Fair enough. It is quite interesting to note how long ago such matters were considered. My apologies for misconstruing the intentions of your post.
It is often in life that one must pay for his errors. It seems that all of you already
celebrate a victory over me. But it does come with a heavy price.
You have revealed your ignorance to me, your arrogance against clear and simple facts that I have presented. You are proud people. But what have you achieved?
What will you achieve? Of what ground does your proudness rise?
What is important to you is that to earn the respect of your colleagues, your
friends, other people. You may have to sacrifice truth in order to be respected
among other people. I don’t need your respect, I seek knowledge and truth.
Lies and ignorance are poor substitutes for them.
When things get more difficult you will have to pay for your errors. You don’t
anymore know what are you dealing with when I introduce infinity. I did not
talk about it yet because there was no need. You should have learned from
your mistakes when things were simple, it is maybe too late to learn anymore.
Let’s look at the number …99999.99999…what does it represent for you?
Do you have any idea what it is?
For you this number represents infinity, and you will end up at the formula
∞+1=∞
and when you insert your infinity into into this
∞=…99999.99999…=…99999 + 0.99999…= ∞ + 1
There it is, your infinities
∞=…99999.99999…
and
∞=…99999
Now, try and seek the errors in these. You should be able to see them. You have
told there are many knowledgeable people here. Lets hear them now.
Lets hear them talking about set theory, the complications and rigor of Cantor.
I would append onto that their incomprehension of, anger toward, and total bewilderment about the concept of infinity.They want common sense.
And they are very, very firm about this.
Perhaps different infinities are the solution.
Maybe there are at least 10 different infinities, …88888.88888…, …77777.77777…, …66666.66666…up to …00000.00000…
Can you order them into increasing order?
Or perhaps it depends on the base we are using, there may be more than 10,
maybe 16 or even more, maybe infinite number of infinities.
Could it perhaps help to resort to your well known truth 1=0.99999999…
That is a bad medicine, since our formula ∞+1=∞ seems to be insensitive
to a subtle difference between 1 and 0.99999…
There should be some way to distinguish between these two numbers, but
where it is, did it already got drowned in the information which flooded in.
OK, I’m done until the next questioner comes along.
0.99999… recurring is the loneliest number you’ll ever do.
Seems like a lot of analysis for an effectively infinitely small difference.
777777 you need to appreciate a bit more about how things work here on the SD. No-one is attacking you. There is no victory to celebrate. We have no idea who you are. You are just some person, one of the seven billion people on the planet, writing ideas and questions. We don’t know where you are from, how old, how educated, even your name. What we do do is look at facts. If you take the time to go back and read through this thread you will see this discussion taken up time and time again with different people, and essentially all the arguments presented more than once. Not going back and reading the thread and just posting is somewhat insulting to those who have been posting on this subject in this thread for some time.
As to infinities - you are close. You list many possible infinities. The critical point to understand, they are all infinity. Infinity + 1 is infinity. Infinity + infinity is infinity. Take all the integers - how many are there? That is infinity. Take all the rational numbers - how many are there? The answer is infinity. Infinity does not work like ordinary numbers. This is important. Really important.
But here you have exactly the point. If there is no subtle difference, everything is consistent. An assertion that there should be a way to differentiate is what is flawed. So far one has not been usefully shown. Just the converse, that if there is some way, you can reach contradictions. That is a reductio-ad-absurdum argument that is conventionally used to disprove a theorem. The moment you use notations like …9 .9999… etc you are implicitly using infinity. How many 9s are present in the decimal representations of:
0.9999…
…99999 and
…9999.9999… ?
Am I the only one who found this funny?
Here’s the proof that 0.999… = 1.
Define: 0.999… = 1
There, done. I’m sure you could do better, but for all intents and purposes, if you make that definition, nothing goes wrong.
