# 1 is 0.9999999999999............

It is 1/9 and that is a frational fraction.

You are right, but what fraction is .999999.... ?

Chinglu, you are mistaken -- I do not need to concatenate an infinite number of 9's to form 0.999... because by assumption the digits in 0.999... are always 9 and already exist at all positions right of the decimal point.

Because of the axiom of completeness for the real numbers, the following holds true for the non-negative real numbers.

$${a_0.a_1a_2a_3\dots}_{\tiny 10} \quad = \quad \sum_{k=0}^{\infty} \frac{a_k}{10^k} \quad = \quad \textrm{sup} \; \left{ x : \; \exists n \in \mathbb{N} \; \wedge \; x = \sum_{k=0}{n} \frac{a_k}{10^k} \right} \; = \; \lim_{n\to\infty} \sum_{k=0}^{n} \frac{a_k}{10^k} \; \in \; \mathbb{R}$$

Therefore $$0.999... = \quad \textrm{sup} \; \left{ 0, \, 1 - \frac{1}{10}, \, 1 - \frac{1}{10^2}, \, 1 - \frac{1}{10^3}, \dots \right} = 1$$.

This follows from continuity and monotonicity of the exponential because $$\forall k\in \mathbb{Z} \; \frac{1}{10^{k+1}} \lt \frac{1}{10^k}$$ and so if $$0 \lt 0.999... \lt 1$$ then $$n \quad = \quad 1 - \left\lfloor \log_{\tiny 10} \left( 1 - 0.999... \right) \right\rfloor \quad \in \mathbb{N}$$ and therefore $$1 - \frac{1}{10^{n-3}} \lt 0.999... \lt 1 - \frac{1}{10^n} \lt \textrm{sup} \; \left{ 0, \, 1 - \frac{1}{10}, \, 1 - \frac{1}{10^2}, \, 1 - \frac{1}{10^3}, \dots \right} \lt 0.999...$$ which is a contradiction that proves $$0.999... \not\lt 1$$.

I looked at your supremum operator and it does not fit in your equation.

You can't compare sequences based on n and then all of a sudden compare them to infinite sequences as you did. The Pinching Theorem requires a comparison to n for all sides in the sequence to assert the conclusion. You violated this requirement.

Your proof has no basis in modern mathematics.

That is 1.

Where is the .999....?

How can Chinglu's ravings be acceptable but Tach's not? I truly don't understand the standards here.

Here is one of may very simple proofs, this is from wiki:

$$\frac{1}{9} = 0.111...$$

$$9 \times \frac{1}{9} = 9 \times 0.111...$$

$$1 = 0.999...$$

It isn't really that hard to understand.:shrug:

How can Chinglu's ravings be acceptable but Tach's not? I truly don't understand the standards here.
do you find chinglu's ravings to be abusive?

Here is one of may very simple proofs, this is from wiki:

$$\frac{1}{9} = 0.111...$$

$$9 \times \frac{1}{9} = 9 \times 0.111...$$

$$1 = 0.999...$$

It isn't really that hard to understand.:shrug:
try
1/9 = 1(0.111...)
9 x 1/9 = 9 x 1(0.111...)
1= 1(0.999...)
1=1

using quantity for quantity...
Mix the context from quantity to value and you will always be able to fudge a result.
I think my proof that 1 = 1 is better than your proof that 0.999... = 1
the proofs mixing context from wiki are flawed... sorry ...and you math guys should be ashamed of yourselves for falling for it. IMO.. [chuckle]

How can Chinglu's ravings be acceptable but Tach's not? I truly don't understand the standards here.

I once asked James R this very same question when dealing with a troll. The reply I got was "We don't ban people for stupidity."

Having said that, I have been known to ban people for pseudoscientific trolling.

do you find chinglu's ravings to be abusive?

