The Most Amazing Number

[slikrapid]

It seems to me that assuming that infinite objects mandatory have to share the same properties than finite one's and nothing else is overly presumptuous.

similar properties, not 'the same' properties
although, in another sense, 'the same' may apply as well: both appear in the mind as a result of thinking, therefore they necessarily have to share the same properties as all other things that appear that way

9*d=0

the solution of this equation is d=0
that means d=0, it does not mean d=0.0...1 or any other number

My argument only relies on some basic calculation rules, nothing more. If it is false, then at least one of these basic calculation rules fails in case of infinite sequences...

your calculation is erroneous, as already shown on several occasions - calculation rules did not fail here

in infinite case the part may be "as big" as the whole.

no, the part is still smaller than the whole, but nevertheless still infinite
it may be easier to think of them as 'countless': there are countless numbers between 0 and 1, but there are twice as much between 0 and 2

I have an other argument to try to convince you that 0.99999....=1

why complicate things further, when i already gave the correct value in the most simple way:

1 - 0.999... = 0.0...1

or written more elegantly:

1 - 0.9... = 0.0...1

[Renk]

no, the part is still smaller than the whole, but nevertheless still infinite
it may be easier to think of them as 'countless': there are countless numbers between 0 and 1, but there are twice as much between 0 and 2

Not always in case of infinite set: In your example, it is possible to match each number between 0 and 1 with an unique number between 0 and 2, and vice versa (mathematicians call that a bijection). In that case, you can mach e.g. every number x between 0 an 1 with the number 2*x, between 0 and 2, reciprocally you can match every number x between 0 and 2 with an unique number (in that case x/2) between 0 and 1. So there are exactly as many numbers in [0,1] than in [0,2] (and than in [0, 100000], and in fact than in the whole set R itself) This size (cardinality) is named aleph zero (I can't write it here).

the solution of this equation is d=0
that means d=0, it does not mean d=0.0...1 or any other number

Precisely. And as d=1-0.9999...., d=0 means 1=0.9999....

View differently, as we have 9*d=0, if d were non zero, then it would be possible to divide by d, leading to 9=0/d=0. Big contradiction again when assuming that 0.9999... is different from 1.

your calculation is erroneous, as already shown on several occasions

This calculation is only: 10*d = 10*(1-0.9999...) = 10*1 - 10*0.9999... = 10 - 9.9999... = 10 -9 -0.9999... = 1-0.9999 = d, so 10*d and 9*d=0. What is so erroneous in there?

why complicate things further

This second argument I presented is not complicated at all. It's only based on logic and on one fundamental property of the set of all "real numbers" (this set happens to be "archimedian"). this second proof is even (in some sense) simpler, more elementary than the first, as almost no calculation rules are implied here, only very basic and fundamental facts about numbers.

when i already gave the correct value in the most simple way:

1 - 0.999... = 0.0...1

or written more elegantly:

1 - 0.9... = 0.0...1

You are naming "0.0....1" what I named "d" (previously I defined d by the distance 1-0.9....). My choice is far more concise . Moreover, as the notation 0.0...1 may have no mathematical sense at all, I'm going to put it in quotes.


In any case, then again: 10*"0.0...1" = 10*(1-0.9...) = (see above) = 1-0.9... = "0.0...1".

So we have 10*"0.0...1" = "0.0...1", which leads to 9*"0.0...1"=0, and then, either "0.0...1"=0, either 9=0. What's your choice?

[slikrapid]

In that case, you can mach e.g. every number x between 0 an 1 with the number 2*x, between 0 and 2,
So there are exactly as many numbers in [0,1] than in [0,2]

matching unique numbers is not the same as comparing them
it is obvious that the infinite set between 0 and 1 is smaller than the set between 0 and 2, simply because you can find numbers in the latter that aren't available in the former, while the opposite does not apply

(1) 9*d=0

(2) d=1-0.9999....

sorry, but there is no d that satisfies both of these equations simultaneously, because the solutions are as follows:
(1) d=0
(2) d=0.0...1

these two values are obviously different

if d were non zero, then it would be possible to divide by d, leading to 9=0/d=0

if d were non zero, the equation would be invalid, only d=0 satisfies the equation - isn't that the goal of solving equations?

