What is the largest number less than 1?

[uart]

This is not the last digit but the limit digit or the unreachable digit of 0.999...

Just one question Organic. Is that a potential unreachable digit, an actual unreachable digit or just a betoid unreachable digit ?

 

[Organic]

Hi suyver,

jcsd's very elegant proof is about the non-existence of the largest number smallest than 1

where 1 is the limit of [0.999...,1.000...).

0.99999...
+
0.09999...
=
1.09999...8

and we use the interval notations not between two numbers but among range of different scales, represented by some number, and in this case the number is [1.0999...8) and the infinitely many digits of 9 cannot exist in the above addition if the limit digit 8 does not exist.

 

[suyver]

Hi Organic,

Question: why are you posting this? All the math shown previously in this thread you either didn't understand or ignore, but I am sure that by now even you must realize that nobody believes that you are correct!

Why don't you just give it up and go do something fun. Maybe read a book?

 

[Organic]

Hi suyver,

Please read this:


http://pespmc1.vub.ac.be/INFINITY.html

 

[suyver]

You're responding in a thread on MATH and you're referring to a site on PHILOSOPHY. Don't you realize how absurd that is?

In mathematics and philosopy we find two concepts of infinity: potential infinity, which is the infinity of a process which never stops, and actual infinity which is supposed to be static and completed, so that it can be thought of as an object.

Only the second kind (in this definition) is meaningful in this debate. Again, using this I can prove that
$$\sum_{i=1}^\infty 9\cdot10^{-i} \; = \; 1$$
but I guess that you won't believe that either...

 

[Organic]

Math is based on different consistant systems of axioms,which are propositions regarded as self-evidently true without proof.

So the "true" of the axioms is out of the scope of any mathematical research, therefore can be examined only by PHILOSOPHY.

 

[suyver]

O, now I see!
Yes, you must be completely right, Organic.


I give up. Anybody else wants a go?

 

[Organic]

Also you wrote:

but I am sure that by now even you must realize that nobody believes that you are correct!

What is the connection between belief and Math ?

 

[uart]

Thanks for finally posting a link to explain that concept that you've been talking about Organic. So now I know what the concept of "potential" vs "actual" infinity is. I must say however that the disinction is much more a philosophical one then a mathematical one.


As for your arguments of "unreachable digits", such as 1.0999...8 , I can't see how this is any different from the usual old argument that 0.9999' can't be equal to 1 becuae it "clearly" differs by 0.000...1

It does not make any sense to talk about having an infinite number of zeros followed by a one, just the same as it doesn't make any sense to talk about an infinite number of nines followed by an eight.

If you want to set it up as a proper limit then that's fine, but the result you will get is the same as everyone has already proven.

0.999...8 = Lim as n->infinity 9 * (10^(-1) + 10^(-2) + ... 10^-n) + Lim as n->infinity 8*10^(-n-1) = 1 + 0

 

[Organic]

Hi uart,

You write:

0.999...8 = Lim as n->infinity 9 * (10^(-1) + 10^(-2) + ... 10^-n) + Lim as n->infinity 8*10^(-n-1) = 1 + 0

If you write 8*10^(-n-1) then you don't understand my argument, which is based on the idea
of the open interval http://mathworld.wolfram.com/Interval.html .

Instead of using it between two different numbers, i use it on one number, represented by base 10 (we can use any other base value instead).

Through this point of view i clime that:

0.99999...
+
0.09999...
=
1.09999...8

and we use the interval notations not between two numbers but among range of different scales, represented by some number, and in this case the number is [1.0999...8) and the infinitely many digits of 9 cannot exist in the result of the above addition if the limit of digit 8 does not exist.

 

[HallsofIvy]

If you write 8*10^(-n-1) then you don't understand my argument, which is based on the idea
of the open interval http://mathworld.wolfram.com/Interval.html .

Instead of using it between two different numbers, i use it on one number, represented by base 10 (we can use any other base value instead).

Then you are using it incorrectly. There is no open interval consisting of one number.

You are still using the phrase "among range of different scales" without defining "different scales". As long as you do not define your terms no one will understand what you are saying.

