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

Billy T, I suspect you confused arfa brane with me.
There is no such thing as infinite summation based on natural numbers.
Chinglu asserts $$0.999... \neq \sum_{k=1}^{\infty} \frac{9}{10^k}$$ because the latter is "not a thing" but gives no alternate definition.

There is in analysis, because analysis deals with the real numbers, not just the natural, rational, or algebraic numbers.

Here's just one textbook on the subject: http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF (This was Google's first hit for "analysis textbook" but you are welcome to try any others.)
... [Explicit refutation that one may not have infinite summation, with an applicable example not involving limits] ...
But as long as you are looking at this textbook you might as well look at Theorem 1.1.3 and Theorem 1.1.6 on pages 4 and 6 which says if $$0.999... < 1$$ then there must be a real number equal to 1 - 0.999... and there must likewise be a rational number between 0.999... and 1. The fact that no such numbers have ever been named seems to indicate a problem with the hypothesis $$0.999... < 1$$.
(This was not offered as mathematical proof but as the beginning of a proof. Had someone named such a purported number, it would lead to contradiction. Also, absence of evidence when there must be evidence if the hypothesis is true and a diligent search has been taken is evidence (but not proof) that the hypothesis is wrong.)
I am looking at page 201 of your link and I am glad the mainstream has not contradicted itself on an infinite series.
This, by my counting, is post #1532 which may be linked to with [post=3141308]a forum-specific markup like this that does not get your post queued for moderation for too many hyperlinks[/post].
Chinglu seems to accept my source as authoritative on the subject of infinite summation whereby $$0.999... = 1$$ follows directly. All the rest of this post is chinglu's failure to grasp a subject he's spent less than 5 minutes thinking about in a language he doesn't understand properly.

Math is not done by "common sense" nor by popular vote, nor by opinion, BUT BY PROOFS.
Also proofs from assumptions and definitions. You cannot argue with a definition, you can only propose alternatives. The definitions and axioms of the real numbers are fixed and while infinite repeating decimals are rational numbers, you may need a generic theory of infinite decimals to convincingly prove this. Particular for an audience that is dubious that 0.999... / 10 = 0.0999...

See my response to RPenner.
There is no such thing in mathematics that permits infinite addition.
This is a stronger claim than chinglu has made above, but is entirely baseless, particularly in light of the above reference. Chinglu doesn't have the right to ignore the mathematical field of analysis just because he is still struggling with set theory.

Sure, #1532 for RPenner. He has not responded.
As I have said before, you have no right to compel a response from me. You simply misunderstood the text.

2nd Thoughts---Renfinement---Set Theory or is set facts---set of facts---?

rpenner.."You cannot argue with a definition, you can only propose alternatives. The definitions and axioms of the real numbers are fixed and while infinite repeating decimals are rational numbers, you may need a generic theory of infinite decimals to convincingly prove this. Particular for an audience that is dubious that 0.999... / 10 = 0.0999..."

Is there actually someone here in this thread taht are "dubious" of 0.999.../10 = 0.999....?

I think it is more, someone(s) are dubious of finite 1.0 = infinite 0.999.....

This is a stronger claim than chinglu has made above, but is entirely baseless, particularly in light of the above reference. Chinglu doesn't have the right to ignore the mathematical field of analysis just because he is still struggling with set theory
.

Yeah the infinite set of 0.999... + 0.999... = 1.99999999999999999999999.....if I did the cal. procedure correctly.

Whats to understand? Infinite set plus infinite set = and infinite value.

Finite does not, and cannot ever equal an infinite. Common sense, rational thought, logical thought etc... overides mathematically illusionary, mental masturbation at some point, doesn't it? Yes it does.

Practicallity = common sense, rational and logical thought i.e. we stop the infinite series set abstraction at 0.9 and we round to 1.0 for puposes of practicality and not absolute truth(s_.