No, but I didn't find Tach's to be abusive either. I admit I didn't read everything he wrote. But all he ever directed at me were mild jibes consistent with the nature of this thread. I gave the same back to him from time to time. I actually have no understanding of why the moderator(s) took the action they did. I'm really puzzled by it. Maybe I missed something terrible that he said. I have no idea.

try
1/9 = 1(0.111...)
9 x 1/9 = 9 x 1(0.111...)
1= 1(0.999...)
1=1

using quantity for quantity...
Mix the context from quantity to value and you will always be able to fudge a result.
I think my proof that 1 = 1 is better than your proof that 0.999... = 1
the proofs mixing context from wiki are flawed... sorry ...and you math guys should be ashamed of yourselves for falling for it. IMO.. [chuckle]

You do realize that when you wrote:

1= 1(0.999...)
1=1

That this means that 1 x 0.999... = 1 or 0.999...=1

So you apparently now agree with the mathematicians. [chortle]

0.999...infinite NOT Equal 1.0 Finite

OK, can you prove the irrational .9999.. is a rational number being 1? That is the issue.

Thx Chinglu, that is really the issue the RP trys to avoid, because he and others with his viewpoint, have no relatively simple explanations that are also rational, logical and common sensical. They have nothing and no proof that they even have a complex proof.

I put the label/identiy of 'irrational' on 0.999.... but I don't really no what the correct label/identity for such an implied infinite number on right side of decimal is.

I do know, from common sense, that 0.999....is not equal 1.0 and no amount of repeated complicated very high mathmatical illusions are going to convince me, nor 90% of the common people, that the infinite 0.999... is equal to the finite 1.0.

RP, has not only does RP not have a simple proof, my guess is that he has no proof at all and is just playing mental masturbations games with us. If i'm incorrect, then please have someone with their illogical viewpoint, actuall post something that 75% of the common people can have a relatively simple chance of understanding.

Please give us the' big dummies guide', to how the finite, 1.0 is equal to the inferred infinite 0.999....

They don' that so that don't offer that. It is situation of the the seemingy highly educated trying to pull the illusionary wool over the less educated persons eyes and hope they will cry out 'yes, I've seen the mathematical light, and you are correct a great mathematical wizard'. NOT.

They are pulling the mathematical illusion over their own eyes at same time the attempt to pull it over ours. imho.

This issue is really a no brainer, yet they appear to want to present them selves as being beyond the simple person and simple logic and simple common ergo the send us to a link of complex mathematics, and words most of humanity has never seen, and will never hear of, and say to us, see, did you read that. that is the proof.

Or they may post a half page of complex algebraic symbols and other who knows what symbols, and again say, see,did you read that, that was the proof. Uh, yeah sure dude, when you come back to Earth with something for 80% of humanity to actually even begin to grasp then you can cry proof because you certainly have not proved anything to me.

r6

You do realize that when you wrote:

1= 1(0.999...)
1=1

That this means that 1 x 0.999... = 1 or 0.999...= 1

So you apparently now agree with the mathematicians. [chortle]
no ... not at all.. look again

1= 1(0.999...)
1=1 of (0.999...)
1=1

0.999... is a value not a quantity
1 is a quantity

When they use these proofs they mix up the quantity vs value thing and end up with a fudge of poodle pie...
as I posted earlier...'tis the same issue

by using the long handed method you force the context to stay consistent through out the routine.

Sorry Origin No Banana For You

Here is one of may very simple proofs, this is from wiki:
$$\frac{1}{9} = 0.111...$$
$$9 \times \frac{1}{9} = 9 \times 0.111...$$
$$1 = 0.999...$$
It isn't really that hard to understand.:shrug:

First of all, Origin has only been a troll to me.

So he I rarely waste any intellect responding, however, since he is indirectly trying-- however lame ---to offer us a simple explanation, as i have repeatedly asked for, he has a attempted to do so, tho it is funny, how one person sends us a relatively simple set of formula--- with no guiding explanation ---,

and another had sent me to a wiki page with immense complexity( mapping ) and complex unheard of terminologies( subjerctive{SP}) and never to be heard of again, by 90% of humanity.

