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== Theory and Reality ==

There are no infinite objects in our world.<br>
In our world it is impossible to create an infinite object.<br>
<br>
The number 0.(9) does not exist and can not exist in reality.<br>
This applies to any infinite number.<br>
Any infinite number is a theory.<br>
<br>
If one day we begin to create something infinite in reality, then we will never finish creating it.<br>
Any infinity achieved is not infinity by definition.<br>
<br>
Therefore, 0.(9) = 1 only in theory.<br>
<br>
When we say that "0.(9) = 1", we mean that the already created number 0.(9) exists, but it exists only in our imagination.<br>
<br>
<br>
Now consider an example:<br>
x = 1/3 (is always)<br>
x = 0.(3) = 0.33333... (only in theory)<br>
y = 3x (is always)<br>
y = 3 * 1/3 = 3/3 = 1 (is always)<br>
y = 3 * 0.(3) = 3 * 0.33333... = 1 (only in theory)<br>
<br>
There is no paradox here. We just did not create an infinite number 0.(3) = 0.33333... in the reality.<br>
<br>
When we say:<br>
"x = 1/3 = 0.(3) = 0.33333...", we just deceive ourselves and do not understand it.<br>
<br>
<br>
One more example:<br>
x = 1/3 = 0.(3) = 0,33333...<br>
y = 3x = 3 * 1/3 = 3 * 0.(3) = 3 * 0.33333...<br>
y = 3x = 1 = 0.(9) = 0.99999...<br>
All this is only a theory.<br>
<br>
And how are things really?<br>
x = 1/3 ≠ 0.(3) = 0.33333...<br>
y = 3x = 3 * 1/3 ≠ 3 * 0.(3) = 3 * 0.33333...<br>
y = 3x = 1 ≠ 0.(9) = 0.99999...<br>
0.(9) ≠ 1<br>
<br>
<br>
I hope that I have completed this eternal argument.

[[User:Kirill Dubovitskiy|Kirill Dubovitskiy]] ([[User talk:Kirill Dubovitskiy|talk]]) 03:52, 6 January 2019 (UTC)

:Never would I disallow you to ''personally'' consider this ''eternal argument'' as ''completed'' by avoiding the use of decimal representations for non-terminating decimals (e.g.:  0.(3), or 0.33333..., or whatever notation), but in very broad, well informed circles these notations are consistently and fruitfully associated to numbers, the existence of which you evidently do not deny (e.g.: 1/3).

:OTOH, you are not given the freedom to simply disallow for the existence of coherent ''theories'', insinuate fallacies, and restrict conceptual ''realities'' to certain physical representations.

:Please, re-read the article's caveats about the range of "[[real number]]s" addressed in this treatment of your ''eternal argument''. As said, you are free to change the ballpark. [[User:Purgy Purgatorio|Purgy]] ([[User talk:Purgy Purgatorio|talk]]) 09:23, 6 January 2019 (UTC)

::"1/3" is a formula, a mechanism, a program, a machine, which is capable of infinitely creating an infinite number: 0.33333333...
::Or just 0.(3) is an ordinary short form.

::Take a piece of paper and a pen and try to create a number completely: 0.33333... (with an infinite number of "3"), probably then you will understand what is being said.

::And also "real numbers" have nothing to do with my evidence.

::And yes, we can say that 0.(3) is also an instruction or program for a machine or a machine for the production of an infinite number 0.33333...
::But then it turns out that:
::1/3 = 0.(3) ≠ 0.33333... or even 1/3 ≠ 0.(3) ≠ 0.33333...
::And therefore we simply agree among ourselves that 0.(3) is just a shorter way of writing an infinite number 0.33333...
::[[User:Kirill Dubovitskiy|Kirill Dubovitskiy]] ([[User talk:Kirill Dubovitskiy|talk]]) 03:52, 8 January 2019 (UTC)



== {{math|0.(9)<sub>''n''</sub> &ne; 1}} for any positive integer {{math|''n''}}, but whether {{math|0.999... {{=}} 1}} is true depends on the definition of {{math|0.999...}}  ==

I happened to know the interesting equation {{math|0.999... {{=}} 1}} through a video on Youtube. I was curious that why people are discussing this for quite a long time since it looks quite obvious that {{math|0.(9)<sub>''n''</sub> &ne; 1}}. 

I have to say that I am not an expert on math. To the best of my knowledge, I am giving the following arguments for the interesting debate on {{math|0.999... {{=}} 1}} which is intuitively incorrect to me.
However, the correctness of this equality really depends on the definition of {{math|0.999...}}.

=== {{math|0.999...}} or {{math|0.(9)<sub>''n''</sub>}}? The potential issue of two previous proofs of the equation {{math|0.999... {{=}} 1}}. ===

Below is a '[[0.999...#Algebraic_arguments|proof]]' of the equation {{math|1 {{=}} 0.999...}}:
{{block indent|1=<math>
\begin{align}
     x &= 0.999\ldots \\
   10x &= 9.999\ldots && \text{by “multiplying” by }10\\
   10x &= 9+0.999\ldots && \text{by “splitting” off integer part}\\
   10x &= 9 + x && \text{by definition of }x\\
   9x &= 9 && \text{by subtracting }x\\
    x &= 1 && \text{by dividing by }9
\end{align}
</math>}}
The issue of the above 'proof' becomes clear if we write it in another way:
{{block indent|1=<math>
\begin{align}
     x &= 0.(9)_n \\
   10x &= 9.(9)_{n-1} && \text{by “multiplying” by }10\\
   10x &= 9+0.(9)_{n-1} && \text{by “splitting” off integer part}\\
   10x &= 9 + (x - 0.(0)_{n-1}9) && \text{by definition of }x\\
   9x &= 9 - 0.(0)_{n-1}9 && \text{by subtracting }x\\
    x &= 1 - 0.(0)_{n-1}1 && \text{by dividing by }9\\
    x &\neq 1
\end{align}
</math>}}


Another well-known '[[0.999...#Algebraic_arguments|proof]]' of the equation {{math|1 {{=}} 0.999...}} is that: 
::Since {{math|{{frac|1|3}} {{=}} 0.333...}} (taught in elementary school), 
::we have {{math|3 &times; {{frac|1|3}} {{=}} 3 &times; 0.333...}} (by algebra), 
::that is {{math|1 {{=}} 0.999...}} (by algebra). 
However, as it has been widely pointed out, is {{math|{{frac|1|3}} {{=}} 0.333...}} correct? It depends on how we define {{math|0.333...}}. Nevertheless, we can say: 1 divided by 3 equals 0.3 with a [[remainder]] of 0.1, which can be written as: 
::{{math|{{frac|1|3}} {{=}} 0.3 R 0.1}}.
Or similarly, we can say:
::{{math|{{frac|1|3}} {{=}} 0.33 R 0.01}},
::{{math|{{frac|1|3}} {{=}} 0.333 R 0.001}},
::{{math|{{frac|1|3}} {{=}} 0.3333 R 0.0001}}.
And generalised as:
::{{math|{{frac|1|3}} {{=}} 0.(3)<sub>''n''</sub> R 0.(0)<sub>''n-1''</sub>1}},

Based on the above, if we agree that {{math|0.(0)<sub>''n-1''</sub>1 &ne; 0}}, it would be clear that {{math|{{frac|1|3}} &ne; 0.(3)<sub>''n''</sub>}}.
Also, by algebra, it can be easily derived that:
:: {{math|3 &times; {{frac|1|3}} {{=}} 3 &times; 0.(3)<sub>''n''</sub> + 0.(0)<sub>''n-1''</sub>1}}
which is
:: {{math|1 {{=}} 0.(9)<sub>''n''</sub> + 0.(0)<sub>''n-1''</sub>1}}
and hence {{math|0.(9)<sub>''n''</sub> &ne; 1}}.

=== What is {{math|0.999...}}? ===

As mentioned at the beginning, whether {{math|0.999... {{=}} 1}} is true depends on the definition of {{math|0.999...}}.

In the wikipedia page of [[0.999...]], it is said that "''0.999... (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).'' "
To me, this explicitly make {{math|0.999...}} as a number (something like {{math|0.(9)<sub> ''n''&rarr;&infin; </sub>}} or {{math|0.(9)<sub>&infin;</sub>}}), which I believe should be a member of the sequence sequence (0.9, 0.99, 0.999, ...) or the set {{mset|0.(9)<sub>''n''</sub> | ''n'' ∈ '''Z'''<sup>+</sup>}}. In this case, {{math|0.999... &ne; 1}} since we have shown that {{math|0.(9)<sub>''n''</sub> &ne; 1}}.

