A simple point of view on 0.999... [base 10] = 1

Doron Shadmi · November 24, 2005

In the attached page http://www.geocities.com/complementarytheory/no1.pdf we can understand that there is a deep connection between our abstract ideas and the ways that they are represented by us.

I'll be glad to know your point of view.

Doron Shadmi · November 24, 2005

The length 0.222... is not different just because it is drawn in a different position of the line! Yes, the purple rectangle indicates a different interval than the green one, but the value 0.222... is the same value wherever we draw an interval that have it as its length.

Using this silly argument we could also say such a thing as:

"8-7 is not the same as 3-2.
Since 123456789
and the purple part is definitively not the same as the green part."

but of course this is wrong: 8-7 = 3-2 = 1 = 0.999...
(where ... means that we take the limit as the number of decimals increases without bounds)

Juma,

All you did is to simply ignore my argument as if it does not exist in front of your mind.

0.222... has an exact position in 2.222... and the same holds for 0.999... in 9.999... and generally any 0.###... (where # is the highest value of any base n>1) has an exact position in #.###...

My new representation method clearly shows that the initial 0.###... entity is definitely not the result of #.000*X/#.000 = 1, and all you do is to eliminate 0.###... by #.###... - 0.###... subtraction, and then you use #.000*X/#.000 = 1 in order to get your requested result, which is clearly not 0.###... entity (marked by purple in my argument), which was simply replaced by you by 1 (which is another entity, marked by green in my argument) in order to get the requested result, which is false if you insist that the initial X=0.###... and the result X=1 are the same mathematical entity.

What I have discovered in this argument is that there is an inseparable connection between our representation methods and our abilities to understand correctly abstract ideas/insights.

It means that the standard linear representation method actually prevents from us to deeply understand abstract thoughts, and new and better representation methods have to be developed, exactly as I did in this particular argument.

Any non-finite collection is incomplete by definition, as I clearly and simply prove in (and this incompleteness is represented by 0.000...1 where 1 is a permanent successor that permanently cannot be included in the non-finite and incomplete .000... sequence, and "..." does not mark our inability to write down a non-finite sequence of zeros, "..." actually says that any non-finite collection is incomplete by its vary own nature):

http://www.createforum.com/phpbb/viewtopic.php?t=39&start=15&mforum=geproject

http://www.createforum.com/phpbb/viewtopic.php?t=45&mforum=geproject

In other words, there is no such a thing like a complete non-finite collection.

You can say "show me a natural number which is not in N".

My answer to this question is very simple:

Natural numbers are defined by their axioms and the existence of each n in N does depend on how much n members are in N.

In other words, if we want to understand what a non-finite collection is, we have to deeply research the Successor concept, as I did in the above links.

0.999... means lim(n->inf) [ sum(i:0->n) [ 0.9 / 10^i ] ].
This representation is not an imprecise approximation of some other number, but it is that number. It is that number because it is defined as the limit of increasingly precise approximations. It is as precise as possible to that number, which is to say, exactly that number.

I agree with you that "It is as precise as possible to that number" that cannot be that number, because any non-finite sequence is incomplete by its vary own nature.

Therefore 1 is not the limit of 0.999... because each "9" represents some scale-level of an endless fractal, and if 0.999... = 1 than this endless fractal does not have non-finite scale levels, which is impossible if 0.999... is a path along this endless fractal.

Since 0.999... [base 10] is a path along an endless (non-finite) fractal (as can clearly be seen in my 0.222... [base 2] which apears in my first message of this thread), then 0.999... [base 10] not= 1

Please explain what 0.00.....1 or 9.000....1 means. If you think that 0.999... is not equal to 1 what number is between them. Between any two distinct real numbers there is always another real number.

ex-xian,

The answer is very simple.

Between any base n>1 0.###... number there exists a base (n>1)+1 0.###... number, for example:

In this example we can clearly see that 0.222... [base 2] is between 0.111... [base 2] and 1.

All the numbers in the above example have infinite precision. The proof that 0.999... = 1 relies on uncontroversial premises of transfinite set theory. That numbers don't have infinite precision in "real life" is sublimely ignored by mathematicians, with a slight narrowing of the eyes, a lift to the nose and a subtle pursing of the lips.

There is no such a thing "uncontroversial premises" of abstarct ideas.