…Seriously, I don’t understand the problem here. 0.999… is not defined in any way. It’s an utterly meaningless concept until you define it, either over the set of Real Numbers, or as I just did above. The end result is the same, and it’s the same in every base - in base-2, 0.111… is simply 1. Oh, and back in base-10, 3.4929999… = 3.493. This is not basic, but it’s so well-established, mathematically, that at this point if you can prove some inconsistency, you could probably win some very prestigious awards by publishing your proofs in a peer-reviewed journal. Hell, the concept applied to base-2 above? Fundamental problem in floating point arithmetic when it comes to rounding errors. This isn’t just some theoretical navel-gazing, this is a real problem in modern computing. At least, I think - I am going to have to retake that stupid Numeric Programming class again next semester…
Out of curiosity, if the quantity of matter in the universe is “1”, how far do you have to extend .999… before the difference is smaller than the smallest possible amount, i.e. smaller than whatever could fit in a cubic Planck length, or other suitably minimal quantity.
187 nines would do it if you’re talking about the known universe. 8.5*10^185 plank lengths in that sucker.
But that’s the point – 0.999… is defined. It is defined as the limit of the sequence {0, 0.9, 0.99, 0.999, 0.9999, 0.99999, …}, and the limit of that sequence happens to be exactly 1 (and what “limit of the sequence” means can also be defined rigorously, in terms of a standard metric which can also be defined rigorously).
One of the root problems people have with the idea of 0.999… being equal to 1 is the concept that the real numbers are “defined”, as opposed to just obviously existing externally to the human mind. As such, one reasons about them in a different way than one would reason about, say, bunnies.
I’d put in a nitpick here about how exactly the definition works. Starting with the rational numbers Q, we’d like to define 0.a[SUB]1[/SUB]a[SUB]2[/SUB]… for a string of digits a[SUB]i[/SUB] as the limit of the sequence 0, 0.a[SUB]1[/SUB], 0.a[SUB]1[/SUB]a[SUB]2[/SUB], … of rational numbers. That doesn’t work, though, because Q is not complete; the limit of a Cauchy sequence of rational numbers is not defined. (And that’s the point; we want the completion of Q.) To get around that limitation, we define a real number to be the sequence (0, 0.a[SUB]1[/SUB], 0.a[SUB]1[/SUB]a[SUB]2[/SUB], …) itself, then kill off sequences that converge to 0 to get something reasonable. Of course, the sequence for 0.999… converges inside the rational numbers, so there’s no difference in that case. (Of course, once we have that construction, 0.a[SUB]1[/SUB]a[SUB]2[/SUB]… is indeed the limit of the sequence of 0.a[SUB]1[/SUB]…a[SUB]n[/SUB], more or less by definition. Also, we want to consider reals outside the interval [0, 1], but using math notation on a message board is hard enough as it is.)
The point is that a real number is something that’s by necessity something more complicated than just a string of decimal digits. Like you said, talking about real numbers is not like talking about bunnies.
Yes, people attack me, and every single thing what I say here. You are aggressive people. Look at last page post #1212, Exapno Mapcase writes that “10) The whole thing is an attack on people just like you who try to argue against the notion that 0.999999… = 1.”
Why do you think that I haven’t read the thread? I have read dozens of similar thread
on many different sites. I am bringing new insights on the problem. If I disagree
with 1=0.99999…does not mean that I have not read the whole thread.
I need to repeat your mistakes. You have not provided a proof that 1=0.999999…
You try to prove that 1=0.99999…by assuming it is true. But assuming something
true does not prove something. You demand a proof from me. Why don’t you yourself
try proving these things? You just pretend that you have a proof, you use all possible
tricks to hide your ignorance. You use tricks in your proofs, you offer illusions.
You should seek for truth, not respect from people.
You did not answer my questions, perhaps it is better for you to ignore them
if they may threaten your position.
I asked you to show why do you think that the numbers
…00000.00000…
…99999.99999…
are the same even though every digit is different. We have learned that every
digit being different does not always mean that the numbers are different, it
depends on the context, and it does open new problems, for example 2=1 is true depending on the context. What is the context that makes the above two infinite numbers same? I know this question is misleading. These two numbers are
not the same, so it is not right to ask why they should be the same. I just insist
you proving that they are the same by providing the context, because you tell that they are the same.
It is just another example of how do you work. You provide no proofs, but you
demand proofs from me.
Another question, a simple one this time, solve x:
-1 + x = 0
I know, you ridicule me because my question is so simple, but is it amazing if
you refuse to provide an answer. Tell me what is the answer, what is the value of x?
Yes, infinity is not a number. But things are not always so simple. Sometimes
infinity needs to be treated as a number to work with it in various formulas.
