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

Discussion in 'Alternative Theories' started by chinglu, Oct 27, 2013.

1. ### UndefinedBannedBanned

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1,695
Hi Trippy.

Would you care to elaborate on why you disagree? With specific reference to the point I made that the axioms as currently obtaining can only lead to "undefined" for the 0/0 construction? Can you explain where the mathematicians have any 'choice' in the matter if they apply said axioms as is?

My mistake, yes. I already apologized, and explained that it was my assumption based on the fact that I already earlier and often effectively did address such questions on such trivial aspects ever before you asked them in that post. Again, I should have made clearer why I assumed them to be throwaway lines at the time. Again, my apologies if you were offended.

Again, I should have made clearer what I meant. Namely, to invoke the unitary states when 'setting up' an exercise/proof is what I meant. That is different to the unitary state being the 'result' of that exercise, as distinct to being the circuitous inevitability of the exercise which had the unitary state built-in and the rest of the manipulations merely followed self-selecting logics from that setup unitary state.

In short, I am looking for a proof that achieves the unitary equivalence as the 'result' and not the 'starting' condition as inbuilt and inevitable via circuitous routes 'back to' that initially invoked/inbuilt unitary state. I hope that is clearer as to the difference between starting/predetermining with the unitary state and ending with the unitary state via a route independent of any staring unitary state conditions inherent from the get-go?

Cheers.

Last edited: Dec 8, 2013

3. ### MonimonikaRegistered Member

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59
You keep stating this as a fact. Despite describing 0.999... as having an "infinite" value, can I assume by "1 is never attained" to mean that 0.999... is always lesser in value than 1?

So basically:

1.0 >.9 [n=1]
1.0 >.99 [n=2]
1.0 >.999 [n=3]
1.0 >.9999 [n=4]
1.0 >.99999 [n=5]
1.0 >.999999 [n=6]
etc.

where "n" is the amount of 9s after the decimal point.

From this the "common sense" assumption is that 1.0 > 0.999..., but wouldn't this also apply to the following?:

0.999... > .9 [n=1]
0.999... > .99 [n=2]
0.999... > .999 [n=3]
0.999... > .9999 [n=4]
0.999... > .99999 [n=5]
0.999... > .999999 [n=6]
etc.

"infinite process is eternally existent i.e. the 9's never stop ergo [0.999...] is never attained"

So, the conclusion is: 0.999... > 0.999..., which either matches with the "common sense" you stated above, or is very much against some other common sense idea.

Is this a simple enough example of how you cannot assume from induction that what holds true for finite values of "n" does not necessarily hold true when the "n" is an infinite value? Can you at least now stop bringing up this particular faulty argument?

5. ### chingluValued Senior Member

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1,637
Wait, why are you claiming the proof is wrong because it does not match reality? What does that mean?

It is a proof in recursion theory and by the way, it is perfectly correct.

You cannot step your way to infinity.

7. ### rr6BannedBanned

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635
Sincerly Not See any Faulty Thx As I "Want to Belive"--Wednesday-Addams Famly Values

As an obvious to me, common sense fact.

Yes, it appears to me, that 0.999...is and infinite value ergo is eternally less than 1. I've not ever really thought about what this looks like with line segments so let please bear with me whilist I try and give you explanation, on what I see in minds eye, if I use line segment subdivison of finite line segment value one, as the starting point.

Subdivide line value one into 9 parts, divide each of those 9 into another 9 segments etc....micro-infinitely ergo eternally. So yes I start with a finite line segment value one, so perhaps this what you and others in that camp are trying to get across to me, if not others.

In this given scenario above, 0.999... is within the context of value one line segment, and actually--- now that I think about it ---I've have stated this in several posts, as the only way we can associate 0.999... with value one is that of micro-infinite subdivision of value one something/whatever to begin with.

Yes I believe all finite irrationals(?) of the 0.9 nature are less than one( 1 ). They each individual part, so if we sum the parts we get one( 1 ). Ha, now this is interesing i.e divide line segment value 1 into nine parts-- 0.9 I guess is what they each are then labeled as ---and add them all together we get 1.

