# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

You forgot to reply to some of my replies to you.

In particular, this one and this one.

Also:

Now I'm getting confused. If 0.999...=1, then does it even mean anything to say, "first number?"

What does the "first number" mean, anyway? You need to specify that you're talking about counting numbers if you want to say "1 is the first number".

I mean, if I list three phone numbers: 555-1234, 555-7632 and 555-2356 then the first number is 555-1234.

On another issue, you're still getting it wrong. If 0.999... = 1 (which it does), then they are two different representations of the same number.

It makes no sense, then, to ask if 0.999... comes before or after 1. They are the same number.

Get it?
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And on the other example you were confused about:

0.25 + 0.25 + 0.25 + 0.25 = 0.999.... = 1

because this is equivalent to

0.24999.... + 0.24999... + 0.24999... + 0.24999.... = 0.999....

Let me give you a few more examples:

99.9999... = 100
0.741239999.... = 0.74124
65.19999.... = 65.2
etc.

See the pattern?

The fact that we can write the same number as 1 or as 0.99... is just an artifact of our base-10 number system. I have shown you that similar things happen in any base.

You can't because you don't understand the notation you are manipulating and so you convey only misunderstandings.

$$S_n$$ refers to an entire family of numbers. $$S_n$$ doesn't have any specific value until n is specified.
For every counting number n, all of the following are true $$S_n = 1 - 10^{-n}, \; S_n \lt S_{n+1}, \; S_n \lt 1, \; S_n \lt S$$
Thus for every number X which is less than 1, it follows that most of the family $$S_n$$ are larger than X. We can be specific and write $$m(X) = \frac{- \ln ( 1 - X ) }{\ln 10}$$ and prove that for every n greater than m(X), $$X \lt S_n \lt 1$$ is true.
Proof:
$$\begin{eqnarray} m(X) & \lt & n & \lt & n+1 \\ \frac{- \ln ( 1 - X ) }{\ln 10} & \lt & n & \lt & n+1 \\ - \log_{10} ( 1 - X ) & \lt & n & \lt & n+1 \\ \log_{10} ( 1 - X ) & \gt & -n & \gt & -(n+1) \\ 1 - X & \gt & 10^{-n} & \gt & 10^{-(n+1)} & \gt & 0 \\ X - 1 & \lt & -\left(10^{-n}\right) & \lt & -\left(10^{-(n+1)}\right) & \lt & 0 \\ X & \lt & 1- 10^{-n} & \lt & 1 - 10^{-(n+1)} & \lt & 1\\ X & \lt & S_n & \lt & S_{n+1} & \lt & 1 \end{eqnarray}$$​
The number 1 is special in the function m() in this proof because if it were any larger number then the proof would be trivial and if it were slightly less than 1, the proof would fail. Thus 1 is the least possible upper bound to the entire family $$S_n$$.
Thus no matter how close any particular member of the family $$S_n$$ is to 1, all of the higher-numbered members of the family (most of them) are closer to 1. That's the meaning of $$\lim_{n \to \infty} S_n =1$$ for a family indexed by counting numbers.
http://en.wikipedia.org/wiki/Limit_of_a_sequence
For the same reasons, $$\lim_{n \to \infty} S_n = S$$ and since equals to one thing must be equal to each other. $$S = 1$$.

Likewise for the family $$T_n$$:
$$T_n = 9 \times 10^{-n}, \; T_n \gt T_{n+1}, \; T_n \gt 0, \; T_n \gt \lim_{k \to \infty} T_k$$, For any X greater than 0, for all values of n greater than $$-\log_10 \frac{X}{9}$$, $$0 \lt T_n \lt X$$ and $$\lim_{n \to \infty} T_n = 0$$​

That's not a case, that's a whole family of cases that says "as n takes on ever increasing successive counting numbers". The full expression for $$\lim_{n \to \infty} S_n = S$$ reads as "The unique limiting value approached by the family $$S_n$$ as n takes on ever increasing successive counting numbers is 1."

There is no case where there is a value of n (which is limited to counting numbers) such that $$S_n = 1$$, but because all such n are finite, neither is there a value of n such that $$S_n = S$$.

