.

Even if you don't understand math you should be able to figure it out from basic logic.  0.0r1 cannot exist.  You can't have an infinitely long string of zeros with a 1 at the end because there is no end.  0.0r means that it repeats to infinite which means you can't have a 1 at the end because there is no end.  How can you put at one at the end of something that has no end?

Then what comes after 0 on a thoereticalnumber line where 0.9r is displayable?

Then what comes after 0 on a thoereticalnumber line where 0.9r is displayable?

First off, any number line that displays 1 is also displaying .999... Second, I don't think there is a number that "comes after" 1 in the sense that you mean. It's my understanding that in between any two rational numbers are an infinite amount of irrational numbers, therefore for any number you can give me I can give you a smaller one.
Of course, I haven't got the slightest idea what I'm talking about, so we'd better wait for Skeptical to get back on.

Right. There is no smallest number after zero. You don't even need to resort to irrationals to see that, since between any two numbers there are not only infinitely many irrationals, there are infinitely many rationals as well. This is pretty obvious in the case of rationals; because if you have any number bigger than zero, you can divide that number by two and get a smaller number bigger than zero.

Erebos, if you're talking about "actual reality" outside of human perception/thinking then you should not be talking (literally).  You should not even be thinking.  Thinking about something separate from yourself is impossible because everything you think and perceive is part of yourself.  We cannot think about anything we cannot think about.  We will never understand anything beyond the human capacity to understand. 

You are trying to define something that is impossible for humans to understand (by your very definition) and then calling us stupid for not understanding you.  How are you somehow outside of human perception, human mathematics (as if there is some other kind), and human understanding?  Maybe you think you are enlightened (or something) but regardless, there is no way to "talk" about such things and expect to say anything meaningful at all.

Also, saying mathematics is not useful in talking about maths and numbers is. . . beyond ridiculous.  Let me ask another question:  how do you know there is anything that is infinite? 

Quote from: kasroa on March 10, 2007, 03:48:47 PM

So what is the conclusion to this thread? Has everyone agreed as to what 0.9r is equal to?

.999... = 1. It was concluded that before the thread even started, just ignorant, stubborn people chose to disagree.


~D-Draw

I AGREEEEEEEEEEEEE

Why can you have o.9r9 but not 0.0r1?

Because .9r9 can reduce to .9r, whereas 0.0r1 is essentially meaningless because with the infinite amount of 0's there will NEVER be an end on which to put the 1. 

But there'll never be an end to put an extra 9 on 0.9r9

But there'll never be an end to put an extra 9 on 0.9r9

That's true, but that extra 9 is a redundant symbol; it can logically be reduced.  If you want to stick with 0.9r9 then yes, it also is meaningless. 

Why can you have o.9r9 but not 0.0r1?

You can't have either, or at least they're not decimal expansions. In theory you could think of them as "sequences" of digits indexed by the ordinal ω+1 instead of ω, but such things are not decimal expansions, and don't represent real numbers.

Because .9r9 can reduce to .9r

No, it can't. The expansion .9r9 is a valid decimal expansion, which represents the number 1, while .9r9 represents no number at all.

0.9r doesn't represent a real number either.

0.9r doesn't represent a real number either.

Yes, it does.

What does 0.111... represent?

Right. There is no smallest number after zero. You don't even need to resort to irrationals to see that, since between any two numbers there are not only infinitely many irrationals, there are infinitely many rationals as well. This is pretty obvious in the case of rationals; because if you have any number bigger than zero, you can divide that number by two and get a smaller number bigger than zero.

no, i didnt mean a number smaller than zero, I meant after as in next on the number line, greater than, etc. What is the very first number larger than zero, on a line where 0.9r is a viable, defined point, one step up from however you choose to name the number one step less than it, how I define 0.9r8.

After reading a bit more I've come to the conclusion that this all boils down to the fact that most people don't know what the definition of a decimal is. I didn't until I read this. To be honest, this debate is not something I've ever come across. If you'd have asked me before I would have said that 0.999r is so close to 1 that it may as well be. Obviously that answer doesn't make any sense now that I've read what a decimal is.

Instead of arguing about it and flexing their epeens, people should just explain the definition of the decimal in a civilised manner and that would end the discussion.

speaking of decimal points, isn't this like saying that 9r is equal to 10r?