Take note BillyT's 2nd thoughts on the finite = infinite line of rationality needing some refinement. Yeah, him and a few others need some 2nd thoughts of refinement in these regards, is my best intuition, guess, assessment of the situation also.:bugeye:

His ... reference does indicate 1 = 0.999... is not well proven
Objecting to this specific language.

http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

Page 201 establishes $$\sum_{n=0}^{\infty} \, r^{n} \; = \; \frac{1}{1 - r} \, \quad \quad \quad -1 \, \lt \, r \, \lt \, 1$$ with proof in example 4.3.3 based on theorem 4.3.5 both on page 204.
So specifically $$\sum_{n=0}^{\infty} \, \left(\frac{1}{10}\right)^n \; = \; \frac{1}{1 - \frac{1}{10}} = \frac{10}{9}$$
Or $$1.111... = 1 + \frac{1}{9}$$.
And justification for writing $$0.999... = \sum_{n=1}^{\infty} \, \frac{9}{10^n} = 9 \times \left( \sum_{n=0}^{\infty} \, \left(\frac{1}{10}\right)^n \; - 1 \right) = 9 \times \left( \frac{10}{9} - 1 \right) = 9 \times \frac{1}{9} = 1$$ comes from theorem 4.1.8 on page 184 along with the definitions at the beginning of section 4.3 on pages 200 and 201.

Likewise, a complete computer-checked proof that 0.999... = 1 from the axioms of the real numbers and set theory is here: http://us.metamath.org/mpegif/0.999....html

I have a post stuck in moderation, so I will write this much:
Is there actually someone here in this thread that are "dubious" of 0.999.../10 = 0.999....?

Several doubt 0.999... / 10 = 0.0999...
for if 0.999... / 10 = 0.0999...
and if 0.999... - 0.9 = 0.0999...
and if equals to one thing are equal to each other, then it follows that
0.999... - 0.9 = 0.999... / 10
or
0.999... - 0.999... / 10 = 0.9
or
0.999... ( 1 - 1/10 ) = 0.9
or
0.999... = 0.9 / (1 - 1/10)
But since
0.9 / ( 1 - 1/10 ) = 0.9 / 0.9 = 1
then it must be true that 0.999... = 1 follows from 0.999... / 10 = 0.0999...

Now, I think it is almost crazy to claim 0.999... ≠ 1 (although I will concede that to those who only have experience with decimals of finite length, 0.999... looks like it should be less than 1), but it is definitely crazy to claim that 0.999... / 10 ≠ 0.0999... . But since 0.999... / 10 = 0.0999... requires 0.999... = 1 then follows that decimals of infinite length play by somewhat different rules than decimals of finite length and there isn't a one-to-one mapping between decimals of infinite length and real numbers.

Thx RP--Banana For You---Complex > Simple Explanation---Yea!!!!!!!!!

rpenner.."I have a post stuck in moderation, so I will write this much:"

If you think you need moderation, no problem, I will try to assist you.

Several doubt 0.999... / 10 = 0.0999...

If there is in thread, it was posted before I entered the thread or after and I never saw such claims of "dobious" in those specific regards.

for if 0.999... / 10 = 0.0999...

I think I did that procedure correctly on my ca. to verify your resultant conclusion.

and if 0.999... - 0.9 = 0.0999...

Is that negative zero.nine > to inifinity?

How do I create a negative 0.9 > to infinity on my cal.?

and if equals to one thing are equal to each other,

What "thing"(s) are you specifically refering too RP?

then it follows that
0.999... - 0.9 = 0.999... / 10

Sorry RP, you lost me with the, oh wait maybe you meant 0.999... minus 0.9.

0.999... minus 0.9 = 0.0999....if I did procedures correctly on cal. adn yes that is equal to 0.999... / 10

or 0.999... - 0.999... / 10 = 0.9 or
0.999... ( 1 - 1/10 ) = 0.9 or
0.999... = 0.9 / (1 - 1/10)

But since
0.9 / ( 1 - 1/10 ) = 0.9 / 0.9 = 1

Let me see if I can do this procedure on cal.