However, after the above definition in the wikipedia page of [[0.999...]], it is also said that "''This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...).''" which is equivalent to the definition of the notation 0.999... as the [[Limit_of_a_sequence|limit of the sequence]] (0.9, 0.99, 0.999, ...). 
There should be no doubt that the limit of the sequence (0.9, 0.99, 0.999, ...) is 1. 
Hence, in this case, since 0.999... is just a notation, there is no problem to say {{math|0.999... {{=}} 1}} which is the same as to say something like {{math|''x'' {{=}} 1}}.
<!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Snowinnov|Snowinnov]] ([[User talk:Snowinnov#top|talk]] • [[Special:Contributions/Snowinnov|contribs]]) </small>
----
:Please, reread the definitions: '''0.{{overline|9}}''' or 0.999... is &ndash;within this article&ndash;  '''not''' defined as {{math|0.(9)<sub>''n''</sub>}}, '''not''' for any natural number {{mvar|n}}, and additionally, '''neither''' ''n''&rarr;&infin; '''nor''' &infin; are ''numbers'' in any contexts referred to within this article, so the notations {{math|0.(9)<sub> ''n''&rarr;&infin; </sub>}} or {{math|0.(9)<sub>&infin;</sub>}} are not covered by the undisputed proposition "{{math|0.(9)<sub>''n''</sub>≠1}} for any natural number ''n''". There are no objections to "{{math|1 {{=}} 0.(9)<sub>''n''</sub> + 0.(0)<sub>''n-1''</sub>1}} for any natural number ''n''", however taking the limit "''n''&rarr;&infin;" takes these notations beyond their capabilities and the rigorous application of formally defined limits must take over. The claim that 0.999... is a member of the sequence (0.(9)<sub>''n''</sub>)<sub>n∈ℕ</sub> is not sustainable, because there is '''no''' such {{mvar|n,}} the limit of the sequence is not contained in it. [[User:Purgy Purgatorio|Purgy]] ([[User talk:Purgy Purgatorio|talk]]) 16:53, 19 February 2019 (UTC)


== All arguments for equality can be defeated, including the limit argument ==
{{atop|I think it's clear that this discussion is no longer productive, and nothing is served by keeping the thread open. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 14:33, 15 March 2020 (UTC)}}
One indication of this article's flaws is that the only official argument for 0.999... = 1 is the limit argument. All other arguments/proofs are straw men; they are false arguments that can easily be defeated. For example, the flaw in the formal proof on the '0.999...' Wikipedia page is that it does not allow x to be specified in terms of its sum to the nth term whereas it does allow 0.999... to be specified in terms of the sum to its nth term. We can use the nth sum of {{math|1=''x'' = 1 – 0.5/10<sup>''n''</sup>}} to give us a value where the nth sum is always half way between 0.999… and 1. Here we have:

{{block indent|1=<math>0\le 1 - (1 - 0.5/10^n) \le 1/10^n</math>}}
for any [[positive integer]] {{math|''n''}}.

This simplifies to:

{{block indent|1=<math>0\le 0.5/10^n \le 1/10^n</math>}}
for any [[positive integer]] {{math|''n''}}.

And we can see that this does hold for any positive integer n. This shows that if we allow x to be treated in the same way as 0.999…, then there are an endless amount of ‘numbers’ between 0.999… and 1. Indeed, by considering the nth sum, the only thing we can prove is inequality. The nth sum of {{math|0.(9)<sub>''n''</sub>}} will never equal the nth sum of {{math|1.(0)<sub>''n''</sub>}} and therefore these two cannot be equal.

Now let's consider the official argument. Limits and convergence were introduced in the early 19th century but they have always had their critics. In the case of 0.999..., the sequence is 0.9, 0.99, 0.999, and so on, and the limit of this sequence is said to be 1. But this sequence is not a finite structure that is continually being extended; it supposedly preexists as a static abstract object containing 'infinitely many' terms where each term corresponds to a digit in the infinite decimal 0.999... In other words, the limit argument requires that an actual infinity of terms must be possible. 

But this type of actual infinity has had counter-arguments going back to the Ancient Greeks over 2,000 years ago. For example, consider a continuous abstract line of length 2 units. If an actual infinity of parts were possible, then the first 1 unit of this 2 units length should be able to exist as the infinitely many lengths 9/10 + 9/100 + 9/1000 + ... corresponding to 0.999... But this causes at least two contradictions. Since the whole line is continuous, then there must be a 'last part' of the infinitely many lengths that is connected to the training length of 1. This contradicts the concept of 'infinitely many' which requires there to be no last part. Also since all parts are connected, if we were to count the lengths, then somewhere the count will need to go from a finite value to an infinite one.

And if the limit argument is flawed (because of its reliance on the validity of an actual infinity of terms), then no valid arguments remain for 0.999... equals 1. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 16:59, 9 May 2019 (UTC)

:You're kind of missing the point.  "0.999..." is a string of symbols (on a page, or a computer display, whatever).  "1" is also a string of symbols.  Under the [[real number]] system, these two strings represent the same number, as do "57 / 57", "4 - 3", and so on.  Your analogies with lengths and lines are completely irrelevant, because that's not how the real numbers are defined.  Your issue with "actual infinity" (which I find to be a meaningless concept anyway, but that's another story), is also irrelevant, because the set-theoretic frameworks under which the real numbers are defined happily handle infinite sets.  And on a side note, saying that something "has had its critics" is empty rhetoric.  The idea that the Earth goes around the sun has had its critics as well, but that hardly stops the rest of us from accepting reality. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 15:30, 7 May 2019 (UTC)