In other words, in my work I clearly show that no non-finite collection has an accurate cardinal.

Therefore the Cantorean transfinite universe does not hold.

For more details, please look at:

http://www.createforum.com/phpbb/viewtopic.php?t=45&mforum=geproject

http://www.createforum.com/phpbb/viewtopic.php?t=39&start=15&mforum=geproject

The problem with this picture is that convenient ". . . " you have.

If we disregard the ". . . " then it is true that for any number of decimal places,

.11 [base 2] < .22 [base 3] < 1,
.1111 [base 2] < .2222 [base 3] < 1,
and .111111 [base 2] < .222222 [base 3] < 1.

But, it is not true that
.11 [base 2] < .2 [base 3] < 1.

The interesting thing that happens is when we use the convenient ". . . ". These numbers converge to each other and all of them are equal to 1. (remember that ". . . " is just short hand for a well defined limit)

This is a simple result of the fact that they converge as you add decimal places. The numbers "go to" 1 as the number of decimal places increases. A good way to understand this is that in order to measure any difference between the [base 2] and [base 3], the decimal must be terminated somewhere and if we do this, we are no longer considering the " . . . ".

1) I am talking always about the highest value of each [base n>1] non-finite sequence.

For example:

The highest value of base 10 is 9,

The highest value of base 2 is 1,

The highest value of base 3 is 2,

...

The highest value of base n+1 is n

2) The examined sequences must have the same number of scale levels.

So, your example does not hold in this case.

No, each one of them has its own unclosed gaps between itself and 1, and these unclosed gaps are the interesting things that can be found on non-finite scale levels along each unique path that exists along the non-finite fractal.

OK, we'll use your picture to try to explain this a little better. I think we have different ideas of what ". . . " really means. Let's consider each "scale level" as you like to call them. We'll just number each scale level in your picture with the naturals like so:

1 ----> .1 [base 2] < .2 [base 3] < 1
2 ----> .11 [base 2] < .22 [base 3] < 1
3 ----> .111 [base 2] < .222 [base 3] < 1

and so on.

It should be clear that there is a difference between all of these numbers (and we can compute this difference if we want to). What it seems like you are saying is that for any "scale level" there must be a difference between all three numbers and I will agree with you. But that is not what ". . . " means.

Consider the function f() that is the difference between 1 and each "scale level":

1 - .1 [base 2] = f(1)
1 - .11 [base 2] = f(2)
1 - .111 [base 2] = f(3)
1 - .1111 [base 2] = f(4)

and so on.

Now there is no "magic" n where f(n) = 0.
In fact, there does not exist an n so large such that f(n) = 0.
But, there is also no number, call it "e," so small that f(m) > e, for any m.

In conclusion, ". . . " does not mean that f(n) = 0 for some n, it means that f(m) < e for any arbitrarily small e.

"..." says exactly what I mean, which is:

Any non-finite collection/sequence is incomplete by its vary own nature, and e is not some number along the non-finite sequence that is smaller than each f(n), but it is a permanent and invariant proportion of self-similarity upon non-finite fractal's scale-levels, and this proportion > 0 , and it determined according to the structure of each non-finite sequence.

I guess ". . . " can mean whatever you want it to.

I agree with you, but if I have a choice then I prefer the simpler, richer and nicer interpretation to "..." .

A part of my work, about this case ( http://www.createforum.com/phpbb/viewtopic.php?t=45&mforum=geproject ), which deals with the Successor concept from a totally new point of view:

My concept of a non-finite collection is based on a "cloud-like" magnitude of any collection of infinitely many elements, for example:

Let us take for example the non-finite collection of the Natural numbers.

The Successor of this collection is notated as +1, because the simplest structure of the Natural numbers is the non-composed and non-finite collection that is notated as {1,1,1,1,1,…}+1, where +1 (the Successor) is the permanent next element, the existence of which was proven by Cantor’s second Diagonal method.

If the Identity map of a non-finite collection does not exist, then its exact cardinality does not exist and the Natural numbers’ cardinality is |N|-Successor, because the Successor is permanently out of our desirable “complete” domain.

Let @ be |N|-Successor

If A = @ and B = @-2^@, then A > B by 2^@, where both A and B are collections of infinitely many elements.

Also 3^@ > 2^@ > @ > @-1 etc.