1/10 = 0.1

1 - 0.1 = 0.9 ergo,

0.9 / 0.1 = 9

H,mm, somehow I think you wanted me to get value 1 and yet I get a value 9.

then it must be true that 0.999... = 1 follows from 0.999... / 10 = 0.0999...

If I followed your lead correctly, then value 9 is my final resultant not 1, as you suggest.

You get a banana for finally offering something that appears to be relatively simple explanatory guide--- thx! ---yet I don't the desired, and sought after resultant of value 1, when following your lead.

Now, I think it is almost crazy to claim 0.999... ≠ 1

For practical( stopping and rounding ) purposes it does, not absolute truth purpose. With your use of the word "almost" you certainly leave the door open to others who are not crazy, and actually sincere on wanting to see a rational, logical, common sense and relatively simple explanatory guide as to how finite 1.0 = infinite 0.999...

(although I will concede that to those who only have experience with decimals of finite length, 0.999... looks like it should be less than 1),

when the cal. fills all of the spaces-- one of my experiences with decimals ---I believe an infinite value is inferred/implied etc...but hey, I don't claim have a degree on what what the decimals on cal. mean.

but it is definitely crazy to claim that 0.999... / 10 ≠ 0.0999... .

Irreseptive of that claim finite value 1 does NOT equal infinite value 0.999...

But since 0.999... / 10 = 0.0999... requires 0.999... = 1

Why you still still believe that the first half of that sentence is equal to 2nd half is mysterty to me, since I thougth I followed your lead on this page, and the math-- cal. procedure ---gave resultant 9.

then follows that decimals of infinite length play by somewhat different rules than decimals of finite length and there isn't a one-to-one mapping between decimals of infinite length and real numbers.

Well, again, to complex me for me all these natural, real unatural etc type of comments and also confusing to me is that now your bringing a non-mapping condition/situation into the conversation, whereas in past I believe you, if not also others, sent me to and/or used complex mathematics of "mapping" as a proof, that finite value 1.0 = infinite value 0.999...

So again, this new pathway only serves to confuse others, tho that may not be your intention. I dunno.

Thx again for finallly posting what appear my long sought after, realtivly simple explanatory guide as attempt prove that finite 1.0 = infinite 0.999....

r6

0.9 / ( 1 - 1/10 ) = 0.9 / 0.9 = 1

Let me see if I can do this procedure on cal.

1/10 = 0.1

1 - 0.1 = 0.9 ergo,

0.9 / 0.1 = 9

H,mm, somehow I think you wanted me to get value 1 and yet I get a value 9.
Whose mistake do you suppose that is?

0.9 / (1 - 1/10 ) has parentheses to indicate the order of operations.

1/10 = 0.1
1 - 1/10 = 0.9
so
0.9 / (1 - 1/10) must be the same number as 0.9/ 0.9 which a good calculator will tell you is 1.

http://www.wolframalpha.com/input/?i=0.9+/+(+1+-+1/10+)

iPhone calculator with memory: [c] [ac] [mc] [1] [-] [1] [÷] [1] [0] [=] [m+] [0] [.] [9] [÷] [mr] [=] (Windows 7 calculator in scientific mode is similar)

iPhone alternate calculation with parentheses: [c] [ac] [0] [.] [9] [÷] [(] [1] [-] [1] [÷] [1] [0] [)] [=]

As to the 0/0 "undefined" being a choice by mathematicians, I again point out that it is not my 'opinion' that the axioms force the "undefined" tag. Mathematicians have no choice in the matter UNLESS the relevant axioms are (as I am involved in doing) modified/enhanced in such a way that such things as 0/0 are no longer "undefined" by them. That's another aspect which I feel is crucial to making the mathematics 'complete' so as to handle more unusual states and be more capable of reality modeling than at present incompleteness demonstrates.
I disagree.

And just to be clear, I didn't mean to imply you were facetious as such, I just thought your 'throwaway lines' were a 'facetious questions' tactic. My mistake. My apologies for any offense, Trippy.
That they were throw away lines is your assumption and your assumption only. They were legitimate questions I was using to illustrate a point.