Here above Origin posts his 3 lines of formula, with no step by step guide as to what his numbers and symbols mean and then asks if it is really that hard to understand?

Yes it is that hard to understand, because I see no proof that finite 1.0 = infinite 0.999....in Origins less info ergo less complex presentation.

I see once again, that Origins states that 1.0 = 0.999....with no explanation of how 0.999...relates to any of his givens above that.

Sorry Origin, you have offerred no proof. That finite 1.0 = infinite 0.999....

PLease give us the step 'big dummies guide' to your given statements that actually show us how infinite 0.999.... becomes finite 1.0.

I have a MS calculator on my computer so please give us the step step instructions that i punch in, to make infinite 0.999....become finite 1.0.

Oh wait, I get it, there is not such buttons on my calculator to push that convert infinite 0.999...into finite 1.0. Duhh! What was I thinking?

My guess is, that if you find a calculator that converts infinite( irrational ) 0.999... to finite( rational ) 1.0, then you have ceased infinity and done a process of renormalization i.e. you have;

1) stopped an inferred infinite 0.999... process, so that 0.999...becomes 0.9,

2) and then round it off to 1, ergo normalized so humans can actually get their heads( minds ) around it.

Again, no simple explanation, certainly not a proof that finite 1.0 = 0.999.....

It will take a better troll than Origin

Stating such does not make it so.

0.999... is not an irrational number. Being equal to 1, it is clearly rational.

Please give us the' big dummies guide', to how the finite, 1.0 is equal to the inferred infinite 0.999....

You have been given about 4 or 5 proofs that show 0.999... = 1. I guess your description of yourself truly fits.

They are pulling the mathematical illusion over their own eyes at same time the attempt to pull it over ours. imho.

Unfortunately your opinion is based on ignorance, so is not worth anything.

This issue is really a no brainer, yet they appear to want to present them selves as being beyond the simple person and simple logic and simple common ergo the send us to a link of complex mathematics, and words most of humanity has never seen, and will never hear of, and say to us, see, did you read that. that is the proof.

Like I said many proofs have been given - your arm waving gibberish is not a proof.

Or they may post a half page of complex algebraic symbols and other who knows what symbols, and again say, see,did you read that, that was the proof.

Just because the proof is beyond your ability that does not make it wrong. It just means it is beyond your ability.

Uh, yeah sure dude, when you come back to Earth with something for 80% of humanity to actually even begin to grasp then you can cry proof because you certainly have not proved anything to me.

Lucky for us science and math is not dictated by what you can understand, if that were true I fear we would be back in the dark ages.

Wheres the proof? No Banana has yet to be Earned

origin.."You have been given about 4 or 5 proofs that show 0.999... = 1."

I've only seen two, and there the two I mentioned.

Like I said many proofs have been given - your arm waving gibberish is not a proof.

You have not given a proof and certainly not one that I nor 90% of humanity can understand, nor repreduce.

Come back and talk to me when you have something is true, you can explain to 90% of humanity so they can understand it, and that we can actually repreduce. You nor anyone else has done that. James R states that is so but he also gives us nothing beyond that. Absolute nothing ergo no simple explanations and certainly no guide. Empty comments are just that. Lame, when it obvious what I'm asking for and no one can deliver.

Just because the proof is beyond your ability that does not make it wrong. It just means it is beyond your ability.

Just because you can post a link to web site, with some mathematics on it, or copy and paste mathematics is not proof and does not mean you, RP aor james are c are correct.

Finite 1.0 does not equal infinite 0.999... and as always, you do not address my comments directly with any relatively simple, rational, logical explanation, because none of you have one, and that is why you offer none.

Your going from 0.111..to 1.0 = 0.999.... appear as nonsensense to me, and I would think most of humanity that does not want to join a clic of people who cannot offer simple explanation and guide for rest of humanity. Sorry but you and others lack the integrity to offer what is asked of you, because you don't have it.