10*d = 10*(1-0.9999...) = 10*1 - 10*0.9999... = 10 - 9.9999... = 10 -9 -0.9999... = 1-0.9999 = d, so 10*d and 9*d=0. What is so erroneous in there?

simplified, the error is: 0.99 * 10 = 9.99, whereas it should be: 0.99 * 10 = 9.9, you have 'one too many nines' after multiplying by 10


1*d = 0.0...1 <> 0
2*d = 0.0...2 <> 0
...
9*d = 0.0...9 <> 0
10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d

obviously, none of these equal zero and they are all different from each other

[Renk]

matching unique numbers is not the same as comparing them

I compare the cardinality of the sets [0,1] and [0,2] (ie the quantity of numbers they are containing) in matching every number of the 1st set with every number of the second set, leading to conclude that these two sets have the same cardinality (each element of one of the set can be "married" to only one of the other set).

When in some population you can marry each man with exactly a woman, and each woman with exactly one man, then in this population there are exactly as many men as women. This is even the very mathematical definition of "as many as".

it is obvious that the infinite set between 0 and 1 is smaller than the set between 0 and 2

"obviousness" is not an argument at all, particularly when considering infinite objects.

simply because you can find numbers in the latter that aren't available in the former, while the opposite does not apply

What you say means that [0,1] is a strict subset of [0,2]. That absolutely not implies that the cardinality of the second set is strictly greater than the cardinality of the first.

10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d

This reasoning assume you are considering a (finite) integer n. But it does no more work here because in your "0.0...1", your "n" in fact equals to "infinite", and "infinite"-1="infinite".
(what I just said is more intuitive than rigorous, due to the fact that your "0.0...1" is only a symbol, and as no clear mathematical meaning).

sorry, but there is no d that satisfies both of these equations simultaneously, because the solutions are as follows:
(1) d=0
(2) d=0.0...1

The fallacy in your reasoning is that you implicitly suppose that what you name "0.0...1" differs from 0. For you it's kind of an axiom. You consider from the outset that="0.0...1" can in no way be equal to 0. But preliminary admitting an assertion is not a good way to prove it.

For my part, I don't presuppose anything concerning the value of "0.0...1". I just consider this object and, in applying very basics calculus rules (or some reasoning involving basic properties of numbers) , I come to the conclusion that your "0.0...1" is in fact 0.

You can refuse to accept it, but, nevertheless, your "0.0...1" has the property that 10*"0.0...1" = "0.0....1", and then equals to zero.

if d were non zero, the equation would be invalid

Exactly. And as the equation is valid, d can not be non zero, ie d=0.

0.99 * 10 = 9.99, whereas it should be: 0.99 * 10 = 9.9

I don't say that 0.99*10 = 9.99, I say that 0.999...... *10 = 9,999.....

What is then the result of the operation 10*0.999..... for you?


All that leads me to think to a 3rd proof that 0.999.... =1:

We have:
1/3=0.333.....

and similarly:
2/3=2*(1/3)=2*0.333... =0.666......

And we have too: 3*(1/3)= 3*0.333....= 0.999....

But on the other hand:
3*(1/3) = 3/3=1.

Hence, 0.999..... =1.

This proof is analogous to the one given by anon previously:

1/9=0.1111.....
Then: 9*(1/9)=0,9999......
But in the same time, 9*(1/9)= 9/9=1.
So 0.999...=1.

The point is, 0.999.... doesn't look like 1, it even doesn't look like an integer at all but it is an integer, and even it is the number 1.

In fact, every decimal number has exactly 2 decimal representations. For example, the two decimal representations of 1/2 are 0.5 and 0.49999.....


You can visit any forum about mathematics, any wikipedia page about these subjects, you will find that 0.999... =1, that there are as many odd integers as there are integers, and that the set [0,1] has as many elements as the set [0,2], with arguments probably close to the ones I presented here.

It can even be proved that there are as many odd (or even) integers as there are decimal numbers (and that, despite the fact that eg the set [0,1] does contain only one even integer, and an infinite quantity of decimal numbers).


Mathematics are about what mathematical objects are, not about what they look like...