 

[Organic]

HallsofIvy

The Indian-Arabic number system, based on some base > 1 and powered by
0 to -n or n, is actually a fractal with -n or n finite levels or
-aleph0 | alaph0 infinite levels, where each level has a different scale, depends on base^power.

Any infinite fraction is some unique sequence of digits along these scales, and there is no mathematical law that does not allow me to use the open interval idea on this range of digits, existing in these infinite levels of scales.

Therefore [1.0999...8) is a legal notation, which is the result of

[0.99999...9)
+
[0.09999...9)
=
[1.09999...8)

 

[NateTG]

Originally posted by Magnus
How can you represent the largest # that is less than 1?

Under convetional ordering, in the reals or the rationals, this number does not exist.

Let's say that there is a real number x with that property.
Then we have x < (1-x)/2 + x <1, which contradicts the desired property of x.

However, if you choose a different ordering on the real numbers then there can be a number x such that x is the smallest number less than one.

 

[Hurkyl]

you don't understand my argument

How can he understand it if you don't understand it?

 

[Integral]

What HallsofIvy was trying to say is that this is an important feature of the decimal representation (or base-n representation) of the real numbers, not an important feature of real numbers themselves.

The countability of digits is certainly is a fundamental feature of the real numbers. I do not care how you represent it. The sum

$$\sum_{i=0}^\infty d_i b^{-i}$$

is the general representation of the fractional part of a real number, b is the base the d_i is a selection from a set of digits. For example
$$d_i \in \{0,1,2,3,4,5,6,7,8,9\}$$ if b =10
or

$$d_i \in \{0,1\}$$ if b =2
The one thing that is consistent across all representations is that the summation index i is countable. Thus any representation of a Real number can have only a countable number of d_i associated with it. Am I still not clear enough?

 

[Hurkyl]

I can represent real numbers with points on a line.
I can represent (some) real numbers algebraically.
I can represent (some) real numbers with combinations of elementary functions.

(assuming the axiom of choice) There are uncountable index sets $I$ such that each real number can be written uniquely in the form

$$\sum_{\iota \in I} c_{\iota} \iota$$

Where all but a finite number of the $c_{\iota}$'s are zero, and the nonzero ones are rational numbers.


The point is, the countability of the digits cannot be a fundamental property of real numbers because digits are not a fundamental property of real numbers.

 

[russ_watters]

Originally posted by Organic
...and there is no mathematical law that does not allow me to use the open interval idea on this range of digits, existing in these infinite levels of scales.

Therefore [1.0999...8) is a legal notation, which is the result of

[0.99999...9)
+
[0.09999...9)
=
[1.09999...8)

As stated earlier, the law there is the mathematical definition of infinity. You have it wrong and as a result your proof is wrong.

Incidentally, this is the Math forum. In it, we discuss the accepted version of how math works. If you wish to invent a new type of math, you need to post it in the Theory Development forum (I think you already have though...).

 

[Integral]

Originally posted by Hurkyl
I can represent real numbers with points on a line.
I can represent (some) real numbers algebraically.
I can represent (some) real numbers with combinations of elementary functions.

(assuming the axiom of choice) There are uncountable index sets $I$ such that each real number can be written uniquely in the form

$$\sum_{\iota \in I} c_{\iota} \iota$$

Where all but a finite number of the $c_{\iota}$'s are zero, and the nonzero ones are rational numbers.


The point is, the countability of the digits cannot be a fundamental property of real numbers because digits are not a fundamental property of real numbers.

I rest my case, you have proven my point. In your own words

$$\sum_{\iota \in I} c_{\iota} \iota$$
Even you use the integers to index your representaion. That is all I am saying, it is fundamental that any representation can be indexed with the integers. You do it in your general representation, I do not care what method you use, a real number has at most a countable number of terms in the sum which represents it.

 

[Organic]

Hi russ_watters,

Please show me why I cannot use the idea of the open interval on a single number, represented by base^power method, for example:

[0.999...8)

[0.999...9)

[0.000...1)

and so on.