That is why James R did not offfer any guiding explanation to anything.

Lucky for us science and math is not dictated by what you can understand, if that were true I fear we would be back in the dark ages.

So lets see Origin, you have wasted yet another whole message with no offering of a simple, explanation guide to how finite 1.0 = infinite 0.999.. and certainly no proof that I nor most of hmanity can understand.

Mathematical illusions is what you and a few others have to offer in trying to convince me that finite 1.0 = infinite 0.999.. thank god humanity is not dependent on learning anything of significance from you as we would still be in dark ages in that scenario.

Please come back and talk to me when your truy can address my comments as stated give us a simple, rational, logical and common sense, explanatory guide of how finite 1.0 = infinite 0.999... and that we can actually repreduce on our end with a calculator.

You, nor anyone else has done that, because, you cannot. That is my best guess and i'm sure most of common sense humanity would say.

If you cant do that, the any other comments are irrelevant, uneccessary and most likely unhelpfull in any way. imho

I will not be holding my breath waiting for any of the above as requested, because my guess is, it doesn't exist i.e. mathematical illusion is mental masturbation and not a proof of anything, except that you fail to offer what is asked of you and others.

r6

I looked at your supremum operator and it does not fit in your equation.
Sure it does. All the $$a_k$$ terms are non-negative because they are digits, right?
$$\sum_{k=0}^{m} \frac{a_k}{10^k} \; = \; \textrm{sup} \, \left{ x: \; \exists n \in \mathbb{N} \; n \leq m \wedge x = \sum_{k=0}^{n} \frac{a_k}{10^k} \right}$$, right?
So $$\sum_{k=0}^{\infty} \frac{a_k}{10^k} \; = \; \textrm{sup} \, \left{ x: \; \exists n \in \mathbb{N} \; x = \sum_{k=0}^{n} \frac{a_k}{10^k} \right}$$, because I just knock out the finite upper bound on the index. Right?
And the completeness axiom says $$\textrm{sup} \, A$$ exists when $$A \neq \emptyset$$ and $$A$$ has an upper bound which is satisfied by $$a_0 + 1$$ but equally well by $$a_0 + 2$$.

You can't compare sequences based on n and then all of a sudden compare them to infinite sequences as you did.
As I did where? Also, why not?
The Pinching Theorem requires a comparison to n for all sides in the sequence to assert the conclusion. You violated this requirement.
Where am I using the pinching theorem? I'm saying that IF $$0.999... \lt 1$$ then there exists an n where $$1 - \frac{1}{10^{n-3}} \lt 0.999... \lt 1 - \frac{1}{10^n}$$ and this contradicts $$0.999... \; = \; \textrm{sup} \, \left{ 0, 1 - \frac{1}{10}, 1 - \frac{1}{10^2}, 1 - \frac{1}{10^3}, \dots \right}$$ which says that for all n, $$1 - \frac{1}{10^n} \lt 1 - \frac{1}{10^{n+1}} \leq 0.999...$$ -- there is no pinching, just contradiction in assuming $$0.999... \lt 1$$.

Then I go beyond what is necessary and tell you how to calculate n if you insist $$0.999... \lt 1$$, so it is no mere existence proof which some people find unsatisfying, but a constructive proof.

Further it is not the only such contradiction that arises if you assume $$0.999... \lt 1$$.

Your proof has no basis in modern mathematics.
All sources on modern mathematics agree $$0.999... = 1$$ so it appears you are trying to invoke an authority you yourself ignore.

You have not given a proof and certainly not one that I nor 90% of humanity can understand, nor repreduce.

I do not believe that 90% of the population is too stupid understand the extremely simple proof I showed. If you wish to claim that your understanding of basic math is this horrendous, well that is your right but don't drag 90% of the population in with you.

Here is a load of proofs, see if there is one that you can understand.

Notice this from the site:

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education.