[slikrapid]

matching every number of the 1st set with every number of the second set, leading to conclude that these two sets have the same cardinality (each element of one of the set can be "married" to only one of the other set).

the matching works only for half of the numbers of the larger set: the smaller set [0,1] perfectly matches with the first half of the larger set [0,1], but the other 'half' of the larger set, namely <1,2] is left unmatched, which is obvious

"obviousness" is not an argument at all, particularly when considering infinite objects.

infinite objects are not immune to simple, obvious arguments about their nature
'obvious' means: easily understood by an average open minded person

This reasoning assume you are considering a (finite) integer n...."infinite"-1="infinite".

n is an infinite number of leading zeroes
n-1 is 'one zero less than' an infinite number of leading zeroes
it is a way to label the 9th neighbor of d, the 10*d, where d=0.0...1

in a way, the infinitely small number is also finite, we just don't know how small it is, other than saying 'it is the smallest' - nevertheless, it also has its immediate neighbors which are all equally far away from each other (this smallest distance is an infinitely small constant of 0.0...1)

you implicitly suppose that what you name "0.0...1" differs from 0. For you it's kind of an axiom.

if we were to apply your idea of 0.0...1 = 0, the whole calculation of infinite numbers collapses to zero, but an infinitely small (non-zero) distance keeps it mathematically consistent

Exactly. And as the equation is valid, d can not be non zero, ie d=0.

you missed the part where both equations (1) and (2) need to have a common solution, in order to both apply (to both be valid)

What is then the result of the operation 10*0.999..... for you?

[EDITED ANSWER:]
the result would be:

0.999... -- n being the number of nines after the decimal point
10*0.999...=9.999...0 -- the result, n-1 being the number of nines after the decimal point


there are still 8 numbers between this result and your previous result:

9.999...0 -- the result, n-1 being the number of nines after the decimal point
9.999...1 -- a number in between, n-1 being the number of nines after the decimal point
...
9.999...8 -- a number in between, n-1 being the number of nines after the decimal point
9.999... -- your previous result, n being the number of nines after the decimal point

We have:
1/3=0.333.....

see, you just assumed that
the problem with 1/3 is that it cannot be precisely represented with decimal numbers (10 doesn't divide with 3 nicely)

In fact, every decimal number has exactly 2 decimal representations. For example, the two decimal representations of 1/2 are 0.5 and 0.49999.....

i can think of a third representation: 0.50...1
but only one of these (0.5) is equal to 1/2, while others are either approximations or nearest neighbors

and using your logic, we can use numerous other representations, like: 0.50...2, 0.50...3, etc. or 0.49...8, 0.49...7 etc. simply because 1*d, 2*d, 3*d, etc. all equal zero - you see the absurdity of such reasoning?

You can visit any forum about mathematics, any wikipedia page about these subjects, you will find that 0.999... =1
Mathematics are about what mathematical objects are, not about what they look like...

that is just new-math, which has the same validity as (orwellian) new-speak - it is simply a sign of the times we live in

[anon]

My my, I've caused quite a racket here... but we can all agree that zero is even, right?

[Renk]

Well, this section is named "Chill & Fun", probably for a reason, and maybe with hardcore mathematics we are going a little "off topic" here. But as a Distinguished Moderator allowed Himself to write a troll-like reply, maybe some rules have been provisionally relaxed. So let me answer, but for ther antepenultimate time, so as not to exaggerate.

the matching works only for half of the numbers of the larger set: the smaller set [0,1] perfectly matches with the first half of the larger set [0,1], but the other 'half' of the larger set, namely <1,2] is left unmatched, which is obvious

"Obvious" doesn't work very well when arguing about infinity. What I spoke about concerning a smaller set as populated as some larger one, was an instance of mathematical results proved by G. Cantor (who by the way died insane). About the same subject, you can undertake some research about "Hilbert's paradox of the Grand Hotel"(*). You will see that common sense or obviousness don't mix well with infinity.


But I prefer to focus on what has become this thread's very subject: The comparison of 1 and 0.999...999....

I said (and proved, in several ways) that 0.999....=1 (or equivalently, 1-0.9999...=0), you on the other hand refuse these evidences, and consider that 1-0.999...= 0.00....001 (which is absolutely not stupid at all, but is nevertheless absolutely wrong, except if you accept to consider that your "0.000...0001" is in fact equal to 0).