So, essentially you are saying that:

$$\lim_{n \to \infty} S_n = S$$ where $$S = 1$$ and $$\lim_{n \to \infty} T_n = 0$$.

Am i right, this time?

Yes, that's right hansda.

0.25 + 0.25 + 0.25 + 0.25 = 0.999.... = 1

because this is equivalent to

0.24999.... + 0.24999... + 0.24999... + 0.24999.... = 0.999....

So a dollar isn't 4 .25 pieces, it's 4 .24999... pieces. No wonder I'm all screwed up.

Let me give you a few more examples:

99.9999... = 100
0.741239999.... = 0.74124
65.19999.... = 65.2
etc.

See the pattern?

Yes. The left sides all have ... at the end, which means, to be completed. Just wondering how you concluded the right side when the left side is not complete. In other words, they all say, "infinite=finite." Got it, James. Thanks for pointing that out. I appreciate it!

The fact that we can write the same number as 1 or as 0.99... is just an artifact of our base-10 number system. I have shown you that similar things happen in any base.

So .9=1 because .1+.1+.1+.1+.1+.1+.1+.1+.1=.9, so .9 is 100%, not 90% as I always used to think.

So, essentially you are saying that:

$$\lim_{n \to \infty} S_n = S$$ where $$S = 1$$ and $$\lim_{n \to \infty} T_n = 0$$.

Am i right, this time?

Yes, that's right hansda.

If you say $$\lim_{n \to \infty} T_n = 0$$ but $$T_n \ne 0$$ for any value of $$n$$ because $$T_n$$ is a term of a geometric series and so it can not be $$0$$.

$$T_n$$ can only tend towards $$0$$ but $$T_n$$ can not be equal to $$0$$.

I think symbolically we can write this as: " $$T_n \to 0$$ as $$n \to \infty$$".

So a dollar isn't 4 .25 pieces, it's 4 .24999... pieces. No wonder I'm all screwed up.

0.24999... = 0.25

They are the same number.

I won't speculate on why you're screwed up.

Yes. The left sides all have ... at the end, which means, to be completed.

No. The dots mean only that there's an infinite string of 9s there.

A number is not a process. It's a number. You don't construct numbers. You don't start building a number and then have your local council sign off on it once it's complete. A number just is.

Just wondering how you concluded the right side when the left side is not complete. In other words, they all say, "infinite=finite." Got it, James. Thanks for pointing that out. I appreciate it!

0.999... is an infinite string of nines after a decimal point, which can also be represented as the single digit "1".

So, yes, in this case an infinite string represents the same number as the finite string.

So .9=1 because .1+.1+.1+.1+.1+.1+.1+.1+.1=.9, so .9 is 100%, not 90% as I always used to think.

Do that again and I'll ban you for trolling. I'm serious.

You know that nobody is arguing that 0.9 = 1. What is being argued is that 0.999... = 1.

Clearly, 0.9 is different from 0.99, which is different from 0.999999, which is different from 0.999999999999, which is different from 0.999... Only the last of these is equal to 1.

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And you've still (conveniently) forgotten post #1041, above.

I know you're trying to drop your silly pizza argument now, but you can't escape that easily. I won't have you forgetting what has gone before, like you forgot about our relativity box discussion. Age is no excuse. You must do better.

0.24999... = 0.25

1=.9

They are the same number.

So are .9 and 1, they are 100%. The same. See?

I won't speculate on why you're screwed up.

You'd ponder that infinitely, and then round it off to, "he's not."

If you say $$\lim_{n \to \infty} T_n = 0$$ but $$T_n \ne 0$$ for any value of $$n$$ because $$T_n$$ is a term of a geometric series and so it can not be $$0$$.

$$T_n$$ can only tend towards $$0$$ but $$T_n$$ can not be equal to $$0$$.