Be aware that #) is not the last digit but an unreachable limit exactly like in [0,1)

 

[suyver]

The open interval [a,b) consists of all numbers that are equal to or larger than a and also smaller than b. This open interval has no largest number. Only a smallest upper bound: b. Look up the difference between maximum and supremum!

 

[Hurkyl]

Originally posted by Integral
I rest my case, you have proven my point. In your own words

$$\sum_{\iota \in I} c_{\iota} \iota$$
Even you use the integers to index your representaion. That is all I am saying, it is fundamental that any representation can be indexed with the integers. You do it in your general representation, I do not care what method you use, a real number has at most a countable number of terms in the sum which represents it.

None of the other ways I represenetd a real number even have a sum in them...

Well, you really want a sum with an uncountable number of terms? Fine, consider this sum of hyperreal numbers:

$$2 = \mathrm{st} \sum_{n=0}^{H} 2^{-n}$$

Where $H$ is a transfinite hyperinteger. There are an uncountable number of terms, and the value of the sum is $2 - 2^{-H}$. $2^{-H}$ is infinitessimal, so when you take the standard part, the answer is $2$.

Alternatively, I believe that one can sensibly define (for some hypersequences) a hyperreal analogue of an infinite sum, so that:

$$2 = \sum_{n=0}^{\infty} 2^{-n}$$

(as a hyperreal sum) converges to $2$. There is a term for each nonnegative hyperinteger, so there are an uncountable number of terms.

 

[Organic]

Hi suyver,

So the fundamental property of the interval's idea can be used on sequence of digits based on base^power method, where the right side of it has no smallest scale (represented by ...), but the 'digit' + ')' are the notation of the sequence's infimum http://mathworld.wolfram.com/Infimum.html

Therefore [0.000...1) is a legal mathematical notation,
and also [0.999...9).

 

[Hurkyl]

Organic: [1.999...8) is not accepted mathematical notation. You're going to have to explain what you mean by this, so you might as well just say the explanation rather than invent (yet again) new notation that nobody understands.

So the fundamental property of the interval's idea can be used on sequence of digits based on base^power method, where the right side of it has no smallest scale (represented by some digit), but the 'digit' + ')' are the notation of the sequence's infimum

What sequence?

For the record:

The infimum of the sequence
0.1, 0.01, 0.001, 0.0001, ...
is 0.

The supremum of the sequence
1.98, 1.998, 1.9998, 1.99998, ...
is 2. (it's infimum is 1.98)

The infimum of the sequence
0.08, 0.008, 0.0008, 0.00008, ...
is 0.

Therefore [0.000...1) is a legal mathematical notation.

Incorrect. That is notation you've invented and have not defined, so there's no way it can be considered "legal mathematical notation".

 

[Organic]

Hi Hurkyl,

First, i corrected my previous message, so please read it again.

The base^power method is infinitely many information cells upon infinitely many scales, where their periodic changes depends on base^power values.

each information cell includes n ordered digits, which represent some base value quantity, for example:

Base 2 notated by '0','1'
Base 3 notated by '0','1','2'
Base 4 notated by '0','1','2','3'

and so on.


any unique number which has aleph0 information's cells upon infinitely many scales, is asequence made of marked digits in aleph0 cells, but the important thing is not the represented digit on each cell, but the the cell's scale.

So by writing, for example [0.000...1) i say that there are infinitely many information's cells marked by 0, that interpolated forever to some cell that marked by 1.

 

[HallsofIvy]

posted by OrganicThe base^power method is infinitely many information cells upon infinitely many scales, where their periodic changes depends on base^power values.

each information cell includes n ordered digits, which represent some base value quantity, for example:

Once again, you are using words that are NOT standard mathematics and that you have NOT defined. There is no way anyone can guess what you mean.

Your assertion that "[0.000...1)" is "legal mathematical notation" is non-sense. The interval notation [a, b) always requires TWO numbers (or points) a and b- you are using only one so this is NOT interval notation and you haven't told us what it means.

By the way, Organic said, a while back,

Math is based on different consistant systems of axioms,which are propositions regarded as self-evidently true without proof.