I have tried to show there is a way. There is an infinitesimal difference between 1
and 0.99999…It can’t be ignored because the infinitesimal does not equal to
zero. The difference just approaches zero. That is the difference. Approaching
but never reaching zero. What you do, is you assume that the infinitesimal is
equal to zero. You have no proof. You should recognize your mistakes.
You should learn from your mistakes. You are too afraid to admit, you are just
humans capable of making errors. In this respect I am not different than you.
I’m not afraid to recognize that infinitesimally small limits can be treated as zero to make calculus work. I want calculus to work. I like calculus.
You’re not. Not even remotely. The issue of 0.999… is not even a “problem”; it’s a trivial result that follows immediately from the definition of a real number, or in any of the dozens of proofs that have been offered in this rambling thread. If you still think you have something truly revolutionary, go ahead and submit a paper to any of the thousands of math journals, or write up some notes for the arxiv, or post your ideas on mathoverflow or stackexchange, or even email a math professor at your nearest university. We’ve tried to explain to you why you’re wrong; maybe people elsewhere will have better luck.
x = 1
Now you solve for x:
1 / 3 = x
very good
x = 1
but
x ≠ 0.99999999…
So why would the situation be different in this case:
…9999999 + x = 0
x=1 and x is not equal to 0.99999999…
Clearly if you write …9999999 + 0.999999… it is different than
…9999999 + 1
So that we know only that true are:
…9999999 + 0.999999…= …9999999.999999…
…9999999 + 1 = 0
It is not true that
…9999999 + 0.999999…= …9999999.999999… = 0 even though
…9999999 + 1 = 0
x ≈ 0.333333333…
This is a critical thing. You can’t do this. If you are trying to manipulate formulae like you want to, you are violating some of the axioms of the system you are using. For arguments including the real numbers this is very much the case. This is no different to rules such as division by zero being undefined. 0.999… is assumed to be a notation used within the real numbers. You thus only get to use the rules for the real numbers when discussing it. You can’t create new rules, otherwise you are trying to argue about a system that isn’t the real numbers.
Cantor showed how you can reason about infinity, and what operations are valid. But they are not the same as those for the reals, and for the most part are not compatible.
If you want to suggest new rules, you would have to show that they are compatible with everything we expect about the real numbers. This means you don’t reach any contradictions. Not just prove that 0.999… isn’t 1. But contradictions that might arise, like proving that 1 = 2, using your new rules would invalidate them.
For instance if …999.999… = -1 :: what is 10 x …999.999… ?
Does -1 = -10 ?
What does 2 x …999.999… equal?
What about 5 x …999.999… ?
What about 2 x ( 5 x …999.999… ) ?
I don’t create new rules.
-1 + 1 = 0
It is you who are creating new rules. You insist that also x=0.99999…is the solution to -1 + x = 0
Try obeying the rules first before you start accusing others of inventing their
own rules.
I really think we are at an impasse. You seriously need to gain a better understanding of fundamental logical argument here.
You are asserting this as a known truth. There is no possible discourse whilst this is so. You need to prove your case, not assert it and say we are inventing new rules. We say that 1 = 0.999… not as an assumption, but as a consequence of the existing rules. You need to prove your case from the same set of rules. If you go back through the tread you will find what we regard as proof.
If you want to add the idea of …999.999… to the argument, you need to define what this means, and show that you can usefully use this notation in the discourse. And that means showing that adding it to the discourse does not cause the entire system to fail due to internal contradictions.
I need to repeat what I have said before. Why do you assume that I don’t understand
what I am telling? You assume so because I disagree that 1=0.99999…
You cannot brainwash me into thinking like you, and agree with you. I have no
a priori assumption that 1=0.99999…is true like you have. I am seeking a proof,
not assuming it is true and proving it is true because we assume it is true.
I must question your understanding too just because you all the time say that I don’t
understand. Would you think I could dare to challenge the whole world if I did not understand what I say?
I can see that your truth that 1=0.99999…is an absolute truth for you.
Therefore it is what you start with, it is your axiom and you feel there is no need
proving axioms. Therefore you never prove it true. You demand a proof
that 1≠0.999999…and when that is given to you, you refuse to accept it.
You deny everything. You must admit your errors and learn from them.
I am trying to tell what is …999.999… but it seems to be almost impossible because of your aggression and denial of everything.