I think symbolically we can write this as: " $$T_n \to 0$$ as $$n \to \infty$$".
From what I recall from Maths class, this is because we cannot actually reach infinity.
But if we could, $$T_\infty$$ = 0.
If it doesn't = 0 then there must be another $$T_n_+_1$$ that is half way between $$T_n$$ and zero, and thus n would not yet have reached $$\infty$$

From what I recall from Maths class, this is because we cannot actually reach infinity.
But if we could, $$T_\infty$$ = 0.
If it doesn't = 0 then there must be another $$T_n_+_1$$ that is half way between $$T_n$$ and zero, and thus n would not yet have reached $$\infty$$

$$T_\infty \ne 0$$, because $$T_\infty$$ is still a term of the geometric series.

No. The dots mean only that there's an infinite string of 9s there.

Yeah, and there is infinite distance in space, but I don't go around claiming that since distance never ends it equals 1, that is absolutely insane!

HOW MANY 9's are after the point, James? When you answer that question in a finite way, then you've done something. To maintain a position that says that the two words "infinite" and "finite" are equal is just pure insanity! Nothing less!

$$T_\infty \ne 0$$, because $$T_\infty$$ is still a term of the geometric series.
If $$T_\infty \ne 0$$, then let's say $$T_\infty = X$$.
According to the geometric series there must be an $$n$$ that exists such that $$T_n < X$$ but $$>0$$
Thus $$n \ne \infty$$

The only situation where this wouldn't be the case is where $$T_\infty = 0$$

If $$T_\infty \ne 0$$, then let's say $$T_\infty = X$$.
According to the geometric series there must be an $$n$$ that exists such that $$T_n < X$$ but $$>0$$
Thus $$n \ne \infty$$

OK

The only situation where this wouldn't be the case is where $$T_\infty = 0$$

I never said $$T_\infty = 0$$.

Undefined:

You've posted a huge long post, yet still failed to answer the question I asked you. I remind you:

1/2 + 1/4 + 1/8 + ... = 1. True or false?

This is a mathematical statement. Either you think it is true, or you think it's false. There's no half-way house. So, which is it?

Let's look at some of what you wrote. I really don't want to wade through all of it.

A bit strange, isn’t, that you demand an answer from someone whom you just ‘sent off the field’’ for three days, so preventing him from responding?

I have not demanded anything of you. Here you are, making post after post in this thread, yet you seemingly don't have an actual point of view on the simple question above. All I see is waffle.

Also, didn't anybody ever tell you that using ALL CAPITALS on the internet is considered to be SHOUTING? Do you want to look like some kind of deranged nut? Because that's the kind of person who mostly uses random capitals on the internet.

And you wanting it both ways by demanding an answer from me that you ‘want’ over what my answer MUST LOGICALLY BE consistent with all the context/discussion/explanations so far in this thread that points to YOUR stance being UNreal and hence UNANSWERABLE in any way that is real and meaningful when the very BASIS of your stance is the whole POINT of this thread/discussion, seems a little disingenuous.

I'm having trouble unpacking what you even mean by all that. All those CAPITALS for emphasis, and you still haven't communicated.

How often have self-appointed ‘experts’ of one sort or another told a ‘crank’ or whomever something to the effect: “It may not be the answer you wanted, but it’s the only answer you are going to get; and just because you don’t understand or don’t like it, tough!”

Very often indeed, I'm sure. What's wrong with that?

So, James, am I going to be banned for NOT ANSWERING HOW YOU WANTED OR LIKED ME TOO?

Not by me. Not at this stage.

I wouldn't have thought that answering a simple question like the one above, about your own personal beliefs, would be as hard as it apparently is for you. Maybe you should just leave this thread alone.

Never mind, though, it’s just the sort of ‘strangeness’ that creeps into any situation where people get ‘all steamed up’ about things because they are so convinced they are right because their circuitously based and argued/’proofed’ philosophy-based, unreal-Axioms-derived, ‘math system’ tells them they are right, despite all the self-evident reality-based observations to the contrary to which these same mathematicians are deaf while they repeat obviously (as demonstrated by reality based logics and arguments already) flawed and incomplete and self-referential ‘statements and claims’ which haven’t YET been ‘proofed’ via any INDEPENDENT REALITY BASED arguments/axioms.

Does this mean that you're leaning towards saying the statement above is false, then? I really can't tell.

But you keep deriding and evading and threatening banning etc OTHERS who keep telling you what you don’t want to hear, while your own arguments are demonstrably and repetitively trivial and beside the point being made to YOU.

