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Icarus
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The original 0.999. thread has grown so large that it is difficult for those without broadband to reply to it. For this reason, I have locked that thread and created this one to continue the discussion. In this post, I will start off by addressing, for reference, three issues related to this topic. The second and third posts are summaries of the original thread that I had created, and reprise here.
What are Real numbers?
One of the problems that people have with the question of the meaning of 0.999... is a misunderstanding of what real numbers themselves are. In particular, many mistake the decimal expressions we write out as being the numbers themselves. For them, to say that 0.999... = 1 is patently ridiculous, since they quite evidently look nothing alike. But in fact, decimal expressions are not the numbers themselves, but are names for those numbers. They are labels which we have attached to underlying ideas to allow us to easily discuss them. 0.999... and 1 are two different names we use for the same concept.
There are many ways of understanding the real numbers conceptually. Probably the most familiar is that the real numbers represent all possible directed distances between points ("directed" means we have both positive and negative distances).
Mathematically, the Real numbers are defined to be "the smallest topologically complete ordered field". To understand this phrase, we must start at the right and work left:
A Field is a triple ([smiley=cf.gif], +, [cdot]), where [smiley=cf.gif] is a set, and + and [cdot] are binary operations on [smiley=cf.gif] which satisfy the following axioms:
Commutivity: For all [smiley=x.gif], [smiley=y.gif] [in] [smiley=cf.gif]; [smiley=x.gif] + [smiley=y.gif] = [smiley=y.gif] + [smiley=x.gif] and [smiley=x.gif][cdot][smiley=y.gif] = [smiley=y.gif][cdot][smiley=x.gif].
Associativity: For all [smiley=x.gif], [smiley=y.gif], [smiley=z.gif] [in] [smiley=cf.gif]; ([smiley=x.gif] + [smiley=y.gif]) + [smiley=z.gif] = [smiley=x.gif] + ([smiley=y.gif] + [smiley=z.gif]) and ([smiley=x.gif][cdot][smiley=y.gif])[cdot][smiley=z.gif] = [smiley=x.gif][cdot]([smiley=y.gif][cdot][smiley=z.gif]).
Identity: There exists 0, 1 [in] [smiley=cf.gif] such that for all [smiley=x.gif] [in] [smiley=cf.gif], 0 + [smiley=x.gif] = [smiley=x.gif] and 1[cdot][smiley=x.gif] = [smiley=x.gif].
Inverse: For each [smiley=x.gif] [in] [smiley=cf.gif], there is a [smiley=y.gif] [in] [smiley=cf.gif] such that [smiley=x.gif] + [smiley=y.gif] = 0, and if [smiley=x.gif] [ne] 0, then there is a [smiley=z.gif] [in] [smiley=cf.gif] such that [smiley=x.gif][cdot][smiley=z.gif] = 1.
Distributivity For all [smiley=x.gif], [smiley=y.gif], [smiley=z.gif] [in] [smiley=cf.gif]; ([smiley=x.gif] + [smiley=y.gif])[cdot][smiley=z.gif] = ([smiley=x.gif][cdot][smiley=z.gif]) + ([smiley=y.gif][cdot][smiley=z.gif]).
Fields are usually denoted by their set, suppressing explicit mention of the two operations.
An Ordered Field is a pair ([smiley=cf.gif], [le]), where [smiley=cf.gif] is a field, and [le] is a binary relation satisfying the following axioms:
Reflexivity: For all [smiley=x.gif] [in] [smiley=cf.gif], [smiley=x.gif] [le] [smiley=x.gif].
Antisymmetry: For all [smiley=x.gif], [smiley=y.gif] [in] [smiley=cf.gif], if [smiley=x.gif] [le] [smiley=y.gif] and [smiley=y.gif] [le] [smiley=x.gif], then [smiley=x.gif] = [smiley=y.gif].
Transitivity: For all [smiley=x.gif], [smiley=y.gif], [smiley=z.gif] [in] [smiley=cf.gif], if [smiley=x.gif] [le] [smiley=y.gif] and [smiley=y.gif] [le] [smiley=z.gif], then [smiley=x.gif] [le] [smiley=z.gif].
Additivity: For all [smiley=x.gif], [smiley=y.gif], [smiley=z.gif] [in] [smiley=cf.gif], if [smiley=x.gif] [le] [smiley=y.gif], then [smiley=x.gif] + [smiley=z.gif] [le] [smiley=y.gif] + [smiley=z.gif].
Multiplicativity: For all [smiley=x.gif], [smiley=y.gif], [smiley=z.gif] [in] [smiley=cf.gif], if [smiley=x.gif] [le] [smiley=y.gif] and 0 [le] [smiley=z.gif] and 0 [ne] [smiley=z.gif], then [smiley=x.gif][cdot][smiley=z.gif] [le] [smiley=y.gif][cdot][smiley=z.gif].
Again, explicit mention of the ordering is suppressed.
An ordered field [smiley=cf.gif] is Topologically Complete if it satisfies the supremum property. Before giving it, a few definitions are needed:
A set [smiley=ca.gif] [subseteq] [smiley=cf.gif] is bounded above if there is a [smiley=b.gif] [in] [smiley=cf.gif] such that for all [smiley=x.gif] [in] [smiley=ca.gif], [smiley=x.gif] [le] [smiley=b.gif]. In this case, [smiley=b.gif] is called an upper bound for [smiley=ca.gif].
A supremum, or least upper bound, of [smiley=ca.gif] is an upper bound [smiley=s.gif] of [smiley=ca.gif] such that for all upper bounds [smiley=b.gif] of [smiley=ca.gif], [smiley=s.gif] [le] [smiley=b.gif].
[smiley=cf.gif] satisfies the supremum property if every subset of [smiley=cf.gif] which is bounded above has a supremum.
(This is where the real numbers differ from the rationals. Every property mentioned before this one is also satified by the set of all rational numbers. But the rational numbers are not topologically complete. For example, the set { p/q | p, q [in] [bbz], p2 < 2q2 } is bounded, but has no rational supremum.)
Finally, we come to the word "smallest". It can be shown that the intersection of topologically complete ordered fields is also a topologically complete ordered field. (Sort of - I'm glossing over a few things with that statement.) The intersection of all topologically complete ordered fields is the smallest such field, which we call the Real numbers.
Some common set notations:
Real numbers: [bbr]
Natural numbers: [bbn] = {1, 1+1, 1+1+1, ...} (the set of all real numbers obtainable by starting with 1 and repeatedly adding 1).
Whole numbers: [smiley=bbw.gif] = [bbn] [cup] {0}
Integers: [bbz] = { x | [pm]x [in] [bbn] or x = 0 }
Rational numbers: [bbq] = { p/q | p, q [in] [bbz] and q [ne] 0 }
Complex numbers: [bbc] = { a + bi | a, b [in] [bbr] and i2 = -1 }
The definition of a limit.
Many confusions about 0.999... centered on a failure to understand the idea of a limit. More is said about this in the Misconceptions post below. For reference, I give Cauchy's definition of the limit of a sequence.
A sequence is a function [smiley=a.gif] from the natural numbers [bbn] into [bbr]. The value of [smiley=a.gif] for a number [smiley=n.gif] is usually denoted [smiley=a.gif][subn] and the sequence by {[smiley=a.gif][subn]}.
The limit of a sequence {[smiley=a.gif][subn]} is a number [smiley=cl.gif] such that for any [epsilon] > 0, there is a natural number [smiley=cn.gif] such that if [smiley=n.gif] > [smiley=cn.gif], then
| [smiley=a.gif][subn] - [smiley=cl.gif] | < [epsilon].
The limit [smiley=cl.gif] is usually denoted by "lim [smiley=a.gif][subn]" or "lim[subn] [smiley=a.gif][subn]" or "lim[subn][to][subinfty] [smiley=a.gif][subn]".
Decimal Notation
Decimal notation is a means of identifying particular real numbers. In brief, it is defined as follows:
Define 2=1+1; 3=1+2; 4=1+3; 5=1+4; 6=1+5; 7=1+6; 8=1+7; 9=1+8.
A decimal expression is a function [smiley=d.gif] from [bbz] into the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} with the property that for some [smiley=cn.gif] [in] [bbz], for every [smiley=n.gif] [ge] [smiley=cn.gif], [smiley=d.gif][subn] = 0. The decimal expression is denoted by adjoining the digits:
[smiley=d.gif][smiley=subcn.gif][smiley=d.gif][smiley=subcn.gif]-1 ... [smiley=d.gif]0.[smiley=d.gif]-1[smiley=d.gif]-2 ...
Where [smiley=cn.gif] is the larger of 0 or the largest index of [smiley=d.gif] for which [smiley=d.gif] > 0.
Each decimal expression is assigned a real number as its value: For each [smiley=i.gif] [in] [bbn], let [smiley=cd.gif][subi] = [smiley=d.gif][smiley=subcn.gif][cdot]10[smiley=subcn.gif] + [smiley=d.gif][smiley=subcn.gif]-1[cdot]10[smiley=subcn.gif]-1 + ... + [smiley=d.gif]0[cdot]100 + [smiley=d.gif]-1[cdot]10-1 + ... + [smiley=d.gif]-[subi][cdot]10-[subi].
Then the value assigned to the decimal expression is lim[subi] [smiley=cd.gif][subi].
Note that by definition, every decimal expression represents a single number, but that there is nothing in this definition to say that two decimal expressions cannot have the same value.
In the Misconceptions post below is a proof that every real number is the value of some decimal expression.
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| « Last Edit: Oct 18th, 2004, 3:00pm by Icarus » |
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Re: 0.999...
« Reply #1 on: May 1st, 2004, 7:19pm » |
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0.999... = 1 Proofs found in the original thread
Arguments depending on the reader knowing how to add/multiply decimals
1) Times 10
Let [smiley=x.gif] = 0.999...
10[smiley=x.gif] = 9.999...
- [smiley=x.gif] = -0.999...
9[smiley=x.gif] = 9.000... = 9
[smiley=x.gif] = 9/9 = 1.
2) Thirds
1/3 = 0.333...
1/3 = 0.333...
+ 1/3 = 0.333...
3/3 = 0.999...
1 = 0.999...
3) Take the Difference
1.000...
-0.999...
0.000... = 0
Since their difference is 0, they are the same.
4) 9*0.111...
1/9 = 0.111...
1 = 9/9 = 9*(1/9) = 9*(0.111...) = 0.999...
(The linked text has the flaw of depending on "calculator expressions", however this can be cleared up into a true proof.)
Arguments based on calculus
5) 0.999... = 9/10 + 9/100 + 9/1000 + ...
By definition 0.999... = [sum][subn] 9*10[supminus][supn] (1 [le] [smiley=n.gif] [smiley=lt.gif] [infty] )
That is,
0.999... = Lim[smiley=subcn.gif][to][subinfty] [sum][subn]=1[smiley=supcn.gif] 9*10[supminus][supn]
= Lim[smiley=subcn.gif][to][subinfty] 9(( 1-10[supminus][smiley=supcn.gif][supminus][sup1])/(1-10[supminus][sup1]) )-1)
= Lim[smiley=subcn.gif][to][subinfty] 10 - 10[supminus][smiley=supcn.gif] - 9
= 10 - 0 - 9
= 1
While some posters discussed this before the linked post, this post was the first to actually lay out the argument clearly (except when the poster got confused and said that the definition he had just given could then be proved using Analysis. Since it was a definition, it is not subject to proof.)
6) Epsilon-delta
0.999... = lim[subn][to][subinfty] 0.999...9 ([smiley=n.gif] 9s) = lim[subn][to][subinfty] 1 - 10[supminus][supn]
The definition of this limit is: The limit is the number [smiley=cl.gif] such that for every [epsilon]>0, there is an [smiley=cn.gif] such that for all [smiley=n.gif] > [smiley=cn.gif],
| (1-10[supminus][supn]) - [smiley=cl.gif] | < [epsilon].
(That [smiley=cl.gif] is unique is easily proven from the definition.)
So let [epsilon] be an arbitrary number > 0. Choose [smiley=cn.gif] [ge] -log10 [epsilon]. Then for [smiley=n.gif] > [smiley=cn.gif], | (1-10[supminus][supn]) - 1 | = 10[supminus][supn] < 10[supminus][smiley=supcn.gif] < [epsilon].
Hence 0.999... = 1.
Pietro K.C. has a similar proof earlier to the one linked. But it was so wrapped up in refutations of other arguments that I choose to link to James Fingas' instead.
7) Method Of Exhaustion
Since every finite stretch of 9s, 0.999...9, is < 1 and these get arbitrarily close to 0.999..., we must have 0.999... [le] 1. But let x < 1, then 1-x > 0 and by the Archimidean principle, there must be an n such that 10-n < 1-x. Therefore x < 1 - 10-n = 0.999...9 (n 9s) < 0.999... . So 0.999... > x for all x < 1. The only possibility left is: 0.999... = 1.
This isn't a calculus-based proof. It is actually a "what calculus is, but from before calculus was invented" proof.
8 ) 1=0.999...910
For all [smiley=n.gif], 1 = 0.99...([smiley=n.gif] 9s)...91, where the notation means that after all the 9s we have a "10" for the last digit (ie, we are extending base 10 notation to include a "10" digit). Letting [smiley=n.gif] [to] [infty] leaves 1= 0.999...
This works, but unfortunately it requires even more of an understanding of infiniteness than the other methods.
Other proofs
9) Nothing Between
Suppose 0.999... [ne] 1. Let [smiley=x.gif] = (1+0.999...)/2. What number is it? By our understanding of decimals it must have a decimal expansion with digits >9. Since there are no such digits in base 10, there cannot be such an [smiley=x.gif]. And so 0.999... = 1.
My thoughts: Kozo's replies to this argument were very salient, even though they were based on non-existant numbers. This argument depends heavily on a deep understanding of the nature of decimal notation that those who expounded it never explained.
In order to truly demonstrate that 1 = 0.999... using this idea, you must also prove that every real number has a decimal expansion, and prove that 0.999... < [smiley=x.gif] < 1 requires the digits of [smiley=x.gif] to be greater than those of 0.999...
To do this properly is a royal mess.
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| « Last Edit: May 8th, 2004, 9:52am by Icarus » |
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Icarus
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Re: 0.999...
« Reply #2 on: May 1st, 2004, 7:20pm » |
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Misconceptions about Decimals and Infinity found in the original thread
1) Infinite Decimals are Approximations
This argument says that, for example, 1/3 [ne] 0.333... because the right-hand side is only an approximation.
Not so. The definition of the expression 0.333... is
0.333... = Lim[subn][smiley=subto.gif][subinfty] [sum][smiley=subk.gif][smiley=subeq.gif][sub1][supn] 3*10[supminus][smiley=supk.gif] = Lim[subn][smiley=subto.gif][subinfty] (0.3 + 0.03 + ... + 3*10[supminus][supn])
The value of the limit is exactly 1/3.
This argument is closely tied to the next two:
2) Limits are Approximations
There are several posts with this misconception. The one linked has both it, and a variant of the "infinite process" misconception, which is described below.
According to this view, limits are nothing more than a "short-hand" for describing approximation schemes. I believe this idea comes from the descriptions used by math teachers to first introduce the idea of limits. Unfortunately, the student never moved beyond these original incomplete conceptions. The basic definition of the limit of a sequence (the particular type of limit needed here) is:
The limit of {[smiley=a.gif][subn]} as [smiley=n.gif] goes to infinity, written as "lim[subn][smiley=subto.gif][subinfty] [smiley=a.gif][subn]", is the real number [smiley=cl.gif] which satisfies the following:
For every [epsilon] [smiley=gt.gif] 0, there is an [smiley=cn.gif] [in] [bbn] such that for all [smiley=n.gif] [smiley=gt.gif] [smiley=cn.gif], [smiley=vert.gif] [smiley=a.gif][subn] [smiley=minus.gif] [smiley=cl.gif] [smiley=vert.gif] [smiley=lt.gif] [epsilon].
Note that by the definition, the limit is not any of the [smiley=a.gif][subn] or all of them, or some "process". The limit is the number [smiley=cl.gif] which the sequence elements [smiley=a.gif][subn] approximate.
3) Decimals and Limits are Processes
The linked post is the first I could find that treats 0.999... as being a process of constantly adding more 9s. In the linked post for (2), limits themselves are also described as processes.
From the definitions for decimals and limits provided, it is evident that decimal expressions such as 0.999... and that limits are defined to be particular numbers, not processes.
4) Infinite decimals and limits are the result of an infinite process
This misconception is very common, and is evident in many posts on both sides. The truth is rather difficult to explain.
Mathematics, and particularly these two situations, contains a concept of infinite processes, but not the actuality of them. This is why we have such notational conventions as the ellipsis (...) and the overline for repeating decimals that is not available in this forum. Conceptually, the ellipsis means that the digits go on forever. In reality, they can't, and in any actual mathematics, they don't - they always end with an ellipsis or similar notational convention.
Since mathematics has no means to actually carry out any infinite processes, we use other means to define and/or calculate the outcomes that such processes would have if they actually could be performed.
The most basic idea we use for this is: if for all potential outcomes of an "infinite process" but one, you can show (by finite means) that the outcome cannot be the result of the process, then the one remaining potential outcome is the result of the process.
This is the basis for the definition of limit (which is due to Cauchy, by the way) that I gave earlier.
In the concept of limits, they involve an infinite process of improving approximation - with the idea that if only you could continue the approximation on indefinitely, the result would be the actual value rather than any approximation.
The actuality of limits is that we show for any particular limit that there is exactly one number [smiley=cl.gif] that can be the result of infinite continuation of the approximating process. We define the limit to be that number.
The same thing goes for the manipulation of decimals. Conceptually, 0.333... + 0.333... requires an infinite number of additions of the form 3+3 = 6. In actuality, it is evident that all the additions are 3+3=6, so the sum must be a number whose decimal expansion is 0.666.... Using the definitions of decimals and limits above (along with certain properties of the real numbers), we can also show that there is exactly one real number with this decimal expansion. All of this is done in a finite number of steps in the actual mathematics. It is only in concept that it involves an infinite number of steps.
5) Infinity is only a concept, not an actual number
This misconception - which is almost universal among those who understand the previous point - takes the previous point too far: the belief is that infinity has no place in actual mathematics.
This is not true. Infinite processes have no place in mathematics other than as a concept. Infinities themselves are well within the domain of mathematical exploration. They are nothing more than a matter of definition. In this post , I gave a short overview of the three types of infinities most commonly encountered in mathematics.
6) Infinity is an actual number
(The linked post does not state infinity is a number like some other posts do, but it the first to attempt treating infinity as a number without any understanding of what it requires.)
Why am I including the opposite side of the argument above as another misconception? Because those who argued this side did not understand what they were saying either. The problem here is the word "number".
Oddly enough, the word "number" has NO MATHEMATICAL MEANING. In mathematics, we have "Natural numbers", "Cardinal numbers", "Ordinal numbers", "Rational numbers", "Real numbers", "Algebraic numbers", "Complex Numbers", "Hamiltonian numbers (quaternions)", "Cayley numbers", "Hyperreal numbers", "p-adic numbers", "Surreal numbers", and a vast host of other types of numbers someone has defined here, there, or yonder. But we do not have a meaning for "number" in and of itself.
Many who have argued in this thread that "infinity is a number" both had no workable concept of what they meant by "infinity", and generally tried to treat infinity as if it were a REAL number (ie. part of the set of Real numbers). There is no infinity in the real numbers. To get infinities, you must expand the real numbers (to the Extended Reals, the Hyperreals, the Surreals, the Long Line, or by other means).
7) There are numbers without decimal representations
I don't know if Kozo actually thought so when he made the post, or was only trying to point out the weakness of the previous "Pro" argument (and it does have serious weaknesses, though it is salvagable), but just for clarity's sake:
Every real number has at least one decimal expression.
Proof: There is for any real number [smiley=x.gif] an integer [smiley=cn.gif] such that 10[smiley=supcn.gif][supplus][sup1] > [smiley=x.gif] [ge] 10[smiley=supcn.gif]. For every integer [smiley=n.gif] [le] [smiley=cn.gif] we can define the two numbers [smiley=a.gif][subn] and [smiley=d.gif][subn] inductively by:
[smiley=a.gif][smiley=subcn.gif][subplus][sub1] = [smiley=x.gif].
For all integers [smiley=n.gif] [le] [smiley=cn.gif],
[smiley=d.gif][subn] = [lfloor][smiley=a.gif][subn][subplus][sub1]/10[supn][rfloor] ( [lfloor] [rfloor] is the floor function - the greatest integer [le] the contents of the brackets)
[smiley=a.gif][subn] = [smiley=a.gif][subn][subplus][sub1] - [smiley=d.gif][subn]*10[supn]
It is not hard to show that [smiley=x.gif] = [sum][subn][smiley=subeq.gif][subminus][subinfty][smiley=supcn.gif] [smiley=d.gif][subn]*10[supn], so [smiley=d.gif][smiley=subcn.gif] ... [smiley=d.gif][sub0].[smiley=d.gif]-[sub1] ... provides a decimal representation for [smiley=x.gif].
8 ) There is a least number greater than or greatest number less than a given real number
If [smiley=x.gif] and [smiley=y.gif] are any real numbers, then ([smiley=x.gif] + [smiley=y.gif])/2 is one of infinitely many real numbers lying strictly between [smiley=x.gif] and [smiley=y.gif]. Thus there is no such thing as the least real number greater than, or greatest real number less than, [smiley=x.gif].
9) Real numbers consist of decimal representations
While no-one has expressed this view explicitly, it is implicit in many of the arguments made (and not just by Kozo, or just by "Con" posters). The crux of these arguments is to introduce a new variation of decimal notation, and then to talk about the new decimal representation as if it were a well-defined real number - without bothering to define what this new variation actually means.
The post linked introduces a new digit, "#", to base-10 decimal notation. Other posts introduce notations with "tranfinite" decimal places (i.e. decimal places with an infinite number of other decimals preceding them). In all of these posts, the poster never bothers to actually tie these notations to any real number. They are simply thrown out, and it is assumed that they somehow represent real numbers. Much of the argument that follows comes from other posters trying to put a definition to the new notation only to be told "no - that's not what it means" (but still without any attempt on the originator's part to provide a meaning).
Real numbers have an existance entirely separate from any means of denoting them. The simplest concise definition of the Reals is "the smallest topologically-complete ordered field". (Of course all of these terms need their own definition before this one is meaningful.) Decimal notation is a defined means of denoting the members of this field. If you introduce new notations for real numbers, then YOU must provide the definitions of the new notations also.
10) There is no such thing as transfinite digits
This misconception is that such expressions as "1.000...01", where the ellipsis represents infinitely many zeros, are not only undefined, but undefinable.
They are undefined, unless the person introducing them also bothers to define them. But they are not senseless. You can define them. For instance, you could define 1.00...01 as representing the real number pair (1, 0.01). Such expressions could represent members of the set [bbr] [times] [0,1), where the digits of the second term are considered to follow every digit of the first (terminating decimals are filled out with 0s).
11) 0.999... is a hyperreal number, not real
This sounds apealling, but only because only us severe math wonks have ever even heard of hyperreals.
The idea is that hyperreal numbers extend the reals, including in "infinitely small" numbers. That's it! we say: 1 - 0.999... is one of these infinitely small numbers, rather than 0!
Sorry, but it ain't so. By definition 0.999... = limit[subn] {0.999...([smiley=n.gif] 9s)}. Each element of the sequence is a real number. By the definition of limits, the limit of a sequence of real numbers is a real number. So 0.999... is real, not hyperreal.
No matter what larger set of numbers we decide to work in, decimal notation is defined for real numbers, and it is in the real numbers alone that this matter is settled.
12) ANYONE WHO DISAGREES WITH ME IS AN IDIOT!
I think it's clear who the real idiot is with these posters! 
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| « Last Edit: May 1st, 2004, 7:26pm by Icarus » |
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Re: 0.999...
« Reply #3 on: May 4th, 2004, 7:28pm » |
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I cant explain anything scientific or complicated like others that are replying to this post, but i will say something that my math teacher told me. If I am standing 1 foot away from a wall and i walk 9/10 of the space between me and the wall, and from there move another 9/10 closer to the wall again and again and again. Theoreticaly, you will never reach the wall. You might be up against the wall, but you will never get there by moving a fraction of the space left.
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towr
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Re: 0.999...
« Reply #4 on: May 5th, 2004, 1:05am » |
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Your teacher must not be an engineer 
And you should always be sceptical at what teachers say. It is true that for any finite number of steps you don't reach the wall, but if you take the limit to infinity then it no longer is a finite number of steps.
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Wikipedia, Google, Mathworld, Integer sequence DB
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Icarus
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Re: 0.999...
« Reply #5 on: May 5th, 2004, 3:30pm » |
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Exactly. All of your positions at which you are stopping correspond to .9 or .99 or .999 or .999...9 for some finite number of nines. No one is claiming that any of these equal 1.
What is equal to 1 is 0.999..., with an infinite number of 9s. This does not correspond to any of the steps in your approach to the wall, but is beyond all of them. Think about it: if you mark any spot short of the wall, your steps will eventually get past it. So, since 0.999... is beyond all of your steps, it also must be beyond all possible marks short of the wall. The only place it can be at is at the wall itself.
0.999... has to be greater than every number less than 1. Since it is also [le] 1, that leaves no value but 1 for it to equal.
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EZ_Lonny
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Re: 0.999...
« Reply #7 on: Aug 11th, 2004, 5:37am » |
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Dear Icarus,
On your explanation about Ordered Fields (F,=<), I think you made a small mistake that could lead to complete nonsense in the rest of your statement.
You say:
For all x,y,z in F; if x=< y and 0<>z, then x.z =< y.z
Did you think of the possibility that x is negative and z is negative and y is positive? So x.z is positive and y.z is negative. Excuse me if I'm wrong but then x.z > y.z.
I havent checked on other flaws in your statement, bus such easy to mistakes may imply more to be made in apparently harder assumptions.
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towr
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Re: 0.999...
« Reply #8 on: Aug 11th, 2004, 9:22am » |
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on Aug 11th, 2004, 5:37am, EZ_Lonny wrote:
You say:
For all x,y,z in F; if x=< y and 0<>z, then x.z =< y.z
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He also included 0 [le] z
on May 1st, 2004, 7:19pm, Icarus wrote:Multiplicativity: For all [smiley=x.gif], [smiley=y.gif], [smiley=z.gif] [in] [smiley=cf.gif], if [smiley=x.gif] [le] [smiley=y.gif] and 0 [le] [smiley=z.gif] and 0 [ne] [smiley=z.gif], then [smiley=x.gif][cdot][smiley=z.gif] [le] [smiley=y.gif][cdot][smiley=z.gif].
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| « Last Edit: Aug 11th, 2004, 9:24am by towr » |
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Icarus
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Re: 0.999...
« Reply #9 on: Aug 11th, 2004, 6:50pm » |
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Nothing I gave in that post is original to me, nor is any of it an assumption - It is a definition. I.e. I am not making assumptions about what the real numbers are, I am telling you what the meaning of "real number" is. This is where they come from.
The particular problem you find comes about, as towr has pointed out, because you have inadvertently dropped part of the defining statement. As it is, this is a well-known property of inequalities: when you multiply both sides of an inequality by the same positive number, the inequality is preserved. That is all this statement says.
As an aside: why did I say (0 [le] z and 0 [ne] z), instead of simply saying (0 < z)? Because I had not defined the symbol "<" yet, and in a development such as this, you don't use anything until it has been defined.
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EZ_Lonny
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Re: 0.999...
« Reply #10 on: Aug 12th, 2004, 8:46am » |
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My mistake
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StatusX
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To those who say you'll NEVER reach one (or never reach the wall): isn't saying "never" just another way of saying "it will take an infinitely long time"?
Of course, you have to combine this with the fact that the terms in the series, (you) get closer and closer to 1 (the wall), otherwise the limit could be anything (anywhere), but it is combining these two concepts that lets you understand an infinite limit.
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sam_i_am
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Re: 0.999...
« Reply #12 on: Sep 25th, 2004, 8:36am » |
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In the movie dumb & dumber, jim carrey asks the girl in the bar about the chances he has to go on a date with her. she replies ' one in a million'. carrey counters, 'atleast i have a chance' and in the end he gets the girl. looking logically at this situation we can conclude that 0.9999..<1 otherwise 1-0.9999..=0 which would definitlely contradict the ending to the movie. 
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rmsgrey
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Re: 0.999...
« Reply #13 on: Sep 25th, 2004, 9:57am » |
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on Sep 25th, 2004, 8:36am, sam_i_am wrote:| In the movie dumb & dumber, jim carrey asks the girl in the bar about the chances he has to go on a date with her. she replies ' one in a million'. carrey counters, 'atleast i have a chance' and in the end he gets the girl. looking logically at this situation we can conclude that 0.9999..<1 otherwise 1-0.9999..=0 which would definitlely contradict the ending to the movie. |
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The movie has nothing to do with it. One in a million (1-0.999999=0.000001) is definitely larger than 0, but 0.999999 is definitely smaller than 0.999...
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llamario
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Wow, that "Dumb and Dumber" analogy could be the worst argument I ever heard. It's just wrong on so many levels.
1) Carrey's reply was "So you're telling me I've got a chance!"
2) Carrey did not get the girl in the end, she was re-united with her husband. (He didn't even get the Hawiian Tropic girls...)
3) 1 minus .999... does equal zero, because there are no real numbers in between them. (See first post in this thread)
4) You are (erroneously) basing your answer on the philosphy of a movie character whose title clearly implies that he is dumb at best, and possibly even dumber than that.
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Icarus
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Re: 0.999...
« Reply #15 on: Oct 18th, 2004, 2:55pm » |
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llamario, have considered the possibility that sam_i_am was, just possibly, less than completely serious?
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Sputnik
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Sorry to burst your bubble, folks, but if (infinity+1) has no meaning then (0.999... * 10) also has no meaning.
Infinitely recurring values cannot be the subject of normal mathematical procedures any more than infinity itself can.
0.999... does not equal 1
It equals 0.999... and no other value.
It is functionally equivalent but NOT equal to 1.
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Icarus
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Re: 0.999...
« Reply #17 on: Oct 21st, 2004, 6:41pm » |
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on Oct 21st, 2004, 2:29pm, Sputnik wrote:Sorry to burst your bubble, folks, but if (infinity+1) has no meaning then (0.999... * 10) also has no meaning.
Infinitely recurring values cannot be the subject of normal mathematical procedures any more than infinity itself can.
0.999... does not equal 1
It equals 0.999... and no other value.
It is functionally equivalent but NOT equal to 1. |
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- Infinity + 1 has meaning. What that meaning is depends on which infinity you are talking about.
- 0.999... * 10 also has meaning - a meaning entirely independent of any definition of "infinity" or "infinity + 1".
- 0.999... and 1 are not "functionally equivalent" (what the heck do you mean by that, anyway?). They are equal. They are two names for the same real number.
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| « Last Edit: Oct 21st, 2004, 6:43pm by Icarus » |
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Emilie
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"
3) Take the Difference
1.000...
-0.999...
0.000... = 0
Since their difference is 0, they are the same.
"
--------------------------
I thought that
1.000...
-0.999...
0.000... = 0.000...1 <> 0.000...0, only approximately
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Sir Col
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Re: 0.999...
« Reply #19 on: Nov 5th, 2004, 4:49pm » |
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1-0.9=0.1
1-0.99=0.01
1-0.999=0.001
If the string of 9's in 0.999... terminates, then the difference, 0.000..., will also terminate with 1. However, as 0.999... is an infinitely recurring decimal, the difference, 0.000..., is also an infinitely recurring decimal, so there is no 1 "at the end".
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Icarus
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Re: 0.999...
« Reply #20 on: Nov 9th, 2004, 5:30pm » |
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This is an example of why I put this one under the heading "Arguments depending on the reader knowing how to add/multiply decimals". (That is not intended as a put-down. The point here is a bit intricate, and there is no shame in never having come across it before.) Before you can trust the argument, you have to know what it is actually saying.
Note that in the notations above the 3 dots represents the digits repeating to infinity. As such, there is no such decimal notation as "0.000...1", for it would require an infinite number of 0s to occur before the 1. If you check the definition of decimal notation given above, you will see that it does not include such transinfinite decimals.
(It is possible to define such notations, but when you then attempt to attach these new expressions to real numbers, the only value that makes sense for 0.000...1 is 0.)
When you manipulate decimals, once you've pushed the calculation "to infinity", there is no more that comes after. If you are not already aware of this, or do not understand it, then this argument is not for you. To make it more exact, I would have to bring in calculus concepts, and simpler versions of calculus proofs are already given.
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Re: 0.999...
« Reply #21 on: Nov 10th, 2004, 5:19am » |
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OK, this may have already been stated, but I figured that it probably counts as a reasonable, not-very-mathematical way of looking at this.
If 0.999... is the same as 1, then it presumably means that there can be no difference of 0.000...1. So, if you subtract 0.000...1 from 0.999... (resulting in 0.999...8 ), then you still have 1. This process could then be repeated, eventually resulting in 0=1
Presumably people do not want this to be the case. Hence I would argue that the difference of 0.000...1 between 0.999 and 1 is important in that respect, and that 0.999 is less than 1
But then, someone has probably already pointed this out, and the discussion suggested that they are some ignorant person who doesn't fully understand the intricacies of infinity, etc. But, hey, I never claimed to be a mathemawhatsit...
[e] damn smileys... [/e]
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| « Last Edit: Nov 11th, 2004, 7:10am by Three Hands » |
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Grimbal
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Re: 0.999...
« Reply #22 on: Nov 10th, 2004, 5:53am » |
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The whole problem comes from the intuitive (but naive) view that a number is defined by its digits. That it is a set of digits. It works for the integers, after all.
1.000... is not 0.999... because 1 is not 0 and 0 is not 9.
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rmsgrey
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Re: 0.999...
« Reply #23 on: Nov 10th, 2004, 6:02am » |
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So what exactly is 0.999...0 - 0.000...1 ?
And is 0.999... the same as 0.999...9 ?
And what is (0.999... + 1)/2 ?
From the definition of Real numbers I was taught (the minimal closure under Cauchy limits of the Rationals) you get that two numbers are the same if, for any finite (rational) value greater than zero, the difference between them is smaller than that.
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Re: 0.999...
« Reply #24 on: Nov 10th, 2004, 6:37am » |
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on Nov 10th, 2004, 6:02am, rmsgrey wrote:So what exactly is 0.999...0 - 0.000...1 ?
And is 0.999... the same as 0.999...9 ? |
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And what exactly are 0.999...0, 0.999...9, 0.000...1 ?
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