Let me try this on cal. H,m no .9 + .9 = 1.8. Oh well scrap that idea. Oh! I see now above, first we divide into 9 parts but when we divide those 9 parts again we have 18ths--- what is the 0.18 ---sorry I'm no good with much math or cal.

Yes that seems correct to me and 9 parts would sum to value 1. That ought to be simple numeration to do on cal. but I'm not sure what it is.

What is happening here, is we are increasing the number of subdivisional parts/segments ergo each time we cycle through and subdivide 9, 18, 36, etc.... we increase the number of parts. So, it appears to me, that as we have less and less 9's on the irrational right side of the decimal place we are getting closer and closer to value one( 1 ).

Wouldn't agree with at leas the latter part of this statement above MK? I put just that part in bold, but I think it is relevant to 0.999... NOT equal to 1.0.

Wait a moment here MK, I think you may have mistyped. That 2nd 0.999... has the dots after it.

See above prior to this last quote you state -.999... > is greater than > .999999 [ n=6 ] or whatever number of 9's, irrelevant but there you did not have the dots( ... ) that represent infinite,

whereas immediately above in you latter statement you have added the dot( ... ) representation of infinite. So if you are in error in adding those dots to the 2nd set of 0.999... ergo if you did mistype and you did not mean to add the 2nd set of dots( ... ) then that is repeat question, if you do mean to say that 0.999... > 0.999... then you have totally lost me as to what your point is and why you would be trying to say infinite 0.999... is or is not greater than infinite 0.999....

0.999... is increase of parts/segement vis multilication-by-subdivision of a finite( whole ) and is less than the whole, because infinite infers and eternally existent operation, ergo we can never sum and ongoing eternal operation of subdividing line segment value 1.

I think I see now what is your guys are trying to do, and that is just to say that a line segment value 1 is subdivided and that the sum of the subdivided parts = 1. That is fine and dandy.

Infinite is not a finite and I think I've seen others already state this, as just have above, that infinite is and ongoing processs, ergo the number of subdivided parts are eternally increasing and you can not sum to a total, an ongoing process that has not finished its process/opration Do you understand that MK?

I repeat again, what those in your camp/viewpoint are actually doing to have 0.999... = 1, is too,

1) STOP the infinite process i.e. then we would have a finite number of 9's ex 0.9^10, and then and only then,

2) can we sum that given value on the right side of the decimal point.

Infinite inherently infers/implies/suggests apriroily means, and eternally existent processs and we can not sum and eternal proccess of subidivision or and infinite process. Do you understand what I'm saying MonoK?

Thx, for reply as it may have helped me to clarify for myself and others what does 0.999... mean in relationship to the #1.

1 = a finite whole and or whole value, whatever value means. Maybe better to just stay with a finite whole.

I hope that helps you and others understand where I'm comming from, and perhaps Monika, you could explain to me--- as stated in my last and previous posting here ---what happen to the dots "..." on that wiki page witing the infinite series blurb, wherein, jsut prior to the last formula after the last "=" sign, the have no dots to represent infinite process,

and whereas just before that same last equal sign, they have the + sign to include the dots "..." that represent infinite processs?

Do you know what I mean MK?

http://en.wikipedia.org/wiki/0.999...

RPenners last reply to me was almost exactly the same as what is posted at in the "infinite series sequence" paragraph and they for no apparrent reason-- to me and no explanation ---exclude the dots of infinity where prior to the equal sign there are the dots being added into the previous formula/equation.

Seems suspicous to me.

R6

8. ### chingluValued Senior Member

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1,637
1) I have shown 2 posts one from RPenner that an infinite series is actually a sequence of partial sums.

Here is just one of them.

An infinite series is an expression like this:

S = 1 + 1/2 + 1/4 + 1/8 + ...
The dots mean that infinitely many terms follow. We obviously can't add up an infinite number of terms, but we can add up the first n terms,

http://www.math.utah.edu/~carlson/teaching/calculus/series.html

Now, on my side there is no justification in Peano arithmetic and no justification under the theory of real numbers. Now, if you know said justification, please provide it.

2) This program will not terminate without a contradiction. So, there is not halting problem, it is not going to halt. This means an recursive definition like addition, division and multiplication will not halt.

The halting problem deals with the fact that there is no algorithm that can decide whether an arbitrary turing prigram will halt or not halt. The problem is undecidable. Or, as put by Stanford,

i.e., the halting problem is unsolvable

http://plato.stanford.edu/entries/computability/#2.2

9. ### chingluValued Senior Member

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1,637
Yea, well one of my degrees is in math and I cannot find a justification of infinite division or infinite addition based on the theory of natural numbers or the theory of real numbers.

Further, there is no justification in abstract algebra. Note, in this document, there is only finite addition and multiplication.

http://web.mit.edu/~holden1/www/coursework/math/abstract_algebra.pdf

10. ### chingluValued Senior Member

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1,637
This has been explain to you over and over. Your confusing is that you do not know what the collection of symbols $\sum_{k=1}^{\infty} \frac{9}{10^k}$ means. It is not an infinite addition. It is a shorthand way of writing,

case n=1, Sum(1)=$9/10$
case n+1, Sum(n) + $9/10^{n+1}$

As I have already explain to you, your own link says the "infinite series" is actually a sequence of finite partial sums.

An infinite series is an expression like this:
S = 1 + 1/2 + 1/4 + 1/8 + ...
The dots mean that infinitely many terms follow. We obviously can't add up an infinite number of terms, but we can add up the first n terms,

http://www.math.utah.edu/~carlson/teaching/calculus/series.html

Now, this has been asked of you several times. If you believe you can add an infinite number of terms, provide the proof contrary to your own link. Show in the definition of the reals, that it includes more than a recursive definition of addition.

Then, if you think you have crossed the boundary of reality and you can add any collection of infinite terms, tell me the answer to this infinite addition.

$\sum_{k=1}^{\infty}k$

This has already been explained to you. You conflated a proof by induction involving k with a supremum operator on an infinite set involving a countable collection of elements connected to the induction proof. It was asked of you to provide a proof template to justify your collection of inequalities. You have still failed to provide this justification yet you continue to claim you are correct.

If you are correct, provide a known proof template and this will be settled.

This has already been explained to you as well. Your link does not prove $0.999... = 1$. There is no such theorem in your link document. So, post the exact theorem where you link proves specifically $0.999... = 1$ in a theorem.

Let's see your "generic theory of infinite decimals", whatever that means, and how that applies to a proof of $0.999... = 1$. This requires infinite addition which you have yet failed to justify.

Yea, you are so smart. So, based on your great talent to judge others, why not show a justification from set theory, Peano arithmetic or the theory of real numbers that infinite summation is justified. That way, I can know the answer to $\sum_{k=1}^{\infty}k$.

11. ### chingluValued Senior Member

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1,637
This post is completely offensive and misleading.

which converges to $\frac{1}{1-r}$ as $r \rightarrow \infty$

Then it goes on an says,

thus, we write
$\sum_{n=0}^{\infty} \, r^{n} \; = \; \frac{1}{1 - r}$

This has nothing to do with equality and is simply stating this definition is in line with the definition of convergence using delta and epsilon.

Finally, the error in your posted computer proof occurs at step 39.

Theorem geoisum1c 9342

This concludes the infinite geometric series is actually equal to its limit. Yet, there is no infinite addition permitted in the foundations of mathematics.

12. ### arfa branecall me arfValued Senior Member

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6,094
Yes, but if we don't add up any of the terms, or say, n of them, is there still an infinite series of terms? That is, after not adding any terms together, or adding n terms together, does "..." still mean that infinitely many terms follow?
I think you will find this is what the link you posted is really about.

And I guess this thread is also about how some people just can't let go of the idea that you have to write something out, in order that it exist. This is redonculous.
Your particular program will never halt, which doesn't necessarily prove anything.
Yea, well

Yea, well, that's just what it is, actually. It says to sum from k = 1 to infinity. A sum involves the operation of addition, so it is, after all, an infinite addition.

13. ### chingluValued Senior Member

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1,637
There you go claiming you can add an infinite number of terms and yet the link said,

We obviously can't add up an infinite number of terms

And I guess this thread is also about how some people just can't let go of the idea that you have to write something out, in order that it exist. This is redonculous.

it shows any recursive procedure based on being defined for all natural numbers can't halt. This includes addition and division.

So, yea, it proves something significant.

I have provided enough evidence that it is a shorthand way of expressing a sequence of finite partial sums. So, stay with the evidence and go from there.

Now, you also have been asked to defend infinite addition at the foundations. You claim it exists and yet you have no mainstream connection to set theory, Peano arithmetic or the theory of real numbers that proves it exists.

People can make up all kinds of stuff. But, there is no justification for infinite addition in mathematics.

14. ### chingluValued Senior Member

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1,637
Joe > bill(n)

Max > bill(n)

Joe >= Max

So, Joe = Max.

How does this work?

15. ### rr6BannedBanned

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635

MK quoted me and added in where i may have left out word(s) as follows.

Let restate that, infinite is an eternally existent process, i.e. the 9's never stop ergo 1 is never attained by a summation of it infintely subdividing parts.s

I was pretty clear about all of this in my reply to MK. Not sure why I wanted to clarify this above quote of me, by her with her 0.999... addendum.

r6

16. ### arfa branecall me arfValued Senior Member

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6,094

So since "We obviously can't add up an infinite number of terms", this isn't a problem because "..." means an infinite series.
This symbol, "..." has a precise meaning, it represents something, and this doesn't change if you perform some operation on the series containing it.

17. ### chingluValued Senior Member

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1,637
Calculus is not defined in terms of infinite sums. It is defined in terms of limits. It predates Cantor's completed infinity.

Anyway, I am not seeing a justification using set theory, number theory or the theory of the reals that justifies infinite summation.

You link is broken. But, even it it was not, the mainsteam also has links that claims an infinite series is a sequence of finite partial sums, which is actually correct.

So, you have proven nothing. Now, if you are correct, simply prove your can perform infinite summation from anywhere in the foundations of mathematics.

Hint, try looking at the infinite union in set theory. Then, also look how sums might be coded in set theory in terms of sets.

You will find the infinite union can only sum up finite partial sums.

Realize, your construction must remain in the model represented by the reals.

18. ### arfa branecall me arfValued Senior Member

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6,094
By "perform infinite summation", I guess you mean actually add the terms together?
But how does not being able to do that stop me from writing an infinite series like 1/2 + 1/4 + ... + 1/2n + ... ?

What's the paradox involved? What I wrote down hasn't involved any actual addition, and there seems to be no compelling reason to "perform" any operation. And yet it's still an infinite series (dammit!). Where did I go wrong?

19. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member

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It looks like chinglu has successfully convinced himself that 1/3 + 1/3 + 1/3 does not equal 1.

20. ### MonimonikaRegistered Member

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The most important bit is underlined and bolded near the bottom. I'd appreciate it if you'd focus on answering that part first before anything else on this post (so you don't confuse yourself).

Seems like I read you right so far.

I'm not talking about picturing what 0.999... looks like or anything like that, so please focus on what my post actually says instead of going off on a tangent. Undefined liked to do this (mixing in irrelevant topics in replies) in order to run away from answering direct questions. This is why Undefined is the first and currently only member on my ignore list. Don't become a jerk like Undefined.

Confusing as your post is, I am going with that second paragraph above to be your true thoughts about the matter of summing sequences (i.e. you think they don't work). I'm not talking about that, so drop this irrelevant tangent. My point is much simpler.

What the heck does "9 parts would sum to value 1" have anything to do with what's in my post? Stop adding irrelevant stuff.

Look at that part you quoted again. Where is there anything about getting closer to the value one (1)? You seem to have missed that there is no "1" being used there. You're confusing yourself with irrelevancies of your own creation. Focus on what I actually typed, okay?

And finally I can get to my point (which you missed due to your irrelevant musings)! I did not mistype anything. I only just did exactly what you do when the left side of the inequality is "1" instead of "0.999...". In fact, I had presented it to you quite clearly:

Given what you've stated:
this means that you made the leap from:

1.0 > 0.999(finite "n" of 9s)99
to
1.0 > 0.999...

You yourself have added the dot( ... ) representation of infinite to the right side of the inequality when you used this argument about 0.999... never reaching the value of 1 as being relevant to 0.999... not equaling 1.

My point was to show that a contradiction would occur, but you've shown me that you are a hypocrite who uses double-standards.

To you, when the left side is "1" it's okay to replace "finite irrationals(?) of the 0.9 nature" on the right side with "0.999...".
But when the left side is "0.999..." suddenly the replacement of "finite irrationals(?) of the 0.9 nature" on the right side with "0.999..." is a mistake to you.

Read the bolded and underlined part again. That's the point of my post. Focus on what it says and answer it. Don't bring up what else you "think" I'm talking about. The rest of your post looks to be irrelevant to my point, so I'm not going to address it unless you can make it relevant.

21. ### rr6BannedBanned

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0.3... + 0.3... + 0.3... NOTE equal 1 Common Sense

Origin, you still don't get it dude.

0.333... + 0.333.... + 0.333... does not equal 1.

You do not understand the differrence between finite and infinite.

We cannot sum an infinite processs, and this is common sense.

We can do begin and infinite summation, but never arrive at a sum. Common sense.

Common sense trumps mathematics when the math given is flawed or just plain ole illusionary, mental masturbation having nothing to do with our eternally existent, finite reality we call occupied space and Universe.

You explain nothing because you offer flawed math that has not valid explanation.

Like wiki, you exclude the "..." dots because they prove your flawed comments.

r6

22. ### rr6BannedBanned

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635
Sorry No Banana For MK

Huh? I will have to break this apart to try and make sense of it MK.

1.0 yeah got that part

Huh? 1.0 is not as you state. Your infering some number value that is 1.9 and the only value I have EVER referenced is 1.0 not 1.9.

So,MK, this this is where you dont make sense and don't appear to be directly/specifically-- and you asked me to addrees this above yours first, so I that is what I'm doing ---addressing ANY comments by in my last reply to you. You appear to me to be going off on some irrelevancy that may be tangental . I dunno.

Yes, 1.0 and 0.999..., NOT 1.9-- as suggest above --- are the two labels under consideration, and you and those in your camp are the ones who are exchanging 1.0 for 0.999... as tho they are equals, when they are not.

Senseless MK, reread what you just stated above. 0.999...is containd both right and left side numbers. Your NOT making rational, logical sense MK.

Huh? Sorry MK, I have no idea of when or where I stated that it is a mistake to replace replace or change, 0.9 to 0.999.....

I'm sorry MK, but this stuff in bold was lacking rigor, rationality, logic and common sense. imho.

I have to go now, Hopefully when i get back to your last reply, I can go through it and see if you actually address my specific comments directly as stated, as i have been fairly clear on the problems with yours or others claiming a "proof". I see mathematically ilusionary mental masturbation been put forward as alledged "proof"(s). NOT.

r6

Read the bolded and underlined part again. That's the point of my post. Focus on what it says and answer it. Don't bring up what else you "think" I'm talking about. The rest of your post looks to be irrelevant to my point, so I'm not going to address it unless you can make it relevant.[/QUOTE]

23. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member

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Oh I get it alright. You apparently did not make it to the 3rd grade, where fractions were taught. So 1/3 + 1/3 + 1/3 does not equal 1? It is very hard at this point to have any sort of meaningful conversation with you. I think you will have to stick with posting meaningless symbols, colored text and gibberish

Your 'common sense' is actually uncommon nonsense. Your 'common sense' tells you that 1/3 + 1/3 + 1/3 does not equal 1. With 'common sense' like that, I am surprised you are able to feed yourself.