Forget my arguments. How do you respond to post #915, above? That has put the nail in the coffin in this thread, as far as I'm concerned. What's your analysis of that?

James, why should I or anyone else even bother playing the ‘unreal/uncomplete math game’ back at you when I have already given you and arfa, Trippy et al the whys and wherefores your math game is based in unreality and philosophy instead of any reality/sane logics/arguments/outputs?

Nobody has asked you about reality and unreality. For my part, I simply want to know whether you agree with the truth of the above mathematical statement. It's a formal mathematical statement in a formal mathematical system. So, within that system, is it true or false?

And, while we're at it, is it true that, in the same system, 0.999... = 1?

One you've answered those questions, then maybe we can start getting all philosophical and start pondering whether this formal mathematics actually works in "real" world, whatever that is.

James, it would help you be less ‘emotionally attached’ to patently incomplete orthodoxy...

Ok. So help me.

Explain to me what is patently incomplete about my orthodoxy. Because it isn't patent to me at the present time.

Look, James, arfa, I already explained to Trippy that all these 1/3 etc TRIVIAL manipulations, constructions/deconstructions of symbolic terms are NOT ‘proof’ of anything at all EXCEPT the notation/convention used and NOTHING ELSE.

Do you include post #915 in that?

They are NOT any actual process/operation completed, they are ASPIRATIONAL STATEMENTS at best, and misleading assumptions at worst (since in many cases the implied operation is never actually started let alone completed to IDENTIFY EXACTLY what those 1/3 etc expressions actually evaluate to EXACTLY.

You seem to think that a number is a process. Or do I have you wrong?

Also, I'm puzzled as to what you think is "real" about a number such as 7 or 1 or pi. I'd say all numbers are abstractions. We can talk about the number 3, but it's an abstraction until we start talking about 3 sheep or 3 dollars or 3 tons of salt.

So, James, everyone, it is STILL the case that no repetition and demands from ‘current unreal maths’ can be ‘answered’ PROPERLY except by INDEPENDENT perspectives/answers which do NOT depend on the unreal circuitous ‘current maths answers/demands’ TRYING TO SHAPE AND CONSTRAIN the discussion/answers to “what is acceptable to current maths” which has been already well shown NOT to be comnplete and hence IN NO POSITION to LIKE or DEMAND anything when it is the INDEPENDENT answers that matter in the final analysis/review going on here and elsewhere IN REALITY TERMS starting premises rather than dead-end unreality philosohical starting terms.

It now sounds like you're saying you have no idea whether 0.999... = 1 or not. Is that correct?

Consider what the THINKING mathematicians themselves have already realized. Apart from the TRIVIALITY of proffered ‘formal proofs’ so far from conventional maths contruct, there is the inescapable fact as Goedel and others have recognized for some time now: One cannot use arguments from WITHIN an abstract philosophical/maths construct to ‘prove’ anything about that same construct; because such proffered ‘proofs’ are NOT INDEPENDENT, but inescapably, circuitously self-referential.

I think you've misunderstood Goedel. Of course you can use arguments within a formal system to prove statements made within that system. If you couldn't all mathematics would be useless. Goedel's theorems are about the completeness and consistency of formal systems.

Get it now? The ONLY INDEPENDENT system is the REALITY; and that is the ONLY FINAL ARBITER construct from within which REAL PROOFS ARE POSSIBLE that are not circuitous and self-referential, BECAUSE there is no abstraction involved, since the objective physical reality is ALL THERE IS and IS intrinsically, logically, physically and demonstrably COMPLETE and CONSISTENT irrespective of any partial/incomplete abstract ‘takes’ from it by unreal philosophical/maths modeling by us.

So, tell me how REALITY would work out whether 0.999... = 1 or not.

Can’t you ‘get’ that Motor Daddy has ALREADY pointed out to you that the TRIVIAL CONSTRUCTION and equally TRIVIAL DEconstruction exercise is VERBOTTEN from first principles? Mere construction of a COMPOSITE UNITARY from OTHER INDIVISIBLE UNITARY identities is a PHILOSOPHICAL HEIRARCHICAL CONCEPTUAL exercise and NOT a mathematical/logical PROCEDURE that proves anything like what youn WANT it desperately to ‘prove’?