So as we can see, in my universe I have both non-finite collections and unique arithmetic between non-finite collections, which its result is always a non-finite collection.

My results are richer than the Cantorean transfinite universe, for example:

By Cantor aleph0 = aleph0+1 , by me @+1 > @ .

By Cantor aleph0<2^aleph0 , by me @<2^@ .

By Cantor aleph0-2^aleph0 is undefined, by me @-2^@ < @ .

By Cantor 3^aleph0 = 2^aleph0 > aleph0 and aleph0-1 is problematic.

By me 3^@ > 2^@ > @ > @-1 etc.

|{{1,1,…}+1, 1,1,1}| > |{{1,1,…}+1}| by |{1,1,1}|.

|{{1,1,…}+1,{1,1,…}+1}| = |{{1},{1}}|•@ > |{{1,1,…}+1}| by |{1}|•@ and

|{{{1,1,…}+1, 1,1,…}+1}| = |{{1},1}|•@ > |{{1,1,…}+1}| by |{1}|•@ but they have different internal structures

( {{1},{1}} and {{1},1} ).

For further information, please read http://www.geocities.com/complementarytheory/Successor.pdf.

In fact:
.222... [base 3] == 2 Sum[1/3^k,{k,1,Inifinty}] == 2 (1/3)/(2/3) == 1
.111... [base 2] == 1 Sum[1/2^k,{k,1,Infinity}] == 1 (1/2)/(1/2) == 1

No dale, in fact .222...[base 3] or .111...[base 2] are two different and non-finite sequences along a non-finite fractal, and no one of tham

is equal to 1.

Please read this short dialog very carefully:

--------------------------------------------------------------------------

"If the zero's continue forever, then you can never stop writing zero's to add a one at the end. There is no end to the string."

Exactly, and 1 at the end of .0000... is actually _1, which is a non-composed segment that no infinitely many zeros can eliminate (for more details please see http://www.createforum.com/phpbb/viewtopic.php?t=39&start=15&mforum=geproject ).

Furthermore, without the permanent existence of _1 upon non-finite scale-levels, as can clearly be seen in base 2 and base 3 examples (the infinitly many right segments), .000... immediately becomes a finite sequence.

--------------------------------------------------------------------------

It can easily be seen at the moment you understand:

I am afraid you don't even understand your own analysis. No wonder you reach incorrect conclusions. Draw yourself one of your little pictures. Try to use the same technique by which you are claiming that .222... [base 3] > .111... [base 2] to show that .111... == .111111... . You will quickly understand your mistake (I hope).

I don't know if you understand what a countably infinite set is, but the number of digits is a countable number. So irrational numbers and repetitive numbers like these have a countably infinite number of digits. Now, a finite number of countably infinite sets is also countably infinite. So if I double the number of digits of .111... it remains a countably infinite number of digits (2 Infinity == Infinity). Therefore any analysis must be invariant when the number of terms is multiplied by any finite integer. Your analysis relys on each series having the same number of terms, you will reach the exact opposite conclusion if the base 2 series has twice as many terms as the base 3 series. Your analysis is variant when the number of terms is multiplied by 2, so it is therefore incorrect.

-Dale

Dale you simply do not understand (yet) what you have in front of your eyes.

0.111...[base 2] or 0.1111111111...[base 2] are exactly the same path along the non-finite fractal, and this path cannot intersect 0.222...[base 3] in any stage along the non-finite fractal; therefore 0.11111111...[base 2] < 0.222...[base 3] and this is an invariant state.

Your problem is that you do not understand the proportion concept, and try to understand my system by the limit concept.

By the limit concept elements become closer to the limit, but this is not the case when you use the proportion concept, for example:

Let us say that we have a sports car (where the name of the back wheels is "epsilon" and the name of the front wheels is "delta") and our mission is to cross the zero point of X,Y-axis with both "delta" and "epsilon" wheels.

We are seated in the car and trying to reach point zero.

We realize that no matter how fast we drive, we are not getting any closer to the zero point, and the reason is: the faster we drive, the smaller we become (as can be seen in the picture below) and we have here a Lorenz-like transformation that has an invariant propotion along non-finite scale lavals.

According to this invariant proportion, nothing gets closer to the Zero point.

Shortly speaking, our mission cannot be completed.

In the same manner set R is an incomplete collection.