::My claim is that a static version (involving no motion or passage of time) of Zeno's most famous paradox invalidates the limit argument for 0.999... equals 1. You claim that this is completely irrelevant because "that's not how the real numbers are defined". You then give a hint about how you think the real numbers are defined by saying "the set-theoretic frameworks under which the real numbers are defined happily handle infinite sets".
::
::My argument addresses the popular approach to defining real numbers as ‘an equivalence class of rational Cauchy sequences’. In other words, a real number is defined as a container of infinitely many sequences, each of which is infinitely long, and where the difference between any two sequences will be a sequence that tends towards zero. Any sequence corresponding to a so-called 'infinite decimal' (such as 0.9, 0.99, 0.999, etc) will be a Cauchy sequence because its elements become arbitrarily close to each other as the sequence progresses.
::
::In other words, you claim that any counter-augment to the definition of real numbers as an equivalence class of Cauchy sequences is irrelevant because you know of some other definition that uses set theory (and which no doubt relies on the axiom of infinity). To use your own analogy, this is like saying we can reject any proof that the Earth goes around the sun if we have our own axiomatic system where one of our axioms says that the world does not go around the sun.
::
::Currently the Wikipedia page for 'real number' says: "The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; <), up to an isomorphism,[a] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent."
::
::I read this as saying that the different definitions are equivalent. So if one of them is invalid then all of them are. Therefore you cannot dismiss a flaw in one of them simply because it is more difficult to locate the equivalent flaw in another one of them.
::
::For pi to exist on the number line, then the number line from zero to pi must be able to be considered as being a series of lines of length 3, 0.1, 0.04, and so on. Furthermore, the line from pi to the point 4 must also be able to be considered as a line on the continuous number line. And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi mentioned earlier.
::
::I also notice that you have not commented at all on my refutation of the first formal proof of 0.999... equals 1. It is a very simple proof and in my opinion, its flaw is very easy to expose. Do you still agree with the formal proof or do you admit that it is flawed?
::
::Your first point is that I am missing the point. You proceed to argue that the strings of symbols "57 / 57" and "4 - 3" represent the same number. This is Platonism and I reject Platonism. I interpret your first example as a ratio, not a division operation. I also consider it to have a generic real-world meaning such as 57 of something are in one category as compared to 57 of something in a different category. For example, you have 57 apples and I have 57 apples. If we replace "57 / 57" by 1 then it tells us nothing about how many apples each of us have. Similarly your second example might relate to the action of taking 3 apples off a table that originally contained 4. 
::
::I believe that mathematics is the use of symbols as shorthand for generic descriptions of the laws of physics. Most mathematicians are Platonists that believe mathematics is something beyond all physics. At this point any condescending mathematician would respond to me by telling me that I don't understand mathematics and I need to go away and read up on it so that I will eventually know better!
::
::All the arguments for 0.999... equals 1 are flawed...
::
::A common argument is that since 1/3 = 0.333… then we can simply multiply both sides by 3 to get 1 = 0.999… This argument requires that we start by accepting that 1/3 equals 0.333… But we cannot start by assuming a rational can equal a repeating decimal because this is precisely what we need to prove.
::
::When we do short/long division for 1 ÷ 3 we follow an algorithm that repeats. We soon see that the trend is a longer (but finite) number of decimal places and a smaller (but always non-zero) remainder. So the long-term trend is a very long decimal and a very small non-zero remainder. The long-term trend is not ‘infinitely many’ digits with a zero remainder.
::
::If we think of 0.333… as 3/10 + 3/100 + 3/1000 + … then the sum up to the nth term is {{math|1=1 / 3 – 1 / 3(10<sup>''n''</sup>)}} and so this is less than 1/3 for all n. This means that the nth sum is a non-zero distance away from 1/3. This holds for ALL of the terms in 0.333… Since no term can possibly exist where 1/3 is reached, and since 0.333… is nothing more than its terms, it cannot equal 1/3.
::
::Then there is the argument that if we subtract 0.999… from 1 we get zero. If we say 0.999… is the series that has an nth sum of {{math|1=1 – 1/10<sup>''n''</sup>}}, and 1 is the series that has the nth sum of {{math|1=1 – 0<sup>''n''</sup>}} then when we subtract 0.999… from 1 we get the series that has an nth sum of {{math|1=0 + 1/10<sup>''n''</sup>}}
::
::If a series like 0.999… is a valid number, then this answer is equally a valid number. We cannot assert that this result must be numerically equal to 0, because that would mean that our starting position is that 0.999… already equals 1.
::
::Then there is the so-called algebraic proof. We start with x = 0.999... then we multiply both sides by 10 and subtract what we started with to apparently get 9x = 9 thus proving x = 1.
::
::The trick used to pull off this illusion is to misalign the series and then to claim that all trailing terms will cancel out, as shown here:
::
::10x = 90/10 + 90/100 + 90/1000 + …
::
::x = 9/10 + 9/100 + 9/1000 + …
::
::The trick is the misalignment of the terms (terms in the ‘x =‘ line above are shifted 1 place to the right). Such misalignment is invalid because if it was valid we could prove 0=1 by taking 1+1+1+… away from itself (try it yourself). If we align the series correctly then we get this result:
::
::10x — x = 81/10 + 81/100 + 81/1000 + …
::
::Another way to appreciate why the misalignment is invalid is to think of 0.999… as the series 9/10 + 9/100 + 9/1000 + … If we multiply this series by a factor of ten then we don’t change the number of terms; we have the same terms (in terms of one-for-one correspondence) as we started with, only now each term is ten times its original value.
::
::The subtraction 9.999… — 0.999… cannot cancel out all the trailing terms unless this one-to-one relationship (between the original and the multiplied series) is somehow broken, and we get an extra term out of nowhere.
::
::Yet another way to show that this algebraic proof is invalid is to consider the general formula for a geometric series, G, with first term ‘a’ and common ratio ‘r’ (since 0.999… is the geometric series with a=0.9 and r=0.1). If we assume that all matching terms cancel out (to ‘infinity’), then the result of the subtraction simplifies to:
::
::(1/r — 1)G = a/r
::
::Substituting a=0.9 and r=0.1, G=x, this evaluates to '9x = 9'.
::
::The important question is were we correct to assume that the trailing terms canel out all the way to infinity? Well, the resulting expression (above) should apply to all geometric series, both converging and diverging, because none of the manipulations used have any reliance on the values of the variables. So if we can find any values for the variables ‘a’ and ‘r’ where the above statement forms a contradiction, then we will have shown our assumption that all trailing terms cancel out was a mistake.
::
::The values a=1 and r=1 make the above statement evaluate to 0 = 1 and so the algebraic proof for 0.999… = 1 must be invalid.
::
::And since the limit argument is defeated by the static variation of Zeno's most famous paradox, we don't have a single valid argument for 0.999... equals 1. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 16:59, 9 May 2019 (UTC)

:::<small>Please indent your replies and avoid adding extra blank lines between paragraph.  I've fixed your last post up, but see [[Help:Talk]] for info on how to use talk pages, thanks.  Also, please try to add in a single post rather than a little bit at a time (you can use the preview button if you need).  This helps prevent edit conflicts.</small> &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 17:07, 9 May 2019 (UTC)
:::
:::You said: {{tq|"I believe that mathematics is the use of symbols as shorthand for generic descriptions of the laws of physics. Most mathematicians are Platonists that believe mathematics is something beyond all physics."}}  I don't like labels like "Platonist".  I certainly don't really consider myself one.  Also, what I described is more like [[Formalism (philosophy of mathematics)|Formalism]], not Platonism.  And it's not even that; it's just really pointing out the distinction between the string "57" and the real number it represents, which is not a string of symbols, but rather an equivalence class of Cauchy sequences of rational numbers (or pick your favorite other construction; it doesn't really matter which).
:::
:::In any case, our personal beliefs are only useful insofar as they provide guidance on what foundational axioms we're likely to work with.  If you have some sort of personal problem with standard set theory, and you prefer to work in some more restrictive setting, that's perfectly fine.  However, you can't then go on to proclaim that others who don't agree to also work in this more restrictive setting are somehow wrong &ndash; that's just silly.  Within normal (ZF) set theory (and even in many other, less common), any equivalent construction of the real numbers is going to include as a true statement the fact that 0.999... = 1.  The arguments are basic and easy to verify.
:::
:::Again, your problem seems to be not with this, but with other foundational issues.  You're then trying to say within your own framework that the statement "0.999... = 1" is false.  Whether that's because of the actual framework, or simply because you're interpreting the statement differently than everyone else is irrelevant.  You can't change the rules and then tell everyone else that they're wrong because they're not following your rules.  If you want to show the falsity of the statement, you have to do it within the rules under which the claim is being made.  Anything else is completely pointless.  &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 17:29, 9 May 2019 (UTC)

::::I'm new to editing Wiki pages so thank you for pointing out my editing mistakes. 
::::
::::Formalism and Platonism are inseparable. Anyone who is a Formalist must also be a Platonist by necessity. Formalism is the viewpoint that 'mathematical knowledge' is gained through using rules to manipulate physical symbols. But any given collection of squiggles on a piece of paper has no inherent meaning. The formalists have to agree on what the different symbols mean. Some symbols might be called 'numerals' and others might be called 'operators' and so on. These meanings have to be conveyed using a natural language, and so the symbols are merely shorthand for some natural language meaning. Sadly natural language can include logical contradictions such as 'a married bachelor' or 'infinitely many' or 'we can physically work with things that are completely detached from physical reality'. But just because we CAN assign a contradictory meaning to a symbol, it doesn't mean that we should. 
::::
::::Formalists maintain that their mathematical objects and rules have nothing to do with the real world. This belief that mathematics is somehow detached from physical reality is Platonism. Therefore if someone claims to be a Formalist then by necessity they are also conceding to being a Platonist.
::::
::::About your philosophy of mathematics you said "it's just really pointing out the distinction between the string "57" and the real number it represents, which is not a string of symbols, but rather an equivalence class of Cauchy sequences of rational numbers". We can easily say these words, but we cannot easily know what they mean. Nobody has any experience of anything that is 'not finite' and so I claim nobody really understands what one of these equivalence classes is. 
::::
::::We can experience endless algorithms such as: While 1=1: Print "Hello". We can also experience a large body of objects that fade out in the distance, and where we can't see an end point. We can also experience the counting of natural numbers. We know that if we are given the symbol for any natural number, then (if enough physical resources are available) we should be able to add 1 to it and construct the symbolic form of its successor. We might think that knowledge of all these concepts somehow enables us to understand what 'infinitely many' means but it doesn't. None of these things can be described as 'not finite'. Nobody has any concept of what 'not finite' means, but we still create definitions and rules and we pretend that this means we can work with the concept.
::::
::::We don't even have a clear unambiguous agreed definition of exactly what mathematics is. I favour Bertrand Russell's description: "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
::::
::::When we say something is mathematically proven, all this means is that a statement is valid according to a certain given set of rules and premises. But these rules and premises are allowed to be meaningless or even completely invalid; they can be any old nonsense. Mathematicians can therefore have great fun publishing loads of meaningless theorems and proofs.
::::
::::You said "any equivalent construction of the real numbers is going to include as a true statement the fact that 0.999... = 1" but you have not responded my points on this subject.
::::
::::You said "Again, your problem seems to be not with this, but with other foundational issues. You're then trying to say within your own framework that the statement "0.999... = 1" is false". I do have many issues with the foundations, but I believe 0.999... does not equal 1 within your framework, not one of my own.
::::
::::To that end, let's discuss actual proofs starting with the 1st formal proof that appears on the main Wiki page. In my opening comment of this thread I claim to have exposed the flaw in that proof. If you can, please explain to me what is wrong with my argument. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 01:08, 10 May 2019 (UTC)