A way of starting from the fractional string itself and making logical actions/assumptions bout it such that the mathematical 'transition' from fractional infinite string to 'unitary' is achieved without the use of trivial 1/1, 9/9 etc manipulations which only introduce what I observe as a bias to the argument/treatment which gives the result you want but is not in the same 'triviality independent' limits method.
Do you understand how nonsensical this statement seems?

You want to prove that 0.999(9) = 1 without ever invoking a 1...

Take a moment to think about that.

But I note, as is well known: "Absence of evidence for X is not proof of absence of X." where usually this is applied with X = God.
I am so sick and tired of seeing this thrown around like this. It's simply not true. Sometimes absence of proof can actually be proof of absence.

I predict the existence of some thing X.
My hypothesis of X predicts the existence of some other thing Y.
If I fail to observe Y then I am forced to one of two conclusions, Either my hypothesis of X is wrong, or X does not exist. Which conclusion is the correct conclusion depends on what other evidence for the existence of X is available and whether or not my hypothesis of X can be re-written in such a way as to no longer predict the existence of Y.

Not to mention the fact that our whole justice system is predicated on the assumption that the absence of evidence of guilt is evidence of the absence of guilt.

Billy T, I suspect you confused arfa brane with me.
I did. Easy for me to do as both of you are so far advance wrt me in formal math that my memory confusion happened.
... Also, absence of evidence when there must be evidence if the hypothesis is true and a diligent search has been taken is evidence (but not proof) that the hypothesis is wrong.)
That seems to me to be a better qualified version of my (and certainly not original): "Absence of evidence for X is not proof of absence of X." But your better version also concludes the absence of X (or Y in Trippy's "two-step" case below) is not proof, just more supporting evidence for whatever was claimed or denied.

In post 1609 trippy said:
"I predict the existence of some thing X.
My hypothesis of X predicts the existence of some other thing Y.
If I fail to observe Y then I am forced to one of two conclusions, Either my hypothesis of X is wrong, or X does not exist."

My conclusions permitted by his logic also include this third: Or I failed to oberve Y because the search fell short of a fully (to borrow your words) "diligent search." Or even if "diligent" was not done with adequate tools, etc.

While agreeing with ALL the various proofs that 0.999... =1, I think some, especially mine, as developed from a starting definition* and not using any of the regular four operations (+, - , x, or /), is both easier for those with math poor skills to follow (if willing to read it) and "second to none" in its rigor. Do you (or equally respected by me arfa brane) see some not well supported step I make in deriving that a Rational Fraction, RF, equal to the Repeating Decimal, RD is abc / 999 if RD has repeat length = 3 such as: 0.abc abc abc... where a, b, & c are integers in the set {0, 1, 2, 3, 4, 5, 6, 7 ,8, 9} (assuming we work in "base 10")
Or RF = abcde / 99999 if the RD has repeat length of 5 ,etc. when RD = 0.abcdeabcdeabcde...
Etc. for other repeat lengths. I made "a" bold as quite often this methodology does NOT produce the least numerator RF. E.g. gives 123/999 = RD of 0.123123123... not 41/333.

I'm sure others have must have given this same result, but I have not seen it, so it was my independent development, which works for ALL cases, including RD = 0.999... which has all possible "repeat lengths." (If we consider it with a repeat length of 4, then RD = 0.abcdabcdabcd... with a=b=c=d= 9 and the result is RF = abcd/9999 = 9999/9999 = 1/1 =1.

* Trying to have everyting fully defined, I even defined by my defined "& operation" (which resembles addition but only permits unitary increase of decrease steps) the meaning or value of the integers 0, 1, 2, 3, 4, 5, 6, 7 ,8, 9.

Last edited by a moderator:
I Need Help With RP's Simple Explanation--Pease and Thank You--

rpenner.."Whose mistake do you suppose that is?"

Ha ha..yeah I figured mine, but I didn't see where. I think I see what I missed first time around.