Actually we reach point zero, if and only if we don’t have a car anymore but a single point, which is a phase transition that cuts the infinitely many smaller states (smaller cars), and we don’t have an incomplete collection over infinitely many scales (infinitely many cars), but a finite collection of many scales (a finit collection of sports cars).

Also Dale, try to understand my picture from a 2-D non-finite fractal point of view, where each 0.###... is some unique 1-D path along it:

A number like .111... is not a fractal.

It is a single non-finite path along a non-finite fractal, as can be clearly seen here:

If you don't understand that simple fact, you cannot understand my argument, and you do not have any meaningful thing to say about it.

How can you possibly justify the statement that I don't understand proportions when my whole argument is based on a geometric series?

Because 0.111... [base 2] is a 1-D unique and non-finite path along a 2-D non-finite fractal, as clearly can be seen in the above diagram.

Therefore it has a self similarity over scales, which is exactly the property of a fractal, and this self similarity is the invariant proportion of 0.111... non-finite path.

Let us go another step and examine very carefully my analysis about any non-finite collection/sequence.

By my analysis, any non-finite collection/sequence is incomplete by definition (for more details please look at http://www.createforum.com/phpbb/viewtopic.php?t=45&mforum=geproject )

It means that a non-finite collection does not have an accurate cardinal, as can be found in a finite collection.

In other words a finite-collection and a non-finite collection are not in the same category, and aleph0 (as an accurate cardinal) does not exist.

Let us notate this non-accurate cardinal as |@| , where @ is a generic notation of a non-finite collection.

In my analysis |@|+3 > |@| , where @ is a non-finite collection/sequence.

Now, you can say: “hay, if +3 are extra scale levels along some 0.xxx… non-finite path, then 0.xxxxxx… > 0.xxx… (which has the property of a mirage-like effect along the non-finite single path).”

I say, you are right, and in that case 0.111111…[base 2] > 0.222…[base 3], but it does not contradict the fact that 0.111…[base 2] < 0.222…[base 3] because in my analysis |@| = |@| (in 0.111…[base 2] and 0.222…[base 3] case).

In other words, you force the transfinite analysis (where aleph0+n = aleph0) on my system, and therefore you do not understand it.

I’ll say it again: if I compare between two elements that have the same form (0.xxx…, for example) then I actually compare between @ and @ where |@| = |@|.

Therefore 0.111…[base 2] < 0.222…[base 3] and they are two different and clearly distinguished elements, that no one of them is equal to 1.

The same holds also in |@|+3 > |@|, where 0.111111… > 0.111… , and they are two different elements, that no one of them is equal to 1.

I have to add that the notation 0.111111...[base 2] == 0.111...[base 2] only if we look at it as |@| = |@| and not as |@|+3 > |@| (where +3 is 3 extra scale levels).

This time Dale please forget about the Cantorean transfinite universe for 10 seconds, and open your mind very carefully to this, again:

0.111...[base 2] or 0.1111111111...[base 2] are exactly the same path along the non-finite fractal (if [base 2] and [base 3] notations are |@| = |@| case), and this path cannot intersect 0.222...[base 3] in any stage along the non-finite fractal; therefore 0.11111111...[base 2] < 0.222...[base 3] (if [base 2] and [base 3] notations are |@| = |@| case) and this is an invariant state.

Your problem is that you do not understand the proportion concept, and try to understand my system by the limit concept.

By the limit concept elements become closer to the limit, but this is not the case when you use the proportion concept, for example:

Let us say that we have a sports car (where the name of the back wheels is "epsilon" and the name of the front wheels is "delta") and our mission is to cross the zero point of X,Y-axis with both "delta" and "epsilon" wheels.

We are seated in the car and trying to reach point zero.

We realize that no matter how fast we drive, we are not getting any closer to the zero point, and the reason is: the faster we drive, the smaller we become (as can be seen in the picture below) and we have here a Lorenz-like transformation that has an invariant propotion along non-finite scale lavals.

According to this invariant proportion, nothing gets closer to the Zero point.

Shortly speaking, our mission cannot be completed.

In the same manner set R is an incomplete collection.

Actually we reach point zero, if and only if we don’t have a car anymore but a single point, which is a phase transition that cuts the infinitely many smaller states (smaller cars), and we don’t have an incomplete collection over infinitely many scales (infinitely many cars), but a finite collection of many scales (a finit collection of sports cars).