So before going any further, let me ask you a question.

Let us call d your 0.000....0001 (ie the difference between 1 and 0.999...). Then, how do you write d/10? d/100? d/1000? What are the decimal representations of these numbers?

And if n is any positive integer, what are the nth digits of your d, d/10, d/100, d/1000?



(*) Or even the Banach-Tarsky Paradox, which is even more disturbing .

[slikrapid]

What I spoke about concerning a smaller set as populated as some larger one, was an instance of mathematical results proved by G. Cantor (who by the way died insane).

so-called proofs that do not align with common logic have probably been improperly set up or defined or calculated

Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often.

obviously, a fully occupied hotel (regardless of the number of rooms in the hotel) cannot accommodate any other guests if we want to keep one guest per room and without introducing other ways of driving the existing guests out of their rooms

this does not change for infinite rooms occupied by infinite guests because whichever room you may choose, it will already be occupied by a guest, therefore no vacancy!

The Banach?Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3?dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".

However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a volume, which happens to be different from the volume at the start.

if the volumes were finite and known in the beginning, why would they somehow become unknown after chopping them up into a finite number of volumes?
if it is impossible to define the volumes of the considered subsets, how can it be possible for them to become definable again simply by reassembling them?


child1: i can chop up a pea and re-ass-emble it into the Sun!
child2: you cannot do that!
child1: yes i can, because i said so!

the mad mathematician: yes i can, because my theorem said so!
the poor students of the mad mathematician: awww

I said (and proved, in several ways) that 0.999....=1 (or equivalently, 1-0.9999...=0), you on the other hand refuse these evidences

well, i could also say that your claims have been disproven a number of times by using both ordinary logic and simple mathematics

Let us call d your 0.000....0001 (ie the difference between 1 and 0.999...). Then, how do you write d/10? d/100? d/1000? What are the decimal representations of these numbers?

And if n is any positive integer, what are the nth digits of your d, d/10, d/100, d/1000?

there are no fractions of d simply because it already represents the smallest infinite number

[Renk]

so-called proofs that do not align with common logic have probably been improperly set up or defined or calculated



obviously, a fully occupied hotel (regardless of the number of rooms in the hotel) cannot accommodate any other guests if we want to keep one guest per room and without introducing other ways of driving the existing guests out of their rooms

this does not change for infinite rooms occupied by infinite guests because whichever room you may choose, it will already be occupied by a guest, therefore no vacancy!

Another way to express this paradox is as follows. Let consider an hotel with an infinite number of occupied rooms, all numbered from 0,1,2,etc
Suddenly, an infinite number of customers, all numbered from 0,1,2 etc shows up.

Well, although all of the hotel's rooms are already occupied, it is possible to find room for every new guest without kicking out any old guests. (something impossible with a finite_number_of_rooms hotel). And the procedure is simple: For each former guest, if the room number he occupies is n, then he goes to room #2n. And for each new guest, if his number is for example n, then he goes to room #(2n+1). Everyone has a room. No one is thrown out. And this despite the fact that all the rooms were initially occupied...

if the volumes were finite and known in the beginning, why would they somehow become unknown after chopping them up into a finite number of volumes?
if it is impossible to define the volumes of the considered subsets, how can it be possible for them to become definable again simply by reassembling them?

Short answer: Because the usual mathematical 3 dimensional euclidian space is not exactly the same as it's physical counterpart. (and the "axiom of choice" does not have the same consequences in each of them).
Long answer: Read and learn the proofs:
http://web.mit.edu/andersk/Public/banach-tarski.pdf
https://hal.archives-ouvertes.fr/hal-01673378/document

In my opinion, the sole mean to fully comprehend this paradox (in such a way this paradox is no more a paradox, but something intuitive and natural) is to place it within the framework of the theory of Topos, invented by the mathematician A. Grothendieck.


Maybe Jesus, by multiplying the loaves and fishes from a single loaf and a single fish, has managed to implement this paradox in your physical reality, showing He was the incarnation of a purely mathematical Entity after all.

child1: i can chop up a pea and re-ass-emble it into the Sun!
child2: you cannot do that!
child1: yes i can, because i said so!