I don't think Motor Daddy has pointed out anything of the kind to me.

Maybe you can explain without all the CAPITALS and waffle.

It all STARTS from that, and any further operations can be trivial or meaningful depending on the exercise and the unitary involved. For example YOU have DECONTRUCTED “1” DAY into 24 hours, thus EFFECTIVELY converting an ODD unitary into a number of EVEN UNITARIES SUB-UNITS. So naturally that TRIVIAL exercise will suit that trivial purpose, but it does not ‘prove’ anything, nor can you base any conclusions at all on it, because it IS trivial and SELF-SELECTIVE in the construction which will ‘output’ whatever YOU built into it ARBITRARILY and TRIVIALLY.

I thought it nicely demonstrated the point that 1 thing can be divided equally into 3 parts, which was an issue Motor Daddy disputed at the time. Now, he appears to be running away from that silly claim. But perhaps you'd like to take up his baton?

It is timely to point out that the phrase: You can’t get THERE from here”, and “You can’t get BACK HERE from THERE”, are quite seriously apt to illustrate where all your and others current mathematical ‘trips’ are NON-STARTER and/or NON-SEQUITURS from any ‘LIMITS based false starts ’ you imagine when you just write down 0.999..., 1/3 etc as if they identify anything real at all.

Nobody's talking "real" yet. This thread is about the mathematical truth or falsity of 0.999... = 1, isn't it?

Merely invoking philosophical INFINITY concept does NOT actually ‘get you there’ from 1/3, or 1/9 etc , because you never can, hence the 0.333..., 0.111... etc.

Nor can you ever ‘commutate’ backwards from infinity of 1/2+1/4+... back to 1 unitary, since you NEVER WERE AT INFINITY to ‘start back from’.

This sounds (a) like you think that 1/2 + 1/4 + 1/8 + ... doesn't equal 1, and (b) like you think a number is a process.

Do you want to address either of these points in more depth?

1=.9

So are .9 and 1, they are 100%. The same. See?

Since your post was 4 minutes after mine, I'm going to give you the benefit of the doubt and assume you hadn't read my post.

Next time, you're done for trolling. Just so you know.

Yeah, and there is infinite distance in space, but I don't go around claiming that since distance never ends it equals 1, that is absolutely insane!

You're obviously confused. Look:

0.9 + 0.09 + 0.009 + ... = 1

See how the terms on the left hand side get smaller and smaller and smaller as you go to the right?

Tell me: why do you think that sum will necessarily be infinite rather than finite? Or is this just another silly Motor Daddy gut feeling with nothing to back it up? Similarly, consider:

1/2 + 1/4 + 1/8 + ... = 1

This is the one that Undefined is struggling so hard with.

Space, by the way, is not the same as a number. See my comments to Undefined, above, for example.

Let's stick to the mathematics for now. We can sort out space later. You have to walk before you can run.

HOW MANY 9's are after the point, James? When you answer that question in a finite way, then you've done something. To maintain a position that says that the two words "infinite" and "finite" are equal is just pure insanity! Nothing less!

This is looking an awful lot like trolling, MD.

There is an infinite number of 9s after the point.
I have never claimed that the words "finite" and "infinite" mean the same thing. If that's what you say you're getting from my posts then I can only conclude that (a) you're stupid; or (b) you're trolling; or (c) you're not really paying attention; or (d) one or more of the above.

If you're confused about something, ask me a question. But don't you dare put words into my mouth again.

I never said $$T_\infty = 0$$.
I know you didn't.
I did.
It is the conclusion of the part you accepted with "OK" plus my previous post.

In the geometric progression there exists a term $$T_n$$ such that $$T_n > T_n_+_1 > 0$$

If $$T_\infty \ne 0$$ then let $$T_\infty = X$$

Now if $$X \ne 0$$ then there must be a term $$T_n = X/2$$ (if talking of the geometric progression of continually adding half the previous term etc).
I.e. there must be a term $$T_n$$ such that $$X > T_n > 0$$

But if $$X = T_\infty$$ then you're saying that $$T_\infty > T_n > 0$$

So what is $$n$$?
How can it be $$\infty + 1$$?