That is definitely NOT true. I can't imagine any mathematician believing that BOTH "given a line and a point not on that line, there exist exactly one line through the given point parallel to the given line" and "given a line and a point not on that line, there exist more than one line through the given point parallel to the given line" are "self-evidently true"!

 

[Organic]

Hi HallsofIvy,

That is definitely NOT true

Please look at: http://mathworld.wolfram.com/Axiom.html


Also see an example of 2^aleph0 information's cells over different scales here:

http://www.geocities.com/complementarytheory/FPoint.pdf

Where [.000...1) is on the interpolation side of infinitely many cells, notated by '0' and approaches some cell, notated by '1'.

By using the idea of open interval on these cells we mean, that '1' can be distinguished from '0' forever, on infinitely many scales (which means: no cell is turned to zero size).


I think i have another idea based on the above.

Let us say that:

T = Math-theory

A = Its consistent axiomatic system

Therefore by writing [T,A) T depends on A but A does not depend on T, which maybe can give a new point of view on Godel's Incompleteness Theorems.


For example:

[0.99999...9)
+
[0.09999...9)
=
[1.09999...8)

The infinitely many '9' notations of the result, depends on adding 9) to 9) of the two digits of the infimum information's cells of the two added numbers.

By this example we can understand that any change in A, immediately changes T but not vise versa.

 

[Integral]

Hurykl,
After sleeping on it and getting out my copy of Royden, I realized that indeed the word "fundamental" cannot be used in reference to any representation of a Real number. While it is true that every point on the Real line has a decimal representation, this is a fact that requires proof using the fundamental theorems of the Real numbers and is therefore not fundamental of itself.

 

[BluE]

Okay, so is there a decimal place on a number line that can represent 0.9999... ?

 

[NateTG]

Originally posted by BluE
Okay, so is there a decimal place on a number line that can represent 0.9999... ?

There is no 'decimal' place on the number line. Decimal refers to representation in base 10.

The location on the number line that corresponds to the decimal 0.999999... is the same as the location that corresponds to the decimal 1, since they are the same real number. (Decimal representations of real numbers are not necessarily unique.)

 

[BluE]

Ah, okay. Sorry, and thanks. And since there is no "greatest number less than one" then that means there is no "greatest number less than x" when x is equal to any real number, right?

 

[HallsofIvy]

Yes, that is true. For x any real number, the set of all real number "less than x" is an open set and has no largest member.

 

[suyver]

Now we can wait for somebody to post that the number x - 0.0000...1 is the largest number in this set and that it is well-defined. And then we can go [zz)]. Honestly, I can't understand the near-infinite patience of some of the Senior Members here, but you have all my respect!

-Freek.

 

[uart]

Here's a question for Organic.

If x = 0.000...1 is valid number and is other than zero, then what is 10 times x equal to ?

Ok, I know that you'll say something like 10x = 0.000...10 where the "..." in the 10x expression represents one less zero than the original x.

So where does that logic get you? In the expression for x there are Infinity zeros (represented by the "...") whereas in the 10x expression there are (Infinity - 1) zeros - and they are different!.

So obviously you must believe that (Infinity - 1) is differnt than infinity. If that is so then just how do you define Infinity minus one ?

 

[russ_watters]

Originally posted by Organic
Hi russ_watters,

Please show me why...

I'm sorry, Organic, I can't help you here. I've already stated that you are arguing against the DEFINITION of infinity. As Halls said, what you posted there is not an accepted mathematical expression.

So now it is quite simply up to you to accept the definition or continue to be wrong.

There is a third choice of course - you could invent a new type of math to replace the entire existing structure. But that would take decades to do (if it could even be made to work - the definition you appear to be advocating is not self-consistent) and even then, its pretty unlikely that you'd be able to get the entire world to adopt your new math. Clearly that is what you are attempting to do - your website is full of things that don't fit with the way math actually works. But it'll be a long and uphill struggle. So it would probably be better to accept math as it is.

 

[HallsofIvy]

"Honestly, I can't understand the near-infinite patience of some of the Senior Members here, but you have all my respect!"

The problem with the inter-net is that the obvious remedy for people like this- beating about the head and shoulders with a two by four- is not applicable.