Thx for bearing with me on this. I'm fighting off a cold so my head is more foggy than usual and I'm tired, and excuse's excuses, excuses...

what i mis-read or interpreted was that you had "= 0.9 / 0.9" and I only took note of the first 0.9

0.9 / (1 - 1/10 ) has parentheses to indicate the order of operations.

1/10 = 0.1

Yeah I did that

1 - 1/10 = 0.9

I did that

so 0.9 / (1 - 1/10) must be the same number as 0.9/ 0.9 which a good calculator will tell you is 1.

I've been using MS cal. all along. So 0.9 / 0.9 = 1, there got it this time. I see that, I think I also did 0.9 / 0.1 as an error also.

So now let me see if I connect this 0.9 / 0.9 = 1 to and finite 0.999...

I got to scroll and find your 0.999...math again so lets see here--scroll --

0.999... / 10 = 0.0999.... and I verified that no problem two sets of infinities---scroll--

0.999... minus 0.9 = 0.0999....ok did that previously--scroll---

Jeesh my head gets confused as tho it is scrolling around in loops :bugeye:

Ok so I'm having making the connection RP. Why you have to introduce finite 10, finite 0.9 and 0.1 into all of this is not easy for me to follow.

Somehow, it appears to me, that, you appear to take 0.999... and convert it-- not equal it ---to 1 via use of finite #10, finite 0.9 and finite #1.

I honestly and need some clarifying help here. What first appears suspious to me, is when the you have the cal. take a finite 0.9 from and infinite 0.999...

And tho we still have an infinite value as 0.0999...I'm wary of a mathematical trick or something I surely don't understand yet.

And this scrolling back-n-forth is only adding to my frustration of not understand the key points of mathematical transition or alledged proof your offer.

I also believe this is and example of what Chinglu called induction i.e. if 0.9 / 0.9 = 1, then 0.999.. = 1.0.

Somehow you reducing/transitioning 0.999... to same state as 0.9 / 0.9

Thx RP, for this simple, perplexing challenge. I will have to get back to you on this one.

r6

http://www.wolframalpha.com/input/?i=0.9+/+(+1+-+1/10+)

iPhone calculator with memory: [c] [ac] [mc] [1] [-] [1] [÷] [1] [0] [=] [m+] [0] [.] [9] [÷] [mr] [=] (Windows 7 calculator in scientific mode is similar)

iPhone alternate calculation with parentheses: [c] [ac] [0] [.] [9] [÷] [(] [1] [-] [1] [÷] [1] [0] [)] [=][/QUOTE]

Thank you, the post is the post where I take issue with Billy T's characterization of my reference as saying the proof of 0.999... = 1 was elusive, which was neither my idea nor the position of that source or a second source.

... I take issue with Billy T's characterization of my reference as saying the proof of 0.999... = 1 was elusive, ...
I don't recall saying that. Not even sure what it means. In post1600 I did say:

{Chinglu} (originally Alpha Baine's) reference does indicate 1 = 0.999... is not well proven, but then continues to note that if that is not true, then there must be a number N such tha
1> N > 0.999... and then remarks that "none has ever been found."

I have not yet read your post commenting on this but think that the above which I did post, must be "my" (but really just reworded chinglu's) statement you refer as being "elusive." I need to go to bed now so will find your post commenting on my "quasi quote" of Chinglu tomorrow to see if I can understand. I have not read the original reference- only trusted Chinglu not to be distorting it.

... there must be a number N such tha
1> N > 0.999... and then remarks that "none has ever been found."

That idea is easily corrected. It's not that we "haven't found" a number strictly between .999... and 1. Rather, we can logically prove that there's no such number. That's a far stronger statement, with an entirely different meaning and philosophical significance. It means that if we insist on holding that there is such a number, then we must be prepared to abandon logic.

By contrast, it's true that "we have never found" a nontrivial zero of the Riemann zeta function whose real part is different than 1/2. We strongly suspect there is no such zero; but we do not yet have a proof. For all we know, somebody will find such a zero tomorrow and become instantly world famous.