:::::There's no flaw in the proof.  The {{mvar|x}} required in the proof is a fixed number; it doesn't depend on {{mvar|n}}.  On the other hand, you're giving a whole sequence of numbers, and trying to sneak in a different value for {{mvar|x}} depending on {{mvar|n}}.  This isn't what's being demanded in the proof, so it doesn't demonstrate anything.
:::::
:::::For what it's worth, I don't think this is really the most instructive or clear proof here.  The one based on the Cauchy sequence definition is much more straightforward.  Proceeding like this also has the advantage that you don't need to consider any special properties of the real numbers to complete the proof.  Instead, all the heavy lifting is done ahead of time when you first show that the Cauchy sequence construction describes a complete, ordered field, as we're looking for.
:::::
:::::Once that's done, all you have to do is decide what's meant by "0.999...".  There's more than one way to proceed here, but most people would agree that the most reasonable interpretation is the real number which is the equivalence class of Cauchy sequences represented by (0.9, 0.99, 0.999, ...).  And similarly, "1" means the real number represented by the Cauchy sequence (1, 1, 1, ...).  To show that "0.999... = 1" then means to show that the two representatives that we've chosen lie in the same equivalence class.  This is done by showing that their termwise difference converges to 0.  Indeed, that difference is the sequence (0.1, 0.01, 0.001, ...).  This sequence does indeed converge to 0 (straightforward exercise for the reader), which means that the two sequences are in the same equivalence class, which means that "0.999..." and "1" represent the same real number, by definition.  Short and sweet.  &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 02:31, 10 May 2019 (UTC)

::::::You said "There's no flaw in the proof. The x required in the proof is a fixed number; it doesn't depend on n.". So your objection to my argument appears to be that the x in the proof is not a real number (because many real numbers obviously CAN be described in terms of n, just like 0.999... can) but that x is a 'fixed number', whatever that is. 
::::::
::::::If I assume that by 'fixed number' you are referring to a fixed point data type, which is essentially an integer that is scaled by a certain factor, then the proof only applies to a subset of the real numbers. So all that it proves is that SOME real numbers cannot be placed between 0.999... and 1. My counter argument still holds that other real numbers CAN be placed between them.
::::::
::::::The description on the main Wiki page is slippery in that it doesn't explicitly describe what type of number x is. You are claiming that it is not any real number but that it is any of a particular subset of the real numbers. As such, it only proves that numbers from that subset cannot be placed between 0.999... and 1.
::::::
::::::Next you said "I don't think this is really the most instructive or clear proof here. The one based on the Cauchy sequence definition is much more straightforward.". By describing it as not the most instructive or clear it sounds like you don't put too much stock in its validity. You previously said "There's no flaw in the proof" and so I think it is important that we get to the bottom of this lack of clarity so that we can both agree on whether or not the proof is valid within your framework of mathematics. Can you confirm what number type you believe x to be? Is it ANY real number? Is it ANY fixed-point decimal (& therefore only a subset of the real numbers)? Or is it something else?
::::::
::::::Moving on. let's consider your preferred argument, which is that the term-wise difference between the two sequences appears to approach zero, and therefore 0.999... and 1 are equal by definition. To the lay person, this is far from a clear and instructive proof. Indeed, it took over 200 years after the introduction of infinite decimals before any of the worlds greatest mathematicians devised this argument. And all that it demonstrates is that if we are inventive enough then we can construct a series of clever sounding definitions so that both 0.999... and 1 happen to fall into the same categorisation. 
::::::
::::::It causes confusion for the lay person because the meaning of terms like 'sum' and 'equals' have been redefined to mean something completely different from the intuitive trivial meanings that we first learn as children. Furthermore it all rests on the validity of the limit argument, which is not accepted by some well known mathematicians such as Professor Normal Wildberger, Dr. Doron Zeilberger and others. Indeed, even the Ancient Greeks had an argument that causes problems for the limit approach which I have explained several times in this thread.
::::::
::::::And so your preferred argument is surrounded by controversy within your own ranks. It is confusing to the lay person and far from clear or intuitive. Even the Wiki page itself suggests that the intuitive explanation is "If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1." and "there is no number that is less than {{math|1/10<sup>''n''</sup>}} for all n". So I think we should focus on the formal proof of this intuitive explanation before we dive into the mire of equivalence classes of Cauchy sequencs and limits. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 11:03, 10 May 2019 (UTC)

{{od}} I'm not sure how much more I can say that hasn't already been said; your objection to the proof isn't valid because you're using a {{em|different}} value of {{mvar|x}} for each {{mvar|n}}.  That's not what's required in the proof, so you haven't demonstrated anything by doing so.  "Fixed number" does {{em|not}} mean "fixed point data type" (whatever that means exactly); it means that it's a single quantity within the scope of the proof, and its value doesn't depend on any other variables (most importantly {{mvar|n}} in this case).  My issues with the proof are with its exposition, not its validity.  It's essentially treating "0.999..." to be the [[least upper bound]] of the set {{math|{{mset|0.9, 0.99, 0.999, ...}}}}.  This is guaranteed to exist because [[Completeness of the real numbers|the real numbers are complete]], and this value is assigned to {{mvar|x}}.  Then, it goes on to prove that 1 is this least upper bound, and hence what's meant by "0.999...".  This is all perfectly valid, even if it's not clear from the write-up here.

Take a step back for a second and forget the math.  How likely do you think it is that entire generations of brilliant mathematicians (far smarter than either you or I) have missed something so utterly basic that it was refuted in a couple of lines by a completely elementary argument?  And these are people that are trained in the study of this subject, to analyze logical arguments critically.  Or maybe is it just a teensy bit more likely that you don't quite understand something fully?  Now, that's not a mathematical argument, but it's worth considering.

There is no controversy among mathematicians any more than there's controversy among Egyptologists that the pyramids weren't built by [[Ancient astronauts|aliens]].  (On a side note, Wildberger is a kind of a crank (which I realize isn't an argument, but I really don't want to get into that here), and I suspect you're misrepresenting Zeilberger's views (who isn't a crank, but probably isn't saying what you think he's saying)).  In any case, back to what I said earlier, even if there are mathematicians that (maybe due to philosophical views) prefer to work in some more restrictive settings which don't admit constructions of the real numbers, it makes absolutely no difference, because {{strong|the original claim isn't being made in the more restrictive setting; it's being made within the usual foundations of mathematics}}.

And finally, your complaints about the Cauchy sequence argument being confusing are toothless (if maybe a bit telling).  It {{em|is}} clear and instructive to someone with the background to digest it.  Any formal proof is going to rely on a either a construction of (like via Cauchy sequences, or Dedekind cuts, or any number of others) or abstract characterization of (as a complete, ordered field) the real numbers.  And any such approach is going to require a comparable level of mathematical sophistication that's not possessed by the layperson.  It takes some work to get there, and you can't expect to learn it all in an afternoon.  But with dedication, it can be learned.  &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 13:39, 10 May 2019 (UTC)
:{{to|PenyKarma}} I agree with everything that has been said by {{u|Deacon Vorbis}}. I would add some more general comments. It seems that you confuse the philosophical concept of truth with its mathematical counterpart. Platonism has to do with the philosophical concept, and has nothing to do with modern mathematics. A mathematical result is true only if it can be proved from the axioms of the theory in which is stated, and this has nothing to do with any physical interpretation. The mathematical notion of a proof is completely formalized, and there are software that allow verifying difficult proofs. On the other hand even the best computer scientists cannot imagine how verifying a philosophical truth on a computer. So involving philosophers about mathematical truth, as you did by referring to Platonism, is a fundamental error.
:My second point is that there are deep philosophical questions about mathematics, about which there is no consensus, even among mathematicians. Unfortunately these questions are rarely discussed by philosophers. One of them is the following: Until the end of the 19th century the development of mathematics was mainly motivated by the study of the physical world. Since the beginning of the 20th century, many mathematical concepts and theories have been developed independently of any application, as there were motivated only by questions of pure mathematics. Nevertheless many such theories appeared later to be useful in physics. One famous example is the use of non-Euclidean geometry by Einstein, but many other examples are available. This set the question of what is the true relation between mathematics and the real (physical) world, and why pure mathematics are so useful. The answer of this important question can certainly not be found by classifying, as you did, thinkers into Platonists, modernists, formalists, post-modernists, etc. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:33, 10 May 2019 (UTC)

::First off, my arguments are not new, they have been around for over 2,000 years. Zeno devised some paradoxes that he claimed showed that time and/or movement could lead to contradiction. Democritus and some others noticed that these problems did not necessarily have to relate to motion or the passage of time, and this led to the foundation of Atomism. It was Democritus and some of his contemporaries that interpreted Zeno's paradoxes as showing that the concept of infinite divisibility leads to contradiction and therefore everything must consist of a finite amount of indivisible parts. These are exactly the same contradictions that come with the notion of pi (or any number) as being a constant on a continuous number line. It means that the concepts of real numbers and the continuum lead to contradiction.
::
::For pi to exist on the number line, then the number line from zero to pi must be able to be considered as being a series of lines of lengths 3, 0.1, 0.04, and so on. Furthermore, the line from pi to the point 4 must also be able to be considered as a line on the continuous number line. And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi. This forms a contradiction because the concept of 'infinitely many' parts requires there to be no last part.
::
::It was in the 16th century when Simon Stevin created the basis for modern decimal notation in which he allowed an actual infinity of digits. Yes they knew about the contradictions of infinite division, but everyday mathematics used in businesses was made much easier by the widespread use of base 10 decimals. To my sceptical eye, it looks like mathematical rigour was sacrificed in favour of ease-of-use.
::
::The original idea behind infinite decimals was that they were the sum of their rational parts. Essentially a real number was defined as being its decimal representation, the two were inseparable. This definition was considered inadequate by many, not least because its lack of uniqueness (as in 0.999... and 1 being the same number). 
::
::It was not until the early 19th century that limits and convergence were introduced. The equivalence class of Cauchy sequences finally gave us a unique construct for any one real number. Since it took over 200 years before any of the worlds greatest mathematicians devised this approach, it was clearly not intuitive at the time.
::
::You said "Take a step back for a second and forget the math. How likely do you think it is that entire generations of brilliant mathematicians (far smarter than either you or I) have missed something so utterly basic that it was refuted in a couple of lines by a completely elementary argument? And these are people that are trained in the study of this subject, to analyze logical arguments critically. Or maybe is it just a teensy bit more likely that you don't quite understand something fully?". You could shorten this to "go away you stupid person".
::
::You said "your complaints about the Cauchy sequence argument being confusing are toothless (if maybe a bit telling). It is clear and instructive to someone with the background to digest it.". I read this as you telling me that I find equivalence classes of Cauchy sequences troublesome because I'm not clever enough to get my head around it. You are right, I openly admit I cannot conceive of infinity.
::
::For the real number 57, its equivalence class will contain the sequence whose nth term is {{math|57 – 1/10<sup>''n''</sup>}} and the sequence whose nth term is {{math|57 – 1/20<sup>''n''</sup>}} as well as infinitely many other sequences. Yes I struggle to get my head around conceiving infinitely many of something, especially when I am aware of the contradictions associated with 'infinitely many' highlighted by the Atomists. 
::
::I am fully aware that on the cosmic scale of cleverness I am a mere infinitesimal distance from the bottom. I'm sure you have already explained to me as simply and clearly as you can why the Atomist argument is flawed, but I persist because I am just too stupid to understand it. I only studied maths up to A-level and then a little more at University whilst studying Computing Science. I guess this is not a good enough maths background to understand why 0.999... equals 1.
::
::It is testament to my stupidity that I apparently don't even understand the elementary proof. The Wiki page introduces the proof thus: "There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. ". I would have expected the formal version to also avoid reference to more advanced topics but you have just told me about it "treating "0.999..." to be the least upper bound of the set {0.9, 0.99, 0.999, ...}" and "This is guaranteed to exist because the real numbers are complete". If the proof already accepts the definitions of real numbers and the completeness of them, then what is left to be proved?
::
::When I try to understand the proof without reference to advanced concepts, it appears to me to be a statement about infinite decimal representation. It is all about what you can fit into n decimal places using a decimal system. If n is 5 then we cannot construct any decimal with 5 decimal places that is between 1.00000 and 0.99999, and this holds for any value of n. This is all it says to me.
::
::I read it as a proof by contradiction where we start by assuming that a unique number is defined as its infinite decimal representation with no leading zeros in front of the units column. This means that since 0.999... and 1.000... have different decimal representations, we assume that they are different numbers by definition. We also assume that an infinite decimal representation is a coherent concept that does not lead to contradiction. We also assume that any fractions (e.g. 1/3) can be fully represented by an infinite decimal.
::
::Then the proof leads to contradiction and so this means that one or more of our assumptions must be wrong. The popular interpretation of the contradiction is that it simply means that our assumption that decimal representation is unique must be wrong. But if we adopt this solution then we will be ignoring the fact that our assumption that an infinite decimal is a coherent concept still has the unresolved issues highlighted by the Atomists. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 18:39, 10 May 2019 (UTC)

:::{{to|Deacon Vorbis}} Please note that I did not try to change your signature, perhaps our edits clashed or maybe I placed my signature in the wrong place? I am trying to revise something that I said 15 days ago, but I am not trying to change the meaning. I'm just trying to add clarification for any 1st time readers. It would have exactly the same meaning but it would read better for new vsitors. It is important that it is easy to understand because it relates to my objection to the proof. Any mathematician would realise that the change is not substantive because there is no change of meaning in terms of the mathematical argument. Would you be happy with this change?... Just before I say:
:::We can use the nth sum of {{math|1=''x'' = 1 – 0.5/10<sup>''n''</sup>}}
:::I'd like to add this:
:::For example. let x = 95/100 + 45/1000 + 45/10000 + 45/100000 + … [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 18:08, 24 May 2019 (UTC)

::::If you want to add to something you said 15 days ago, after it's been more than responded to, then add it at the bottom.  Hopefully, any first time readers aren't misled by the nonsense that you continue to spout.  I've been more than patient explaining where you're mistaken, but you refuse to listen, having already convinced yourself of your own inerrancy.  I'm done here.  &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 18:14, 24 May 2019 (UTC)

:::::You said "your objection to the proof isn't valid because you're using a different value of x for each n."
:::::But it is obviously a different value because it is the nth partial sum of 95/100 + 45/1000 + 45/10000 + 45/100000 + … just like the proof uses different values (i.e. partial sums) of 9/10 + 9/100 + 9/1000 + ...
:::::I took your comment on board and I said that the only other way I could interpret the proof is if it only relates to decimal representations where n is the nth decimal place. In that case my objection is that we cannot assume that all rationals (or sums of rationals) can be represented by a decimal representation. That would be to assume things like 1/3 equals 0.333... and this is precisely equivalent to what we need to prove.
:::::Now you have resorted to insults and you ended with "I'm done here". Given the tone of your last comment I'm glad your done. You think my lack of intellect is justification for you to insult me. You are wrong. There is no excuse for your behaviour. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 18:51, 24 May 2019 (UTC)
:OK, there's nothing going on here. This article is about 0.999... in the [[real number]]s; the real numbers contain no infinitesimal, which leads inexorably to the conclusion that 0.999... is equal to 1. If you wish to work in some other philosophical system, feel free, but not here. [[User:Jpgordon|--jpgordon]]<sup><small>[[User talk:Jpgordon|&#x1d122;&#x1d106; &#x1D110;&#x1d107; ]]</small></sup> 20:35, 24 May 2019 (UTC)

{{green|For pi to exist on the number line, then the number line from zero to pi must be able to be considered as being a series of lines of lengths 3, 0.1, 0.04, and so on. Furthermore, the line from pi to the point 4 must also be able to be considered as a line on the continuous number line. And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi.}} Nope. {{ping|PenyKarma}} You’re never going to be taken seriously here since literally none of what you said there is [[well-defined]], making your entire argument just [[hand-waving]].—[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 21:50, 25 May 2019 (UTC)
: You see Jasper Deng, this is why this argument never ends.  What you have basically said here is "I don't understand your argument, therefore you are wrong."  It's perfectly clear to me what PenyKarma is saying, so I can't imagine why you would think your reply is persuasive.  [[User:Algr|Algr]] ([[User talk:Algr|talk]]) 14:51, 30 January 2020 (UTC)
::{{ping|Algr}} [[Strawman argument]]. We cannot even evaluate the truth of statements that are [[not even wrong]]; notice how I made no explicit pronouncement on the truth of his statement.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 09:19, 19 February 2020 (UTC)
::: [[User:Jasper Deng|Jasper Deng]] At least a strawman argument claims to try to understand what was being said.  You haven't even done that.  "Not even wrong" is more appropriate to your statement because you don't actually say anything about .999...  You are just engaging in fancy name calling.  [[User:Algr|Algr]] ([[User talk:Algr|talk]]) 19:27, 20 February 2020 (UTC)
::::Nope. "Not even wrong" applies wholeheartedly to PenyKarma's argument since they are devoid of rigorous meaning. Specifically, "And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi." is meaningless; in fact, the second half of it is self-contradictory in any reasonable interpretation. What does he mean by "line"s? There's nothing about the real line that asks for this. {{ping|Algr}} Considering that you have for many years demonstrated that your understanding of this subject is woefully inadequate to converse here, please stay out of any further conversations here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 21:28, 12 March 2020 (UTC)
:::::My line argument is easy to understand. The complaint about what I mean by 'line' is just nitpicking because I did not use the more precise expression 'closed line segment' (which includes both end points).
:::::The closed line segment from 0 to 3 shares just one point with the closed line segment from 3 to 3.1. Apart from the overlapping point, these two line segments equate to the single closed line segment from 0 to 3.1. If the decimal value for pi can exist on the number line, then it follows that each of the line segments that I described earlier (0 to 3, 3 to 3.1, 3.1 to 3.14 and so on, forming infinitely many line segments) must also be able to exist as their static start and end points must exist on the number line. 
:::::It then follows that the line segment from pi to 4 must share the point 'pi' with just one of the infinitely many line segments described earlier. In other words, it must connect to a last line segment within the infinitely many line segments. This forms a contradiction as 'infinitely many' requires there to not be a last line segment. The same argument could be made with 0.999... instead of pi. The concept of an infinite decimal always leads to contradiction. 
:::::Those who have an unshakable belief in the mystical concept of mathematical infinity will always construct slippery, murky, and over complicated arguments in a futile attempt to justify it. These arguments include all the so-called proofs for 0.999... equals 1. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 14:07, 14 March 2020 (UTC)
::::::{{ping|PenyKarma}} Here's your fallacy then: you have discovered that the union of infinitely many (even countably many) [[closed set]]s is not necessarily closed and there are no reasons to believe otherwise. Their union is a ''half-open interval'' including 0 but not pi, since by definition of a [[set union]], pi would have to belong to at least one of the sets in question, but it does not. There's nothing paradoxical about that and it does not disprove the idea of "infinity". Any closed interval from pi to some greater number will have empty intersection with this half-open interval and yet no number in the union of this interval with all those intervals will be omitted (so in your example, the union of all these is still the closed interval from 0 to 4). But the ''[[least upper bound]]'' of the union of all the intervals you mentioned that are less than pi is still pi, and that is the ''definition'' of a decimal representation. Sorry, but you're wrong again!--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 20:43, 14 March 2020 (UTC)
:::::::{{ping|Jasper Deng}} In my example, all of my lines are closed line segments with a well defined point at each end. They are ordered and, going from left to right (in relation to their mapping on the number line), the end point of one line is also the start point of the next line.
:::::::But with your half-open interval argument you appear to be claiming that none of the infinitely many closed line segments (from 0 to pi) in my argument can contain the point pi. In other words, you are saying that if the infinite decimal corresponding to pi could exist, then the sum of all its digits would not reach pi. You are effectively saying that pi does not equal pi. This supports my claim that infinite decimals cannot exist.
:::::::My line argument makes sense to many non mathematicians. They can see that there is an obvious contradiction. The counter arguments presented by mathematicians are always something like your least upper bound interpretation of a decimal representation. They are nothing more than slippery wordplay. If you could actually determine the least upper bound (which you can't in this case because of its infinite nature) then you are back where you started with an infinite decimal. And so you can't actually describe the infinite decimal for pi this way as it is a circular argument at best. The messy and complicated counter arguments might sound clever but they resolve nothing. The contradiction is still there. It is clear and obvious, unlike the counter arguments. [[User:PenyKarma|PenyKarma]] ([[User talk:PenyKarma|talk]]) 00:35, 15 March 2020 (UTC)
::::::::{{ping|PenyKarma}} Sorry, you completely ignored the part about the [[least upper bound]]. If you want more detail on that, see the [[Dedekind cut]] [[construction of the real numbers]]. In particular, if one bounded set is the closure of another, then their least upper bounds are equal. In particular, <math>\sup [0, \pi) = \sup [0, \pi] = \pi</math>. That your view is absurd is demonstrated by the effect of changing base to base 2, 3, etc. "Pi" is not equal to the value of any finite truncation of its decimal expansion but is the ''supremum'' of the set of all such expansions. You also clearly have no clue what you are talking about when you say "reach"; "reach" here means "converges to" and for an increasing [[monotonic sequence]] like this one that means taking the supremum ''which need not be a member of the sequence itself''. This is how decimal expansions work and therefore, your line argument is nothing but complete bullocks in the real numbers. There is absolutely ''nothing'' whatsoever that requires the union of all these closed line segments to be closed. We are not bound by physical limitations on however many "lines" there are. So please, stop wasting your own time on this useless argument and learn some actual real analysis. I stand by my earlier dismissal of your argument even more after this nonsense).--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 01:28, 15 March 2020 (UTC)
::::::::{{ping|PenyKarma}} And "[[least upper bound]]" is not "slipipery word play". You are so blinded by your refusal to actually learn real analysis it's not even funny. Completeness (which the least upper bound is one form of) is one of the most fundamental properties of the real numbers. If you are going to reject that, then you cannot possibly be talking about the real numbers. In that case, please do us a favor and leave, because there is nothing more to be discussed.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 01:36, 15 March 2020 (UTC)
{{abot}}

== Yet another anon ==
The page on 0.999... is very biased. They say that 0.999... equals one, which in actuality is not true. The fact is that people are continuing to believe so-called math experts, just because they have fancy doctorates. I wish that the page on 0.999... would present both sides of the issue. It is wrong for the website that is supposed to promote free knowledge to be so openly taking sides in a debate that is still very much open, especially since there are many reasons why 0.999... does not equal one.  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/24.127.161.155|24.127.161.155]] ([[User talk:24.127.161.155#top|talk]]) 16:23, 7 January 2021 (UTC)</small> <!--Autosigned by SineBot-->

:Are there any reliable sources for your claim? <span style="font-family:Avenir;">[[User:MEisSCAMMER|MEisSCAMMER]]<sup>[[User talk:MEisSCAMMER|(talk)]]</sup><sub style='position: relative; left: -3em; font-size:.5em;'>Hello!</sub></span> 23:48, 17 February 2021 (UTC)
::I guess [[user:24.127.161.155]] is either a troll or a jokester. Still, I'd like to support the view - by about 1%. There are more than one intuitive concept or formal construction of the set of "all numbers", and there are other notations than decimal. The lead appropriately mentions decimal, but it says that in math, 0.999... denotes the decimal blah blah, which, arguably, is untrue; e.g., there is also, in math, a hexadecimal number 0.999... (having the value 3/5, or 0.6 in decimal notation, a number having no other hex representation than 0.999...). I guess there is no construction of numbers of any merit where 0.999... decimal represents another number than 1, but I do not know. Still, what I am driving at is this: The lead may be a little too sweeping in firmly stating that 0.999... is 1. Adding a few words to set the context in which this is true (and perhaps making it clear that this context is for al practical purposes the only one worth considering) would, in my opinion, be an improvement.--[[User:Nø|Nø]] ([[User talk:Nø|talk]]) 09:40, 18 February 2021 (UTC)
::PS. For the record, I just now spotted this reference desk discussionj initiated by same IP user: [[wp:Reference_desk/Archives/Mathematics/2021_January_11]]. --[[User:Nø|Nø]] ([[User talk:Nø|talk]]) 09:45, 18 February 2021 (UTC)
:::Right, if there were any reliable sources, we'd be happy to include the proposition, but as there are none, we can't include it. (Also, for your concern about different number systems — it does state "''This repeating decimal represents the smallest number no less than every '''decimal number''' in the sequence (0.9, 0.99, 0.999, ...)''..." in the lead section.) <span style="font-family:Avenir;">[[User:MEisSCAMMER|MEisSCAMMER]]<sup>[[User talk:MEisSCAMMER|(talk)]]</sup><sub style='position: relative; left: -3em; font-size:.5em;'>Hello!</sub></span> 22:18, 18 February 2021 (UTC)
::::I don't know if that's a reply to me or to the OP. Obviously, as for my post, relaible sources say exactly what I say: Within a certain construction of numbers and a certain number notation, 0.999... is equal to one. All I am proposing is making this context more clear. The trick - and I am not sure how to do this elegantly - is to make it clear, at the same time, that this context is pretty much the only one worth our time.--[[User:Nø|Nø]] ([[User talk:Nø|talk]]) 10:54, 19 February 2021 (UTC)
:::::What I'm saying is that the context is already clear; see my comment above, it states "''This repeating decimal represents the smallest number no less than every '''decimal number''' in the sequence (0.9, 0.99, 0.999, ...)''..." in the lead section. <span style="font-family:Avenir;">[[User:MEisSCAMMER|MEisSCAMMER]]<sup>[[User talk:MEisSCAMMER|(talk)]]</sup><sub style='position: relative; left: -3em; font-size:.5em;'>Hello!</sub></span> 23:17, 19 February 2021 (UTC)
::::::I think that the original comment is very right. 0.999... is not equal to one, and even if it wasn't, Wikipedia should discuss both sides of the issue.[[Special:Contributions/2601:40E:8180:9BF0:6C6C:32D6:51A6:37A2|2601:40E:8180:9BF0:6C6C:32D6:51A6:37A2]] ([[User talk:2601:40E:8180:9BF0:6C6C:32D6:51A6:37A2|talk]])  <!--Template:Undated--><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|undated]] comment added 12:40, 24 February 2021 (UTC)</small> <!--Autosigned by SineBot-->
:::::::The other side of the issue is discuted in details in {{slink|0.999...|Skepticism in education}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 13:21, 24 February 2021 (UTC)
::::::::I still think a more explicit reference to the construction of the real numbers in the lead would be good. As for the OP, there are two types of replies:
::::::::#Follow this proof why 0.999...=1 blah blah ...
::::::::#This train of thought should make it clear to you where your own reasoning goes wrong blah blah ...
::::::::Type 1 is easy; type 2 is next to impossible. Perhaps something like this would be sort of convincing:
::::::::*0.999... does not represent 0.9, as 0.999... > 0.9.
::::::::*0.999... does not represent 0.99, as 0.999... > 0.99.
::::::::*0.999... does not represent 0.999, as 0.999... > 0.999.
::::::::*0.999... does not represent 0.9999, as 0.999... > 0.9999.
::::::::*0.999... does not represent 0.99999, as 0.999... > 0.99999.
::::::::*...
::::::::*0.999... does not represent any number smaller than 1 by any finite amount.
::::::::The way real numbers are constructed (at least since 1849), there is no number that is smaller than 1, unless by a finite amount. This shows that 0.999... >= 1. I do '''not''' suggest actually including this in the article.--[[User:Nø|Nø]] ([[User talk:Nø|talk]]) 14:19, 24 February 2021 (UTC)
::::::I understand your argument, and your reasoning makes a lot of sense. I still believe they are not equal, however. 0.999... is 0.000...1 away from one.That is infinite 0's, then a 1. It isn't able to be fully shown by our current number system, but that is the best I can do to explain the difference. Also, using the same logic as your argument, each nine in 0.999... fails to be equal to one, so do you think that the "infinith" nine somehow makes it equal to one?  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2601:40E:8180:9BF0:6C6C:32D6:51A6:37A2|2601:40E:8180:9BF0:6C6C:32D6:51A6:37A2]] ([[User talk:2601:40E:8180:9BF0:6C6C:32D6:51A6:37A2#top|talk]]) 19:53, 24 February 2021 (UTC)</small> <!--Autosigned by SineBot-->
:::::::The "infinite 0's, then a 1" number ''does not exist''. By definition, infinite zeros would not have an end, so it would be impossible to place a "1" after it. If you mean an infinitesimal, those also don't exist, see [[Infinitesimal]]. I think you're confusing infinity with a ''number'' as opposed to a concept which means ''forever''. <span style="font-family:Avenir;">[[User:MEisSCAMMER|MEisSCAMMER]]<sup>[[User talk:MEisSCAMMER|(talk)]]</sup><sub style='position: relative; left: -3em; font-size:.5em;'>Hello!</sub></span> 13:03, 1 April 2021 (UTC)
:::::::I'm sorry it took me so long to respond. I am not proposing that 0.000...1 is a very intuitive way of writing the difference between 1 and 0.999..., but it is the best I can do. I think you may be confusing 0.999... with the limit of 1/9+1/09+1/009... Just because an endless series of nines after a decimel point is close to 1, albeit very very close, doesn't mean they are equal. Denying the existence of a gap between 0.999... and 1, just because they are very close, is like being a humongous giant and denying the existence of a a space between different sheep in a herd of sheep. The space may be small, unimaginably small to some, but it is still there. [[Special:Contributions/2601:40E:8180:9BF0:9CD6:55A7:F59B:1A01|2601:40E:8180:9BF0:9CD6:55A7:F59B:1A01]] ([[User talk:2601:40E:8180:9BF0:9CD6:55A7:F59B:1A01|talk]]) 01:21, 20 June 2021 (UTC)
::::::::When you say "the limit of 1/9+1/09+1/009", ... there are two mistakes here: first, 1/9 and 1/09 are the same- you probably meant to say "the limit of 0.9 + 0.09 + 0.009 + ...". Secondly, "0.999 repeating" and the limit of the infinite sum "0.9 + 0.09 + 0.009 + ..." are the same thing, namely, 1. [[Special:Contributions/37.186.17.188|37.186.17.188]] ([[User talk:37.186.17.188|talk]]) 22:54, 29 February 2024 (UTC)
:There isn't really any debate in math. It's like saying there's a debate about the earth being flat: sure, there is, but those people are idiots. Likewise, people who don't understand that 0.999... equals one lack a basic understanding of math.
:And yes, please, let's believe the experts. There are no "two sides", it's a false equivalency. As Asimov has said, "Anti-intellectualism has been a constant thread winding its way through our political and cultural life, nurtured by the false notion that democracy means that 'my ignorance is just as good as your knowledge.'" [[Special:Contributions/37.186.17.188|37.186.17.188]] ([[User talk:37.186.17.188|talk]]) 22:49, 29 February 2024 (UTC)

{{unindent|:::::::}} "I think you may be confusing 0.999... with the limit of 1/9+1/09+1/009..." Presumably you mean 0.9 + 0.09 + 0.009 + 0.0009... No, I am not confused, because that's what 0.999... is, per the article. <span style="font-family:Baskerville;">&#9816;[[User:MEisSCAMMER|ME]]<sub>[[User talk:MEisSCAMMER|is]]</sub>[[Special:Contributions/MEisSCAMMER|SCAMMER]]</span> 23:53, 27 February 2022 (UTC)

== Small note about the "rigorous proof" ==
Near the end "This implies that the difference between 1 and x is less than the inverse of any positive integer." This is not true. That symbol is less than or equal, which means that the difference between 1 and x can be equal to the inverse of any positive integer, which includes options that are not 1, so the proof is not complete. It may be correct, I am just a student, but it needs further explanation if so.  <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Tyguy338|Tyguy338]] ([[User talk:Tyguy338#top|talk]] • [[Special:Contributions/Tyguy338|contribs]]) 21:31, 30 September 2021 (UTC)</span> <!--Autosigned by SineBot-->
:{{ping|Tyguy338}} If a quantity is less than or equal to the inverse of any positive integer, then for any positive integer, add one and take the inverse of that: the result will be smaller than the inverse of the original integer. Put another way, if we know that something is less than or equal to a half, and also less than or equal to a third, then it's clearly less than a half. I think this follows so naturally it doesn't need to be spelled out. [[User:MartinPoulter|MartinPoulter]] ([[User talk:MartinPoulter|talk]]) 11:07, 1 October 2021 (UTC)

== Why does this exist? ==

Why do we have a page to hear the rambling of people who deny a fact universally accepted by mathematicians? There are no similar pages for creationists at [[Talk:Evolution]], for relativity deniers at [[Talk:Theory of relativity]] or for anti-vaxxers at [[Talk:Vaccine]]. [[User:OneToZero|OneToZero]] ([[User talk:OneToZero|talk]]) 11:40, 3 November 2021 (UTC)
:There are 11!!! archive pages. How many man-hours have serious editors wasted here? [[User:OneToZero|OneToZero]] ([[User talk:OneToZero|talk]]) 11:43, 3 November 2021 (UTC)
::<small>Hush, don't wake the dragon...</small> - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 12:00, 3 November 2021 (UTC)
:::I have read the most recent deletion thread for this page. The following were stated as reasons for its existence:
:::* Deliberately inviting objections so that regulars can better write the article content for mathematical novices. Objection: Most of the arguments here are by people who "just don't get it", that is, they do not logically comprehend the proofs. The current article is simple enough to maintain its usefulness to non-mathematicians while compromising neither rigor nor depth. As mathematics educators know, it is impossible to fill encyclopedia articles or textbooks to answer every misconception or counterargument of a student. There is a freely editable reference desk for readers who need help. Other discussions, moreover, have descended into philosophical discussions full of original research where the central issue is infinity. Clearly this odd forum has been a waste of time.
:::* Keeping bad proposals and arguments away from the main talk page. Objection: Such a talk page does not exist for most other mathematics- or science-related articles. There is not a huge "Arguments" talk page for calculus deniers or non-Euclidean geometry deniers or set theory deniers, sadly for regulars who wish to flaunt their superior intellect. (I have seen at least one comment about debating math deniers being an intellectual sort of entertainment.) Wikipedians are too careless in spending hours debating teenagers who type up "proofs" and ¡¡profound!! philosophy in a minute before clicking "Publish changes". Overall, content proposals without citations of reliable sources are unlikely to succeed on Wikipedia, yet we entertain hundreds of talk page sections without formal, valid proofs.
:::I propose stricter criteria for keeping talk page sections per [[WP:NOTFORUM|NOTFORUM]]. Objections based on philosophy are automatically deleted unless the poster can cite a reliable source. Discussions that evolve into philosophical ones are automatically closed. Any argument that repeats an old one is deleted. Any poster who does not write a proof formally is asked to write one, and if he chooses not to, his section will be closed. With all of the bad discussions closed down, there is no more need for a separate talk page. [[User:OneToZero|OneToZero]] ([[User talk:OneToZero|talk]]) 16:25, 3 November 2021 (UTC)
:::: {{u|OneToZero}}, you can start another MFD if you like, if you think anything has changed.  Otherwise please just let it lie.  Other editors' time is a resource that belongs to them exclusively, not to you even slightly; you don't get to count it in any way whatsoever as an argument about the existence of the page.  --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 18:28, 3 November 2021 (UTC)
::::: A deletion discussion would be in the wrong order. The problems that result in this page must be addressed first, and then this can finally be closed down. It would be worse than the status quo if this page were only deleted and all the useless discussions would swamp the talk page. MfD is not the place to change the policy of a talk page. [[User:OneToZero|OneToZero]] ([[User talk:OneToZero|talk]]) 00:18, 4 November 2021 (UTC)
:::::: So far I see no need to change anything.  The existing framework works just fine.  Moving new posts to the arguments page is easy and most of the time non-combative.  Then whoever doesn't want to see them is under no obligation to, and the main talk page is usable for its intended purpose. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 02:37, 4 November 2021 (UTC)

== "Well established" claim is a blatant lie.  ==

1. The "Rigorous Proof" section contains no citation. Not one. If this is so "well establish" where are your citations?

2. Without even one credible citation of the rigorous proof, this page is nothing more than propaganda. 

3. It's maintainers blatantly censor Anon contributions, and do not admit a controversy exists, in spite of the very existence of THIS "Arguments" page and widespread strong evidence that every so called rigorous proof is in reality yet another circular argument. 

4. This is also the case for the proof offered in the "Rigorous Proof" section. It is also clearly circular, but who dares to try contradict this overtly false reasoning parading in plain sight.

5. The worst most abusive feature of this "0.999...=0" page is that any attempt to put up a similar page debunking the purported "Rigorous Proof" on this page (eg. "0.999... < 1") would last about 5 minutes. That censorship is the hallmark of book burners. [[Special:Contributions/2003:EB:A714:EC00:1517:3B2E:6E6F:B6C4|2003:EB:A714:EC00:1517:3B2E:6E6F:B6C4]] ([[User talk:2003:EB:A714:EC00:1517:3B2E:6E6F:B6C4|talk]]) 20:58, 15 July 2022 (UTC)

== .999... can't always equal 1 ==

If you graph an exponential decay equation where the asymptote is y=1, but you get a y value that equals .999... that can't be incorrect. [[Special:Contributions/96.237.229.98|96.237.229.98]] ([[User talk:96.237.229.98|talk]]) 23:04, 4 October 2022 (UTC)
:1 and .999... are different ways to write down the same number. If the asymptote of a function is 1, then the asymptote is also .999... . If the function is equal to .999... at some point, then it is equal to 1 at that point. [[User:MartinPoulter|MartinPoulter]] ([[User talk:MartinPoulter|talk]]) 20:12, 5 October 2022 (UTC)
::Not true. The symbol "1" represents unity. Any other symbol or notation, by it's very nature (not being "1"), represents something else (irrespective of how clever the explaination, it's very existence undermines it's purported purpose). I've been told elsewhere that Wikipedia is a "serious" project only to come across silly articles like this. [[User:Iluvlawyering|Iluvlawyering]] ([[User talk:Iluvlawyering|talk]]) 06:28, 25 April 2023 (UTC)
:::If you can find a number, other than 1, to which the series ( 0.9, 0.99, 0.999, 0.9999, 0.99999, etc... ) gets, so to speak, "closer and closer without ever reaching it", then you have a point. The symbol "0.999..." is shorthand for "''the smallest number to which that series gets closer, whithout ever needing to reach it''", and that number is 1. There is nothing to discuss about that. It's in the article, in the last part of section [[0.999...#Infinite series and sequences]], the only relevant section in the article. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 08:26, 25 April 2023 (UTC)
:::"Any other symbol or notation, by it's very nature (not being "1"), represents something else" A truly amazing statement, and hard to treat as serious. There are many other systems of symbols for representing numbers. Look into [[Eastern Arabic numerals]], [[Suzhou numerals]], and [[List of numeral systems|the rest]]. [[User:MartinPoulter|MartinPoulter]] ([[User talk:MartinPoulter|talk]]) 10:13, 25 April 2023 (UTC)
::::No what was said is perfectly valid. You do not ask a man for "0.999... burger" you ask him for 1. The two symbols here represent the same *quantity* but by tbe mere fact of their being different expressions means they are not the same in totality. Same in quantity, different by number (same *in* number). [[Special:Contributions/2600:6C4A:4C7F:D426:B9D1:9261:4B80:6C99|2600:6C4A:4C7F:D426:B9D1:9261:4B80:6C99]] ([[User talk:2600:6C4A:4C7F:D426:B9D1:9261:4B80:6C99|talk]]) 13:14, 10 September 2023 (UTC)
:::::Your confusion is the standard one between '''numbers''' and '''numerals''': "0.999..." and "1" are different numerals, which represent the same number. Just as 壱 does, for example. [[User:Imaginatorium|Imaginatorium]] ([[User talk:Imaginatorium|talk]]) 15:17, 10 September 2023 (UTC)

[[Category:Wikipedia mathematical arguments]]