Try to get your head out of the first millenium BC and join us in the third AD.

Try to understand the non-finite from a different point of view, which is much more interesting than the Cantorean point of view.

"Proportions are smooth, not rough."

No collection of infinitely many elements over non-finite scales can be considered as a smooth (non-composed) element.

For example, please look again at the sport's car diagram:

The invariant existing proportion is represented by the existing red-line, and this red line does not exist when we are only in X,Y zero point, so in order to define this proportion, we simultaneously need at least two different states, where one state is (for example) X,Y zero point, and the other is not the X,Y zero point.

In other words, the existence of any proportion is based on a phase transition from X,Y zero point to a non-X,Y zero point, and these phase transitions can be found on infinitely many scale levels along the red line.

A phase transition is the anti-thesis of smoothness; so any existing proportion is based on roughness where this roughness is a 1-D (linear) roughness.

This roughness is the existence of a non-finite collection of clearly distinguished (based on non-finite phase transitions) sport’s cars over non-finite scale levels, and this non-finite collection of sport’s cars is definitely a non-finite fractal.

If you do not believe me, then try to zoom in-out and you will immediately see the self similarity over scales, which is the minimal condition of being a fractal.

The same holds for 0.111…[base 2] (where each 1 is equivalent to some sport’s car).

You look at this path from a 1-D point of view, and I gave you the chance to understand it as a single path along a 2-D non-finite fractal, where the fractal is the base value expansion method itself, as can be clearly seen here:

but it seems that it is beyond your ability.

How can you possibly claim that I was taking it out of context?

By forcing the Cantorean transfinite universe on my system, you are out of the context of my system, where its non-finite is definitely not based on the Cantorean transfinite universe.

Therefore you get totally wrong conclusions about my arguments, because you are using the wrong tool (the transfinite Cantorean universe) in order to examine them.

So, despite of my request to you to first understand my non-finite universe, before you air your view about it, you totally ignored my requests and got wrong conclusions about it.

So for the last time, please forget about the Cantorean transfinite universe and let yourself to be opened to my non-finite universe.

If you cannot do that, then please say it clearly.

Thank you.

Am I the only one that read that as "Moronic-Mathematics"?

If you are looking at the mirror, then you get what you see.

Maybe I can help you to see something else.

In this article http://infinitesimal.iqnaut.net/ we can find this:

Alternatively, we can have synthetic differential geometry or smooth Infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., NOT (''a'' ? :evil: does not have to mean a = b. A nilsquare or nilpotent Infinitesimal can then be defined. This is a number x where x ² = 0 is true, but x ? 0 can also be true at the same time. With an Infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above.

In other words:

Number x where x ² = 0 is true, but x ? 0 can also be true at the same time (no excluded middle law).

Monadic-Mathematics does not need this non-elegant approach because it associates between two different categories, which are the non-local (which its minimal representation is a non-composed segment) and the local (which its minimal representation is a collectionsequence of points).

This association is a common mathematical realm for elements with an accurate location along the Real-Line (number 1, for example) and elements that do not have an accurate location along the Real-Line, as can be clearly seen in the 0.111...[base 2] and 0.222...[base 2] example:

This is a new notion of the Real-Line, and until this very moment no one of the participators of this thread examined it really carefully.

Turtle · November 24, 2005

Mmmmm.... hold on a minute. At the beginning where you say .222 Base three = .999... Base Ten, but think about it this way. Without a repeating decimal write .2 & say it's Base three notation; since in Base three every change in position is a power of three then .2 is 2*10^ -1 (note that the 10 here is not *ten* but *three* in base three. i.e. a one in the threes column & zero in the ones column) That's looks to me like the value then of .2 in base three is two-thirds & that is not equal to nine-tenths which is the value of .9 in base ten.

___Kinda fuzzy mind yet this morn, but it doesn't look correct what you stated? :evil:

Doron Shadmi · November 24, 2005

I am sorry Turtle but I do not understand your reply.

Do you say that 0.111...[base 2] = 0.222...[base 3] = 0.999...[base 10] = 1 ?

Turtle · November 24, 2005

I am sorry Turtle but I do not understand your reply.

Do you say that 0.111...[base 2] = 0.222...[base 3] = 0.999...[base 10] = 1 ?

___No, I say they are not the same. :evil:

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