But we can prove from first logical principles and the properties of the real numbers, that there is no number strictly between .999... and 1. If you believe in the real numbers and you believe in logic, you have no alternative but to accept that.

Of course one is always free to say: "I reject the real numbers," or even "I reject logic." But that's the only way you can claim there might someday be a number strictly between .999... and 1.

I hope this point is clear.

About numbers and representations of numbers

To the question: are numbers more fundamental than their representation? Does this imply that numbers exist (somewhere) independently of any such representation, which might better be characterised as a language over an alphabet.

Take for example, the rather clunky Roman numeral system, the symbols are: {I, V, X, L, C, D, M}. There's a set of rules for stringing these together, the first three counting numbers in modern form, {1,2,3}, are {I,II,III}, so only use the I numeral (here, you have a singleton mapped to the length of the string). Then you combine these three strings with V, once on the left for 4: IV, then 'nonce' for 5: V, then VI, VII, VIII, (like: {V} x {I,II,III}), then the same again but with X, and so on, it gets a bit awkward and ambiguous.

The Romans 'mapped' their symbols thusly:

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1000

And needed a rule for 'subtracting' a numeral so that for example IX means 10 - 1, (here, you are free to translate into Latin).

But it was fit for the purpose; it doesn't seem to have any room for fractions, though.
The point being that we tend to use number representations which are 'useful', which is why arabic is nowadays the convention, and we're comfortable with fractions.

But I think it isn't hard to show that any representation is very much a language over an alphabet, these days infinite length strings are known to exist, and we can still represent them, since their existence "somewhere" outside such a representation isn't really a mathematical problem.

My conclusions permitted by his logic also include this third: Or I failed to oberve Y because the search fell short of a fully (to borrow your words) "diligent search." Or even if "diligent" was not done with adequate tools, etc.
Explicit in my assertion is the condition that Y has not been observed when it should have been, a corrollary of which is that the search was a dilligent search. IE: Y was searched for under conditions which the hypoothesis unequivicoally states that it should be found using tools that the hypothesis unequivocally states should be able to find it.

... It's not that we "haven't found" a number strictly between .999... and 1. Rather, we can logically prove that there's no such number. That's a far stronger statement, with an entirely different meaning and philosophical significance. ...
while sleeping, my brain worked* a little more on this "Is there such an N?" and concluded NO (correctly as all who understand math will agree) but it also made the following puzzle about the extension (by induction, regression etc.) of the finite sum to an infinite number of terms:

rpenner in post 1603 first proves the validity of some intermediate steps I have replaced with ... in the following:
$$0.999... = \sum_{n=1}^{\infty} \, \frac{9}{10^n} . ... = 1$$ I'm sorry but can't get it to be what I want all on one line.
I don't do $$so copied above from post 1603 and modified with the three dots, but now made a different mod, with a slight change. My change made the funny little "italic like v" sitting on top of the big summation sign. Not exactly what I expected, but ok so long as you understand the "script v" to be a finite number growing larger with n increasing. I defined a finite sum < 1 , with funny little "script v" finite number of terms, the value of which is the function f(n): f(n) =$$0.999... = \sum_{n=1}^{\n} \, \frac{9}{10^n} < 1$$and then define a slightly larger function of n, N(n), by: N(n) = 1 - 0.5[1-f(n)] which is: 1 > N(n) > f(n) for all n. Now the puzzle is that the very same recursive, or induction, etc. procedure that permit the claim that$$0.999... = \sum_{n=1}^{\infty} \, \frac{9}{10^n} .... = 1$$(Ignore three dots after the small n. With them in my poor "tex", at least it is all on one line.) also produces an N which is between 1 and 0.999... I think this at least causes some doubt about applying that induction, recursive, (or what ever you wish to call it) procedure to extend the finite sum to an infinite number of ever smaller terms converging to the limit 1, without also producing that troublesome "inbetween N." I.e. how does the extension procedure destroy the existence of this "in-between N", as n approaches infinity so that no N exist such that: 1>N>0.999...? * BTW it is a fact that in some phases of sleep, the rate of energy use is as high as the peak rate during the awake state, so the idea that sleep is required to "rest the brain or save energy" is not much accepted. Nearly two decades ago, Crick (or perhaps Watson) of "twistied helix DNA" fame suggested we sleep to reorganize thoughts, memories, things learned during the day etc. when awake. In part this may have been related to fact too many "learing examples" when training up a neural network** computer can reduce its performance - over load or confuse it. It is primary, IMHO because of Crick's DNA fame that his POV about why we sleep is most accepted, but in truth we simply don't know why we sleep, especially as sleep is very common way down the animal kingdom. Sharks must forever swim as they don't have gill flaps to force water thru the gills like fish that can be stationary do. They too must sleep, but they do it "half a brain at a time." I have my own idea as to why sleep is common: When sleeping the normal processing of sensory inputs is greatly suppressed, and motor commands even more so. I. e. In your dreams you can fight fierce battles with no related moves of your body. Likewise you can walk thru walls, get out of speeding cars, run thru fires, fly thru the air (that is quite common) etc. because evolution did not need to prevent such things. There are some restriction on your thoughts in sleep. For example you can not be in two different places in same dream at same time (and a few others which I forget) but in general you are with fewer restriction on your thought and logic. I think at least part of the reason we sleep it to "think-out side the box." On many occasions, like last night, I fall asleep, while thinking about some problem that has me "boxed in." When I awaken, I can often see a "hole in that box" I could not discover while awake. I. e. I think sleep evolved, at least in part to allow the mind to work on problems with less logical restrains or "known facts" limiting the process. That is what happened last night. I found what at least appears to be an "extension by induction etc." problem as the n approaches infinity for the convergent finite sum: 0.999999999999999999999999 (with a finite but large number of 9s) not explicitly written here. SUMMARY: I hope someone more clever than me, will explain why N ceases to exist as 9.999... becomes 1. Until they can, there does seem to be at least some doubt that the induction (or regression etc.) process can be applied and infinite number of times. ** I hate that name for what should be called a "connection machine" or "connection computer." There are no neurons in any man-made (artificial) "neural network computer." In fact most are just simulated (emulated?) and trained up in modern digital computers. Some are then made into real electronic hardware, as for the special job they were train up on they are faster and much cheaper more compact and lighter weight. Most modern "clever torpedoes" use them for their intelligence for these reasons. I lost the battle to get the correct name used more than 4 decades ago.$$

Last edited by a moderator:
Wiki > RPenner > Simple > Drops Infinite Symbol "..." > Lack of Rigor/Explanation

It is great to some one other than me use the word logical, now if can see others begin to use rational and common sense maybe we can arrive at absolute truths and not just at 'for all practical purposes' type thinking

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

I finally went to above link and see where RP got the info for his last reply to me. Here is what I deduce from the first too relatively simple alledged proofs they give, the first one is the one Origin repplied to me with, and it is easy enough for to follow those on cal. and for the life of me, how they arrive at 0.999... = 1.0 is not really show at all. It is just not there.

With 3rd or 4th relatively simple--- "infinite series" ---given, alledged proof/expalnation, and it iappears to be similar to the one RP recent supplied to me recently, I find similar lack of rigor, or whatever we want to call it, that actually proves 0.999... = 1.0,

and one key point in that formula set, that stands out irrespective of the rest is that the last thing they add( + ) is the three dots( ... ) that represent infinite process/procedure/number/value/number set/recursion or whatever we want to call it, so, they add it and then on the other side of the equal sign it just vanishes on the the other side of the equation i.e. they just drop it as tho it is irrelevant for some reason.

Yeah it is irrelevant for practical purposes, but not for absolute truth. How can you have rigor if you just dropp the symbol( ... ) for infinite?

Also I just now see they state 2^ powering and 3 powering, and I dont recall RP stating that in the same equations he replied to me with. I suppose I could go back and do the math again, with the powering, but don't see how that would change my above, that they drop the "..." without explanation. Duhh!

Clear( if not simple ) explanation > clear( if not simple ) communication > clear( if not simple ) resolution or so we would hope.