So, as said previously, the only $$X$$ such that $$T_\infty = X$$ such that there is no $$T_\infty > T_n > 0$$ is when $$X = 0$$.
i.e. when $$T_\infty = 0$$.

$$T_\infty \ne 0$$, because $$T_\infty$$ is still a term of the geometric series.

Actually because $$\infty$$ is not a counting number; so, whatever $$T_{\infty}$$ is supposed to mean, it is not part of the family of values $$T_n$$.

If $$\infty$$ were a counting number then $$10^{\infty} \gt \infty^10 \gt \infty + 1 \gt \infty$$. Actual number systems that use numbers larger than any finite number have infinite families of the infinitely large, but don't use the symbol $$\infty$$ because that symbol already has too many different meanings.
Specifically in the hyperreals where H is any number larger than all finite numbers, $$\lim_{n \to \infty} (1 - 10^{-n}) = \textrm{st}\left( 1 - 10^{-H} \right) = 1 - \textrm{st}\left( 10^{-H} \right) = 1 - 0 = 1$$ because by definition $$\lim_{n \to \infty} A_n$$ is equal to the "standard part" of the expression $$A_n$$ as n is replaced with any infinitely large number so long as that "standard part" doesn't depend on which infinitely large number is being used.

I know you didn't.
I did.
It is the conclusion of the part you accepted with "OK" plus my previous post.

In the geometric progression there exists a term $$T_n$$ such that $$T_n > T_n_+_1 > 0$$

If $$T_\infty \ne 0$$ then let $$T_\infty = X$$

Now if $$X \ne 0$$ then there must be a term $$T_n = X/2$$ (if talking of the geometric progression of continually adding half the previous term etc).
I.e. there must be a term $$T_n$$ such that $$X > T_n > 0$$

But if $$X = T_\infty$$ then you're saying that $$T_\infty > T_n > 0$$

So what is $$n$$?
How can it be $$\infty + 1$$?

So, as said previously, the only $$X$$ such that $$T_\infty = X$$ such that there is no $$T_\infty > T_n > 0$$ is when $$X = 0$$.
i.e. when $$T_\infty = 0$$.

What point you are trying to prove here?

Actually because $$\infty$$ is not a counting number; so, whatever $$T_{\infty}$$ is supposed to mean, it is not part of the family of values $$T_n$$.

If $$\infty$$ were a counting number then $$10^{\infty} \gt \infty^10 \gt \infty + 1 \gt \infty$$. Actual number systems that use numbers larger than any finite number have infinite families of the infinitely large, but don't use the symbol $$\infty$$ because that symbol already has too many different meanings.
Specifically in the hyperreals where H is any number larger than all finite numbers, $$\lim_{n \to \infty} (1 - 10^{-n}) = \textrm{st}\left( 1 - 10^{-H} \right) = 1 - \textrm{st}\left( 10^{-H} \right) = 1 - 0 = 1$$ because by definition $$\lim_{n \to \infty} A_n$$ is equal to the "standard part" of the expression $$A_n$$ as n is replaced with any infinitely large number so long as that "standard part" doesn't depend on which infinitely large number is being used.

May be you are right but do you think $$T_n$$ being a valid member of a geometric series can ever be $$0$$ for any value of $$n$$?

Only in math systems where $$\infty + 1 = \infty$$ like the extended real line. In such number systems $$0 = \frac{9}{\infty} = \frac{9}{10^{\infty}} = 9 \times 10^{-\infty}$$ But it is still bending the rules since the family of $$T_n$$ was only defined for n in the counting numbers. Since $$\infty$$ is not a counting number, this $$T_{\infty}$$ is still not part of the family being considered.

So just in number systems where $$\infty + 1 = \infty$$, we can correctly write $$\lim_{n\to \infty} S_n = S_{\infty} = S = 1$$ while in the ordinary reals or hyperreals we can write $$\lim_{n\to \infty} S_n = S = 1$$ and still be correct.