There is no such way as "it's true because I say it is true" in mathematics. Mathematics is not Politics. Something is mathematically true when it is formally, rigorously and logically deduced and proved, in a verifiable manner.

there are no fractions of d simply because it already represents the smallest infinite number

This reasoning is rooted on an unproven hypothesis such as "the smallest strictly positive number exists" (your d). But such a "number" doesn't exist (ie it is not definable in a way that does not give rise to any contradiction), because otherwise there would be a little room between 0 and d, and in that little room there would be the middle between 0 and d.

[anon]

Well, this section is named "Chill & Fun", probably for a reason, and maybe with hardcore mathematics we are going a little "off topic" here. But as a Distinguished Moderator allowed Himself to write a troll-like reply, maybe some rules have been provisionally relaxed. So let me answer, but for ther antepenultimate time, so as not to exaggerate.

Believe it or not, there isn't complete agreement on the evenness of zero either (though I personally believe it is even, hence my comment).

As for "off topic" in the Chill & Fun section, that's what you get when two distinguished members are allowed to participate.

[slikrapid]

Another way to express this paradox is as follows.

juggling mathematical expressions cannot substitute an error in logical thinking

Short answer: Because the usual mathematical 3 dimensional euclidian space is not exactly the same as it's physical counterpart. (and the "axiom of choice" does not have the same consequences in each of them).

no scientist fully understands properties of the physical space, therefore any scientific conclusion about it is necessarily erroneous - not to mention purely hypothetical imaginary paradoxes

Maybe Jesus, by multiplying the loaves and fishes from a single loaf and a single fish, has managed to implement this paradox in your physical reality, showing He was the incarnation of a purely mathematical Entity after all.

events described in sacred scriptures are certainly sources of inspiration for scientists/theorists but due to their lack of Faith, they manage to misinterpret the events and thus go on making continuous errors, some of which may include creating nonsensical paradoxes

Jesus did not multiply the loaves and fishes from a single loaf and a single fish, they were separate material manifestations from the surrounding material energy which is what this universe is made of - this energy can be transformed into any material shape or structure by able or worthy individuals, naturally only if God (aka God the Father) allows it to happen - as it is known, Jesus was (is) a saintly Son of the Father and therefore his wishes (for additional loaves and fishes) or Faith (in having a sufficient number of loaves and fishes) were pure and thus successful in any situation

so there is no need to manufacture some erroneous construction like 'incarnation of a purely mathematical Entity' even though God originally invented mathematics and logic and everything else you could (or couldn't) possibly think of

Mathematics is not Politics.

mathematics is handled by humans - humans may be considered political (homo politicus) and therefore there will be politics in the field of mathematics

Something is mathematically true when it is formally, rigorously and logically deduced and proved, in a verifiable manner.

well, as shown before, the logical errors are pretty obvious for many of these so-called axioms or paradoxes

This reasoning is rooted on an unproven hypothesis such as "the smallest strictly positive number exists" (your d). But such a "number" doesn't exist (ie it is not definable in a way that does not give rise to any contradiction), because otherwise there would be a little room between 0 and d, and in that little room there would be the middle between 0 and d.

simple logic tell you that if you have a group of different numbers, there ought to be a smallest among them - and i am certain that if nonsensical paradoxes can be 'proven' mathematically, then this can also be proven just as easily

that little room between 0 and d can be defined as 'the smallest non-divisible room' - of course, this can be disregarded and you can invent your own interpretation, where for example there is still some divisible room left inside to manufacture another paradox, say like 'The Paradox of Dividing the Non-divisible Room' - in this day and age (of corruption) it will probably earn you fame and rewards

Believe it or not, there isn't complete agreement on the evenness of zero either (though I personally believe it is even, hence my comment).

zero behaves differently than all other even numbers and therefore ought to be given a special 'third' category

[Renk]

Believe it or not, there isn't complete agreement on the evenness of zero either.

I knew that some people are considering 1 as a prime number (what it is not) but I wasn't aware of any controversy about evenness of 0. I just performed some search, showing there seems to exist some debate about that, but without finding the slightest argument in favour of the oddness of zero.

For me an integer number n is even if it is divisible by 2, or equivalently if there exists some integer p such that n = 2p. And an integer number n is odd if it is not even, or (equivalently as each in integer is either equal to 0 or to 1 mod 2), if there exists some integer p such that n = 2p+1.

Clearly, 0=2*0, with 0 being an integer number, so there exists an integer number p such that 0 = 2p, which qualifies 0 as an even number. On the other side, for every integer number p, 0 is not equal to 2p+1, showing that 0 in not an odd number. I can't see what is debatable in all that.

Except maybe if the people considering that 0 is not even, are refusing to consider 0 as an integer number in the first place???

juggling mathematical expressions cannot substitute an error in logical thinking
no scientist fully understands properties of the physical space, therefore any scientific conclusion about it is necessarily erroneous - not to mention purely hypothetical imaginary paradoxes

Remember that, as stated on the 2nd page of the MIT paper to which I gave a link above (that you haven't studied, or even read), the Banach-Tarsky paradox is in fact a theorem, and not a formal intrinsic mathematical contradiction. It is called a "paradox" only because the result proved by Banach and Tarsky is "highly counter intuitive".

In this context, I said that the usual mathematical 3 dimensional euclidian space ("UM3dES") was not exactly the same as it's physical counterpart just because mathematically you can duplicate an object in 2 other objects each having the same volume as the former, which seems to be impossible in the physical world (reason why achieving something like this is considered as a miracle). But you are right, we don't know all about your physical world, and maybe (maybe!) doing this kind of thing is quite feasible in our physical world -which would then be less distant from the UM3dES than what I was saying.

But in either case, the result proved by Banach and Tarsky remains true, because rigorously logically and formally proved (if not, show me where are the errors in the paper I gave to you), showing that obviousness and "common sense" are not criteria of mathematical truth. Something can be mathematically true and completely defying common sense in the same time.

Jesus did not multiply the loaves and fishes from a single loaf and a single fish, they were separate material manifestations from the surrounding material energy which is what this universe is made of - this energy can be transformed into any material shape or structure by able or worthy individuals, naturally only if God (aka God the Father) allows it to happen - as it is known, Jesus was (is) a saintly Son of the Father and therefore his wishes (for additional loaves and fishes) or Faith (in having a sufficient number of loaves and fishes) were pure and thus successful in any situation

And so (maybe by manipulating what physicist name nowadays dark matter and black energy), He found a way to physically succeed in doing what Banach and Tarsky proved mathematically possible 2000 years later. But this part of my post was just a bit of humour. Let us not now enter into a theological/mystical/metaphysical debate.

Anyway, let me just say that I find this part of your post very beautiful and poetic. You also seem to have a deep Faith, deeply rooted in your soul, and I respect that.

mathematics is handled by humans - humans may be considered political (homo politicus) and therefore there will be politics in the field of mathematics

Humans are not reductible to homo politicus (as they are not reductible to homo oeconomicus either).
By the way, tell me what exactly is the political content (leftist? rightist? centrist?) of the following assertions:

a) "2 is an even integer"
b) "3 is a prime number"
c) "the sum of the 3 angle of any triangle is equal to 180° (or equivalently Pi radians)"
d) "The perimeter of a circle is its diameter times the number Pi"
e) etc etc etc...

well, as shown before, the logical errors are pretty obvious for many of these so-called axioms or paradoxes. (...) simple logic tell you that if you have a group of different numbers, there ought to be a smallest among them - and i am certain that if nonsensical paradoxes can be 'proven' mathematically, then this can also be proven just as easily

You talk a lot and often as a mantra about errors of logic, but you haven't proved the existence of any, especially concerning Hilbert's Grand Hotel and the Banach Tarsky paradox. In fact, what you qualify as "logical" is only what your common sense considers as true. But by doing so, you make yourself deaf and blind to any radical novelty and you make it impossible for you to make any evolutionary leap. Just because something shocks your intuition or threatens the way you view and comprehend the world doesn't mean that this something is illogical or false. And just because you don't take the trouble to study a theorem that defies what you call common sense, doesn't mean this theorem is false either.

that little room between 0 and d can be defined as 'the smallest non-divisible room' - of course, this can be disregarded and you can invent your own interpretation, where for example there is still some divisible room left inside to manufacture another paradox, say like The Paradox of Dividing the Non-divisible Room

No, because your "Paradox of Dividing the Non-divisible Room" is not proved.You tell about it, but you have not proved its existence. In fact, you are contradicting yourself: First you refuse to accept the fact that 0.999...=1 only because this assertion is not compatible with what you name common sense and logic, ultimately leading you to then have to admit the existence of a "number" that can't be divided by any other number than 1 and itself, and to admit the existence of an interval that has no middle point, what completely defies obviousness and common sense as well...


Moreover, all these completely not common sensing things that you have made yourself necessary to admit, are not proved in any way, unlike the Banach-Tarsky theorem. Unlike Hilbert's Grand Hotel problem, and unlike the identity of 0.9999... and 1.

And what you say is non proved, and even non provable, because rooted on the intrinsically self contradictory "number" d (=0.0000...1).

You already told me that this d admitted no fraction. So, explain me:

1) What is the smallest positive number divisible by 10 (let name it r)? How do you write it? What is the decimal representation of this number? And if n is any positive integer, what are the nth digit of this number?

2) Now let consider your d. What is the decimal representations of 10*d? Of 100*d? What are the decimal representations of these numbers? And if n is any positive integer, what are the nth digits of these numbers?

[slikrapid]

the MIT paper to which I gave a link above (that you haven't studied, or even read)
if not, show me where are the errors in the paper I gave to you

why would i bother reading such nonsense? to see how math can be used and abused?

because mathematically you can duplicate an object in 2 other objects each having the same volume as the former, which seems to be impossible in the physical world (reason why achieving something like this is considered as a miracle)

linguistically you can invent words without meaningful physical counterparts
artistically you can display things normally impossible in the physical world
philosophically you can establish imaginary concepts
historically you can invent false events
theoretically you can support erroneous concepts

showing that obviousness and "common sense" are not criteria of mathematical truth. Something can be mathematically true and completely defying common sense in the same time.

on the contrary, in order to apply mathematical expressions properly, their usage and meaning needs to be understood: obviousness and common sense is certainly needed
improper usage may lead to paradoxes

maybe by manipulating what physicist name nowadays dark matter and black energy)

dark matter and dark energy are imaginary scientific concepts used as a crutch to support their flawed theories, as in: 'we are missing something in our calculations... lets call it the dark energy'

tell me what exactly is the political content (leftist? rightist? centrist?) of the following assertions:

mathematical models can be written in such a way to support a certain political agenda, even though the mathematical expressions used therein are politically neutral

In fact, what you qualify as "logical" is only what your common sense considers as true. But by doing so, you make yourself deaf and blind to any radical novelty and you make it impossible for you to make any evolutionary leap. Just because something shocks your intuition or threatens the way you view and comprehend the world doesn't mean that this something is illogical or false. And just because you don't take the trouble to study a theorem that defies what you call common sense, doesn't mean this theorem is false either.

that is exactly the problem with mathematics (or mathematicians) nowadays: they are often divorced from logic and reality, playing like children, building their imaginary castles in the air, which is simply a sign of the times in the age of ignorance

In fact, you are contradicting yourself: First you refuse to accept the fact that 0.999...=1 only because this assertion is not compatible with what you name common sense and logic, ultimately leading you to then have to admit the existence of a "number" that can't be divided by any other number than 1 and itself, and to admit the existence of an interval that has no middle point, what completely defies obviousness and common sense as well...

on the contrary, i have already shown on multiple occasions that 0.999 does not equal 1, be it mathematically or logically - seems to me you are the one constantly trying to find some way to support this idea

what i did there is pointing out how generally by abusing mathematics no concept is safe from ending up perverted and how any perversion can be accepted as valid these days

1) What is the smallest positive number divisible by 10 (let name it r)? How do you write it? What is the decimal representation of this number?
2) Now let consider your d. What is the decimal representations of 10*d?

this was already answered before:

1*d = 0.0...1 <> 0
...
10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d

Of 100*d?

100*d = 0.0...100 <> 0 - this one has (n-2) leading zeroes, it is 100 times larger than d

what are the nth digits of these numbers?

in every one of these numbers: all digits are 0, except for one of them, which equals 1