So, in review and summary of the above, both Origins given wiki explanation and RP's given quasi-wiki explanation t both fall short of a proof--- at least that is how at appears to me, and I really have no idea how others can see those at true proofs ---since they just don't do that. Sincerely lacking some key steps of clear explanation ergo no clear communication has occurred with me.

r6

No 'n" Between Finite and Infinite-Logica and Rationall Math Simpleton--

No 'n" Between Finite and Infinite is rational, logical and common sense conclusion to me. And despite RP's concerns-- and several others in other threads ---I'm not nearly crazy either.

I'm glad to see others( trippy? or someguy? ) begin to use terms logic or logical, as associated with our thought process's on this issue Now if they can begin to use terms rational and common sense then we may arrive at a common resolution/agreement to this issue. Not likely tho, and more so with wiki info that clearly is lacking rigor. or lacking something. imho

It is interesting the BT, has offered us a formula/equation that infers/implies there must exist an "N" between infinite 0.999... and finite 1.0. Now that may be a crux/connundrum if his equation/formula are valid.

Untill I have the dropped infinite symbol of dots "..." explained to me, I'm forced to once again appeal to common sense from those more educated in mathematics, and ask that at least admit, the illogic(not common sense ) of stating that infinite 0.999... = finite 1.0. I think I recally RP actuall stating something like that.

rational col-.irrational column
-------1------.---0
-------0------.---999...

Infinite value 0.999...approaches value 1, but it common sense tells us, that and infinite process is eternally existent i.e. the 9's never stop ergo 1 is never attainded. This is rational, logical common sense conclusion, irrespective of mathematical illusions that drop symbols "..." of infinity for no apparrent reason and with no/nada/zip explanation for doing such.

At least we can give credit to the wiie page acknowledge infinite set of 9's early on, and before their going on into illusionary mathematical and alledged proofs. Who wrote that page anyhow? Origin? RP? Or others who also drop the infinity representation without explanation or clear/simple explanation?

As for sleep, I used to use the analogy of Nortons Disk Doctor process on Macs OS 10> i.e. when optimising the disorganized blocks of data, on the screen we saw teh cursor doing many passess across the screen in various colors, to assess where blocks of data have been placed, and reorganize thos blocks into a more orderly pattern at least on the screen in with the color patterns.

I like BT's explanation also, wherein the brains ideas can get out of rational box and let itself go. A release of sorts. Isn't that what some mind altering substances allow us to do? To see our ordinary reality in extraodinoray way.

I did my share of mind altering substances beginning at age 16, but I never had dreams of my self having anti-gravity abilities--- floating above the ground ---untill my 40's. It was a very cool as it was like a wave-linear floating 4ft off ground and moving forward along side river, above a pathway. This happen a few times in 40's but it upon awakeing I became a little unerved when thinking, i sure dont want this to transcend into my conscious reality

Those floating dreams mostly subsided and in my 50's I can only recall one of those occurring a year or so back. That is fine by me. Keep them suppressed is ok by me, because I recall my bad( tripped out )--- i.e.drugh induced pyshcohis ---at 17, wherein it was as tho I was acting out a subconscious dream state during my conscious awake state, and it was only my friends/buddies who go hold of me kept me in my car where i would not be running out into traffic.

As to coming to a resolution via my dreams, that has never happen to me. I have awoken and then saw more clearly what I need to do, and only attribute that to being more refreshed. I have no doubt the importance of sleep and especically REM sleep, where males always have and erection, and humans can not recall dreams during REM sleep.

Here is where I see and analogy to our finite vs infinite issue. What I have experienced in my dreams, is that on many occasion, I have just been shot, or impacted the Earth and I awake. To me this like an infinite condition being put to my brain, and it resolves this infinity by becoming conscious.

Infinites = unconscious dreams ergo Billy T" outside of a finite box

Finites = consicous states associated with our cosmic limits of a physical box, that has the metaphysical laws/principles written on the inside if not also or rather than outside of the box.

r6

Last edited: