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Kozo Morimoto
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Re: 0.999.  
« Reply #125 on: Nov 24th, 2002, 10:09pm »

After the first few posts and reading so many responses which were against me, I realized that I was barking up the wrong tree.  After extensive googling, all the literature that I could find indicated that I was wrong.  So I decided to play the Devil's Advocate and try to provoke the fellow forum participants into think a bit more.
 
The responses I got were basically canned responses and it didn't look as though most of them actually thought about what was being discussed.  Answers like 'plug these numbers in calculators etc' were fallacious arguments and I wasn't satisfied with the provided 'proofs'.
 
There were the few exceptions and I would like to thank them for persevering with me in answering some of questions.
 
The conclusions I came to from this discussion were:
a) this is an artifact of our decimal numbering system and can not be avoided, and it happens with other base n numbering systems as well.
b) that starting with 0.9 and keep adding 9s on the end does not ever become 0.999...  0.999... is a totally different beast and need to be treated accordingly.
c) that my understanding of infinity far far from complete.  Obviously my engineering degree didn't cover enough of the esoterics of pure mathematics and concentrated too much on calculus.  I definitely need to do more personal research into infinity
d) infinity is like a different country - they have different rules over there
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Re: 0.999.  
« Reply #126 on: Nov 25th, 2002, 8:37am »

on Nov 23rd, 2002, 1:52am, jon g wrote:
just drop it, you all have gone too far. Why don't you answer a real question like what is the meaning of life or is there a god?

 
the answers to those questions are already well known!
 
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Re: 0.999.  
« Reply #127 on: Nov 25th, 2002, 2:34pm »

can either of you grasp infinity, maximal or minimal?
 
i didn't think so.
 
try to think of 0.999 as an ongoing process
 
it is a number constantly striving to be 1 but IS NOT!
 
however, since we can't even grasp the difference between
0.999 and 1 this difference doesn't have any
significance what so ever when dealing with final math.
so except for when dealing with math philosophy
there is really no need to use this difference.
 
since this difference is only philosophical it really doesn't bother us when making even the most complex and high resolution scientifical calculation.
 
from the infinity's point of view there is no difference between
17 and 999^9999999999999 at all
so in the same exact way, from our "human" point of view
there is no difference between 0.9999 and 1.
 
hope you enjoyed my thoughts on the matter.
 
peace   Smiley
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Re: 0.999.  
« Reply #128 on: Nov 25th, 2002, 6:16pm »

0.999... is not a process, it is the final result of a process. It is not 0.9 or 0.99 or 0.999 or ..., or even all of them taken together. It represents instead the final result of all their striving. As such it is not different from 1 at all, but is 1, under a pseudonym.
 
As for 17 and 999^9999999999999, what they look like from infinity depends on what infinity you are looking from, and how you are looking. If [omega] is the least infinite ordinal, then 17 + [omega] = [omega] and 999^9999999999999 + [omega] = [omega], but [omega] + 17 [ne] [omega] + 999^9999999999999 (addition is not commutative for infinite ordinals). Some differences never go away.
« Last Edit: Aug 19th, 2003, 7:04pm by Icarus » IP Logged

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Re: 0.999.  
« Reply #129 on: Nov 26th, 2002, 3:51am »

there are SO many posts after reading the first 2 pages I'm just stupified and want to take a crack at this...
 
I dunno if anyone brough this up into words, but I know in writing many have said it, but cannot grasp...
 
We live in a mathmatic world of logs of 10....
 
which means inorder to conserve symbols to represent values, we repeat every 10 by digits.
 
i.e. 1 2 3 4 5 6 7 8 9 (1)0
 
now, lets say we live in a log 15 world
 
i.e. 1 2 3 4 5 6 7 8 9 a b c d e (1)1  <  where 11 = 15 and e = 14 as we know it today...
 
now apply this stupid .9999..9999  to this topic, the next thing would be .aa, then .a1 .a2 .a3 .a4 .a5 .a6 .a7 .a8 .a9 .aa .ab .ac .ad .aeeeeeeeeeeeeeee THEN .b and so .9999inf is NOT 1......
 
you guys with your infinity...  there is no VALUE to anything multiplied to infinity, since it has no end, and since it don't stop, there is no measureable vaule, it is constantly changing, so you can't calculate it.
 
also...
 
10x /= 9.9999..999  = x
 
because if you did
 
10x = .99999...9999*10 then you're 1 decimal behind and the number isn't .99999..99999  any more, it's actually .999999999..9999  minus the last 9 because you just moved it's decimal place with log 10....
 
10x /= 9.999999..999  is also wrong because if you used anything other than 10 you'll get some screwy number that isn't even .9999999...999  ne more because you are breaking algebra's rules under logs of 10.... have you guys ever wonder why you multiply by 10??
 
i.e
 
8x = 7.99992 n some stupid crap if you did that...
 
again, if you look at the math in the sense of logs of 15 where we move every decimal place 15 digits deep, we won't have this discussion but rather .eeeeeeeeee...eeeeeee   = 1?
 
then we'll do  
 
(remember 11 means 15 in log 15) 11x = e.eeeeeeeeee..eeeee
(this is false btw, just as false as the 10x equation due to the fact that anyother number than (11 or 15 in our world) will produce something other that .eeeeeeeeinf)
 
now imagine (if you can) a log infinity....
 
this means we'll have a different symbol for each numeric value possible which is infinity itself...  WE WON'T HAVE DECIMAL POINTS....
 
 
 
I'm sorry, but it don't take no rocket scientist to see 0.999999999999999999999999999999999999999999999999999999999999999999fore ver /= 1
 
if you can't look at this and see, you're too caught up in logs of 10....  Decimals are so overrated...
 
Kozo mad props for haning in so long LOL, I almost believed it myself when theys aid they were =
 
My answer...  0.9inf < 1
« Last Edit: Nov 26th, 2002, 4:03am by KAuss » IP Logged
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Re: 0.999.  
« Reply #130 on: Nov 26th, 2002, 5:13am »

on Nov 26th, 2002, 3:51am, KAuss wrote:
I dunno if anyone brough this up into words, but I know in writing many have said it, but cannot grasp...
 
We live in a mathmatic world of logs of 10....
 
which means inorder to conserve symbols to represent values, we repeat every 10 by digits.
 
i.e. 1 2 3 4 5 6 7 8 9 (1)0
 
now, lets say we live in a log 15 world
 
i.e. 1 2 3 4 5 6 7 8 9 a b c d e (1)1  <  where 11 = 15 and e = 14 as we know it today...

As far as I know, the string of digits "11" in base 15 represents the decimal number 16, while 15 is writen (10)15. After all, you have 10 digits in base 10 and should have 15 digits in base 15, so you'd better use the 0.
 
on Nov 26th, 2002, 3:51am, KAuss wrote:
now apply this stupid .9999..9999  to this topic, the next thing would be .aa, then .a1 .a2 .a3 .a4 .a5 .a6 .a7 .a8 .a9 .aa .ab .ac .ad .aeeeeeeeeeeeeeee THEN .b and so .9999inf is NOT 1......

 
I think I don't get your point there. As you suggested further down your post, the question in base 15 is whether
  0.eeeee... = 1
 
on Nov 26th, 2002, 3:51am, KAuss wrote:
you guys with your infinity...  there is no VALUE to anything multiplied to infinity, since it has no end, and since it don't stop, there is no measureable vaule, it is constantly changing, so you can't calculate it.

 
As Icarus (and others I think) has pointed out, 0.999... (indefinitely) is not changing, it's a way of denoting the value of a limit. And that is what you calculate and compare to 1.
 
on Nov 26th, 2002, 3:51am, KAuss wrote:
now imagine (if you can) a log infinity....
 
this means we'll have a different symbol for each numeric value possible which is infinity itself...  WE WON'T HAVE DECIMAL POINTS....

 
This won't qualify as a number system, I guess.
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Re: 0.999.  
« Reply #131 on: Nov 26th, 2002, 5:45am »

well, if you start looking at other numbering-systems.. let's say base-n than the analog problem is not  
lim(t->inf) sum(9*n-i, i, 1, t)  
but
lim(t->inf) sum((n-1)*n-i, i, 1, t)  
which is the same as  
lim(t->inf) 1 - n-t = 1 , (since lim(t->inf) n-t = 0)
 
It might be interesting to take a look at base-1, you only have one number to represent numbers with
so you get one = 1, two = 11, three = 111 etc  
zero would be an empty string, but if you represent it with 0, than you can get 0.000000... = 1 (since 1 - 1 = 0, analog to (n-1) in above equations)
:p
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Re: 0.999.  
« Reply #132 on: Nov 26th, 2002, 6:03am »

Welcome - just a few things:
 
on Nov 26th, 2002, 3:51am, KAuss wrote:
now, lets say we live in a log 15 world
 
i.e. 1 2 3 4 5 6 7 8 9 a b c d e (1)1  <  where 11 = 15 and e = 14 as we know it today...
 
now apply this stupid .9999..9999  to this topic, the next thing would be .aa, then .a1 .a2 .a3 .a4 .a5 .a6 .a7 .a8 .a9 .aa .ab .ac .ad .aeeeeeeeeeeeeeee THEN .b and so .9999inf is NOT 1......

 
In base 15, the next number after e is 10 (which has the value 15), and then 11 (which has the value 16). No big deal.
 
In base 15, 0.999... has a value equal to 9/14 (base 10 numerals there!). There is no "next" value after it, or after any real number for that matter. But I agree that 0.a > 0.999...
 
But of course, showing that 0.999... in base 15 doesn't equal 1 says nothing about it in base 10. To answer the original question, we should consider base 10, or consider 0.eee... in base 15.
on Nov 26th, 2002, 3:51am, KAuss wrote:
you guys with your infinity...  there is no VALUE to anything multiplied to infinity, since it has no end, and since it don't stop, there is no measureable vaule, it is constantly changing, so you can't calculate it.
... you're 1 decimal behind ... you just moved it's decimal place with log 10....

 
Yeah, you really can't think of this problem in terms of multiplying or dividing finite decimals, because 0.999... is not such a thing, and it's easy to misapply reasoning.
 
That is, just because I could never work out 10*0.999... on paper, since it has an infinite number of digits, does not mean that 0.999... cannot be assigned a value.
on Nov 26th, 2002, 3:51am, KAuss wrote:
My answer...  0.9inf < 1

 
I ask you to consider my favorite counter-question to this then:
Let 0.999... = x; x < 1. But for any x < 1, I can construct a number y = 0.999...9 with enough 9s to make it bigger than x. So x < y < 1. But x has more 9s than y - how can it be smaller?  
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Re: 0.999.  
« Reply #133 on: Nov 26th, 2002, 1:38pm »

I'm lacking the ability now to switch text to numbers and vice versa after reading this theard LOL..
 
but yeah, this forum is pretty cool, lots of smart people haning around talking about just anything to the bitter end LOL
 
anyway back to the number thing...
 
Like I said, minus all the other things...  Can you imagine if we have an infinte base numbering system?  Where we HAVE no decimals... then the number would mearly be a number before another number....  it wouldn't ever get carried over...
 
The thing with the number growing is sheer fact that this number will never stop growing...  1 is a constant...  Someone pointed out that they'll intersect like a lot of you said, but if this is the case, then the rate of growth must be constant, and if whenever it intersects, after the intersection, it must grow larger than 1....  So I don't see how a number that is ever growing can equal to anything that has a solid value where it's value don't change...
 
Another thing I don't get is...  What is this talk about getting bigger and bigger??  Did we ever throw TIME (which don't exsist btw) into this equation?  We look at this as a repetitive process instead of an instant process...  This number is as developed as any, but it don't = x since x is not ever changing...
 
the thing with x < y < 1.........   What I don't get is, how can you get y = 1 more 9 than .999...inf?  There is something bigger than infinte?  I always thought you can't alter infinte, since it represents everything in the number line...  So how can you have 1 more than something that is everything?  so x = y /= 1 cause we can say this..  x = y /= 1 = a = b = c
 
  / .999inf
     /
-------  1inf
  /
/ .999inf
 
?? ^ that makes no sense...  The angle isn't like that obviously but almost parallel, but by what you all are saying with limit, (which deprives the piont of saying infinite since it don't end) will hit and STOP growing...  I don't see how...
 
 
 
I have totally lost my sense of reason....  LOL  I'm now like an aimless arrow in society...
 
 
Did Pi ever stop producing decimals yet?  Our number system sucks Smiley
 
One last thing I still don't get....  If .999inf is the process to 1 or whatever, then what is the process right AFTER 1? The number that is still 1 but it's always infinite X bigger than 1? just like how .999inf is 1 but it's always infinite Y smaller than 1....
 
« Last Edit: Nov 26th, 2002, 1:50pm by KAuss » IP Logged
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Re: 0.999.  
« Reply #134 on: Nov 26th, 2002, 2:48pm »

on Nov 26th, 2002, 1:38pm, KAuss wrote:

 
The thing with the number growing is sheer fact that this number will never stop growing...  1 is a constant...  Someone pointed out that they'll intersect like a lot of you said, but if this is the case, then the rate of growth must be constant, and if whenever it intersects, after the intersection, it must grow larger than 1....  So I don't see how a number that is ever growing can equal to anything that has a solid value where it's value don't change...

 
0.999...9 will never equal 1 no matter how many 9s there are - it will never "intersect." It will just get closer and closer to 1. "To 1". "1" is where it's going but never gets to. "0.999..." is where it's going but never gets to... you can think of the limit either way. Indeed, they are equal!
 
I agree with your comment about time - this shouldn't really be though of as a process. Or you can, but you have to recognized that 0.999... is the stationary limit of that process.
 
on Nov 26th, 2002, 1:38pm, KAuss wrote:

the thing with x < y < 1.........   What I don't get is, how can you get y = 1 more 9 than .999...inf?  There is something bigger than infinte?

 
Well yeah, that's the contradiction that you can derive from the assumption that 0.999... < 1. So therefore 0.999... = 1.
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Re: 0.999.  
« Reply #135 on: Nov 26th, 2002, 6:33pm »

on Nov 26th, 2002, 1:38pm, KAuss wrote:
What I don't get is, how can you get y = 1 more 9 than .999...inf?  There is something bigger than infinte?  I always thought you can't alter infinte, since it represents everything in the number line...  

 
There are several types of infinities. And you can define more freely. Here are the three best known:
 
Continuum Infinities
These are infinities at the "ends" of the number line, represented by [infty] and -[infty]. These are defined by adding a couple extra elements to the set [bbr] of real numbers, which are defined to be respectively greater than, and less than, every real number. The result of including these infinities is to make the real numbers topologically (that is, as far as limits and continuity are concerned) equivalent to a closed interval. A similar construction adds a single infinite point to the complex plane, making it topologically equivalent to a sphere.
 
Cardinal Infinities
Two sets are said to equipotent ([sim]) if you can match up their elements in a one-to-one fashion, with nothing left over from either set. For example, the set of Integers [bbz] and the set of Even Integers [smiley=bbe.gif] are equipotent ([bbz] [sim] [smiley=bbe.gif]), because we can match each integer [smiley=x.gif] with the even integer 2[smiley=x.gif]. But {0} [smiley=nsim.gif] {1,2}, because if you match 1 with 0, there is nothing left to match 2 with, and vice versa.  
 
We can assign to every set an object, called the set's cardinality, so that two sets have the same cardinal if and only if they are equipotent. (How this assignment can be done delves far deeper into set theory than I want to get into here.)
 
The set-theory definition of finite and infinite (from which all other definitions of these terms derive) is that a set is finite if is not equipotent to any proper subset of itself. A set is infinite if there is a proper subset to which it is equipotent. The example I gave shows that [bbz] is infinite. A cardinal is finite (infinite) if it is the cardinal of a finite (infinite) set. The finite cardinals are just the Whole numbers: 0, 1, 2, 3, ... .
 
The question then arises, how many infinite cardinals are there? The answer is: infinitely many. If [smiley=ca.gif] is an infinite set, then it can be proven that P([smiley=ca.gif]) [smiley=nsim.gif] A, where P([smiley=ca.gif]) isthe set of all subsets of [smiley=ca.gif]. The smaller infinite cardinals are expressed using the hebrew letter Aleph [smiley=varaleph.gif]. The smallest is [smiley=varaleph.gif]0 ("Aleph-nought"), which is the cardinality of the Natural numbers [bbn] (also the Whole numbers [smiley=bbw.gif], the Integers [bbz], and even the Rational numbers [bbq]). The next smallest infinite cardinal is called [smiley=varaleph.gif]1 ("Aleph-one"), etc. The Real numbers [bbr] are equipotent with P([bbn]), which as I stated earlier is NOT equipotent to [bbn]. Thus there are no more Rational numbers than there are Natural numbers, but there are more Real numbers. It is not known if the cardinality of the Reals is [smiley=varaleph.gif]1. (Actually it is much more complicated than just "not known", but this is too long already).
 
Addition and multiplication are defined for cardinals as follows: If sets [smiley=ca.gif] and [smiley=cb.gif] are disjoint, then Card([smiley=ca.gif]) + Card([smiley=cb.gif]) = Card([smiley=ca.gif] [cup] [smiley=cb.gif]), and Card([smiley=ca.gif]) [times] Card([smiley=cb.gif]) = Card([smiley=ca.gif] [times] [smiley=cb.gif])  ([smiley=ca.gif] [times] [smiley=cb.gif] is the set of all ordered pairs with the first element from [smiley=ca.gif] and the second from [smiley=cb.gif]). For infinite cardinals, this turns out to be uninteresting: If at least one of [smiley=x.gif] or [smiley=y.gif] is infinite, then [smiley=x.gif] + [smiley=y.gif] = [smiley=x.gif] [times] [smiley=y.gif] = max([smiley=x.gif], [smiley=y.gif]).
 
Ordinal Infinities
This is the set of infinite numbers I refered to in my reply to "Guest" above. Their definition is more complicated than Cardinals, but they have a much richer structure. Each ordinal corresponds to a means of "Well-ordering" a set.  
 
To well-order a set, you define an order operation [smiley=prec.gif] that is
  • Anti-reflexive: [forall] [smiley=x.gif],  [smiley=x.gif] [smiley=nprec.gif] [smiley=x.gif]. ([forall]="for all")
  • Anti-symmetric: [forall] [smiley=x.gif], [smiley=y.gif], if [smiley=x.gif] [smiley=prec.gif] [smiley=y.gif], then  [smiley=y.gif] [smiley=nprec.gif] [smiley=x.gif].
  • Transitive: [forall] [smiley=x.gif], [smiley=y.gif], [smiley=z.gif], if [smiley=x.gif] [smiley=prec.gif] [smiley=y.gif] and [smiley=y.gif] [smiley=prec.gif] [smiley=z.gif], then [smiley=x.gif] [smiley=prec.gif] [smiley=z.gif].
  • Well-ordered: every non-empty subset has a least element.
 
[bbn] is well-ordered by "<", but [bbz], [bbq], and [bbr] are not, since none of them have a least element.
 
A well-ordering [smiley=prec.gif] on a set [smiley=cs.gif] is considered equivalent to a well-ordering [smiley=lessdot.gif] on the same or another set [smiley=ct.gif] if all the elements of one set can be matched in a one-to-one fashion with all the elements of the other set (as with cardinality) but also in a way that preserves the order: if [smiley=a.gif], [smiley=b.gif] [in] [smiley=cs.gif] and [smiley=a.gif], [smiley=y.gif] [in] [smiley=ct.gif] with [smiley=a.gif] [smiley=leftrightarrow.gif] [smiley=x.gif] and [smiley=b.gif] [smiley=leftrightarrow.gif] [smiley=y.gif], then [smiley=a.gif] [smiley=prec.gif] [smiley=b.gif]  [bigleftrightarrow] [smiley=x.gif] [smiley=lessdot.gif] [smiley=y.gif]. Note that two well-orderings can be equivalent only if their sets are the same size (have the same cardinality).
 
Well-ordering is very restrictive. There is only one well-ordering (up to equivalence) for all finite sets of the same size. Infinite sets have more freedom. For instance, I can define the following ordering [smiley=prec.gif] on the Natural numbers: [smiley=x.gif] [smiley=prec.gif] [smiley=y.gif] if ( 1 < [smiley=x.gif] < [smiley=y.gif] ) or ( 1 = [smiley=y.gif] < [smiley=x.gif] ).  [smiley=prec.gif] is a well-ordering of [bbn] which has a maximum element (1), while the standard ordering < has no maximum element.
 
Ordinals are assigned to well-orderings so that two orderings have the same ordinal only if they are equivalent. Ordinals can also be added ord([smiley=x.gif]) + ord([smiley=y.gif]) = ord([smiley=z.gif]) where [smiley=z.gif] is the ordering on the disjoint union of the sets of [smiley=x.gif] and [smiley=y.gif], in which everything in [smiley=y.gif]'s set is greater than everything in [smiley=x.gif]'s set. Ordinal addition is generally NOT commutative (ord([smiley=x.gif]) + ord([smiley=y.gif]) [ne] ord([smiley=y.gif]) + ord([smiley=x.gif])). You can also define a multiplication, but I will not go into it.  
 
When the sets are finite, there is only one well-ordering, and thus only one ordinal, so finite ordinals are just the Whole numbers again. With infinite ordinals, things get more interesting. The ordinality of [bbn] is usually denoted by the greek letter small omega [omega]. The ordinality of the order I gave above (that moves 1 to the opposite end of the number line) is [omega]+1. The rule for adding infinite ordinals is that if 0 < [smiley=x.gif] < [smiley=y.gif], and [smiley=y.gif] is infinite, then [smiley=x.gif] + [smiley=y.gif] = [smiley=y.gif], but [smiley=y.gif] + [smiley=x.gif] > [smiley=y.gif]. The smallest ordinals follow this pattern:
0, 1, 2,... [omega], [omega]+1, [omega]+2,... [omega]+[omega] = 2[omega], 2[omega]+1, 2[omega]+2,..., 3[omega],..., [omega]2,..., [omega]3,...
 


Finally now I can answer KAuss's question: how can you have infinitely many 9s, and then another 9?
 
An ordinary decimal number (we will only look at the fractional part) is just a sequence, that is, a function from the natural numbers, into the set of digits: {0,1,2,3,4,5,6,7,8,9}. To get another 9 after infinitely many, we need a different domain for our function than [bbn]. Instead, we choose functions from a set of ordinals which includes infinite ordinals. So, the first part of 0.999...([infty] 9s)...9 gives the digits corresponding in our sequence to 0, 1, 2, ... , while the final 9 corresponds to [omega].
How do you get a number out of this? If you are talking about Real numbers, then you can't (other than the same number as represented without the final digit). But it is possible to simply treat these sequences as a set of numbers themselves, which contains the Reals, plus additional "pseudoreal" numbers. Which is what I was refering to in my previous posts.
 
Kozo: You did a great job of being the devil's advocate. That's why we keep refering to your posts! Cool
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Re: 0.999.  
« Reply #136 on: Dec 10th, 2002, 11:18am »

the difference between .99999... and 1 = .000...1
the difference between .99999... and .99999...8 = .000...1
so if .9999... = 1 then .999999...8 = .9999999 which is 1 so  
.99999...8 =1
 
if .999999999 was 1 then thats what it would be. ever wounder why it starts with a 0.? its like saying 1=2 or 1 = 1,000,000  
 
what is infinity? a number that really dosnt exist so as .999 never gets to an infinate number if 9's (you can always add 1 more) It never reaches 1 so isnot 1.
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Re: 0.999.  
« Reply #137 on: Dec 10th, 2002, 11:47am »

on Dec 10th, 2002, 11:18am, guest535 wrote:
what is infinity? a number that really dosnt exist so as .999 never gets to an infinate number if 9's (you can always add 1 more) It never reaches 1 so isnot 1.

 
*sigh* Before posting to a thread, do try to read the whole thread...we've been through this before.
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Re: 0.999.  
« Reply #138 on: Dec 10th, 2002, 12:35pm »

on Dec 10th, 2002, 11:18am, guest535 wrote:
the difference between .99999... and 1 = .000...1

There's your problem - you write "0.999..." but you are thinking of something like "0.999...9" - that is, you are referring to some decimal with a finite number of 9s, but this riddle is about the slippery notion of "0.999...".
 
In your equation, the number of digits on the right must be the same as the number of digits on the left, and you indicate a finite number of digits on the right.
 
If you'd like to argue that the right side is "an infinite number of 0s, and then a 1," which is not a possible decimal number, but still something whose value can be reasoned about, then you can... but you'll be hard pressed to show it is anything other than 0!
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Re: 0.999.  
« Reply #139 on: Dec 12th, 2002, 7:01am »

Pleaaaase, i do not know how thsi goes on, if the correct answer was there from the really beginning: 0.99999...=1, no doubt (in base ten of course, that is the math we are doing in another base like you said yo can equally prove that o.eeeee...=1)
The answer is basic from middle school, i knwo other people have said it, but here it is again:  
Lets call the number we are trying to calculate A (A=0.9999999.......)
then 10A= 9.999999......
when substarcting both, we get that 9A = 9
So A is equal to 1.
With the same argument you can say that 0.888888888.... is equal to 8/9 or that 0.345345345345345345....... is equal to 345/999. These numbers all belong to the rational numbers, unlike pi for example or sqrt(2) that cannot be expressed by a fraction.
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Re: 0.999.  
« Reply #140 on: Dec 14th, 2002, 7:46pm »

Ok, if a frog starts ten feet away from a pond and jumps halfway to the pond each time he jumps, he will never reach the pond. .999.... will never equal 1 because it is simply not 1. continue adding 9's and you get closer, but youre never there. called an asymtote. i'm a beginner at this, and correct me if i'm wrong, but isnt this the same as saying 1 is equal to 2 if you just imagine it to be?
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Re: 0.999.  
« Reply #141 on: Dec 14th, 2002, 8:52pm »

Sure, you can think of 0.999... as the limit of the sequence 0.9, 0.99, 0.999, .... Connect the points (1, 0.9), (2, 0.99), (3, 0.999), ... on the plane and you'll have a graph of something that approaches 1 asymptotically as x goes to infinity, I think you'll agree.
 
Put briefly, I think the point is that 0.999... *is* also that asymptote, that value that is approached. It's not a y-value in that sequence. Once you agree that that's what  0.999... is, you have to agree that it's 1.
 
If you don't buy that - look back a page or two to see what kinds of contradictions you get if you assume that 0.999... < 1.
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Re: 0.999.  
« Reply #142 on: Dec 15th, 2002, 11:05pm »

Since people are still bringing up numbers like "0.999...8" or "1.00...1", remember: Talking about an infinite series of 9s followed by an 8 is like talking about a man who lived for for an infinite number of years and then died. Smiley In other words, in the words of every math teacher I ever had, "there's no such animal". (Why do math teachers like that phrase so much?)
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Re: 0.999.  
« Reply #143 on: Dec 19th, 2002, 7:21pm »

My goodness you crazy people, it's so easy.  Simply define this as the sum of an infinite geometric series and you will see that it equals one.
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Re: 0.999.  
« Reply #144 on: Dec 19th, 2002, 7:33pm »

I would like to clarify this topic for all those who believe that infinity is not a number.  Anyone who has read a lot about math knows what I am about to say:      
 
INFINITY=LIMIT AS X->0 OF 1/X=THE MAPPING 1/Z APPLIED TO THE ORIGIN IN THE COMPLEX PLAIN=MANY OTHER DEFINITIONS AS WELL.  
 
So,  INFINITY IS A NUMBER YOU STUPID PEOPLE!!!!!  From now on, philosophers should not try to talk about math, because philosophers aren't mathematicians.  When somebody uses philosophy to talk about math, everything gets screwed up.  For example, somebody I know now thinks that the Pythagorean Theorem "Isn't really true but works anyway" because they read that in a book by a stupid philosopher who knows nothing about math.  This philosopher says that irrational numbers like sqrt(2) don't exist.  This kind of stuff is stupid, I know how to define sqrt(2) using an infinite series for sqrt(1/2) from the binomial theorem.  PHILOSOPHERS SUCK AT MATH.  I'm not saying philosophers suck, I'm saying they suck at math.  This was a very long, angry message.  Ok, bye.
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Re: 0.999.  
« Reply #145 on: Dec 20th, 2002, 12:22am »

So to recapitulate, infinity is a number because you say so, and anyone who doesn't agree is stupid, and that's is you're whole argument..?
Well, you sure aren't a philosopher, they at least know how to argue a point, whether it be true or false..
 
Also note that the way you define infinity you made it the same as minus infinity.. You need to specify if you take the limit approaching from above 0, or from below 0. Else, if I'm not mistaken, that answer is undefined, and thus neither.
 
Philosophers have their place in every science, including math. And also someone can be both mathmatician and philosopher, the two concepts aren't mutually exclusive.
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Re: 0.999.  
« Reply #146 on: Dec 20th, 2002, 9:48am »

  Agreed, just look at Wittgenstein. I myself find it very hard to divorce completely from philosophy a subject so platonic as mathematics. Your reasonableness is refreshing, towr.
 
   As for infinity being a number because of the above definition... um... what is a number? Just because something is defined, it's a number? Well, I define the number P (for "Pietro's number") to be all the trajectories in space-time of all the ripples made in ponds by canadian frogs that jumped in them during a partial ecclipse of the sun.
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Re: 0.999.  
« Reply #147 on: Dec 20th, 2002, 10:12am »

Pietro,
 
I'm shocked! Shocked And I'm sure that every right-minded French-Canadian is appalled at your definition of P, Pietro's number!
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Re: 0.999.  
« Reply #148 on: Dec 20th, 2002, 3:49pm »

Not being French or Canadian I guess I don't have to be shocked.  
 
So Pietro, I am curious about your new number! Does it have any interesting properties? From the definition it has to be closely related to 42. Wink
 
Goodyfresh: you shouldn't insult people for their arguments when it's clear your have not read those arguments! If you will actually read the thread, you will discover that the series representing .999... has been discussed many times!
 
As for your definition of oo, It does not comprise a usable definition at all, though it is a derivable property from the definitions of some infinities. There are plenty of different infinities. Check my post on Nov. 26 to see a summary of a few of the many types of infinities out there. Whether an infinity is a number is a matter of choice. The concept of "number" is not well defined in mathematics. The word is used to describe many things you would not normally think of as numbers. However, the concept of a "Real number" or a "Complex number" is well defined, and no infinity fits in it. When you add infinities to these sets, you get the "extended Real numbers" or the "Riemann Sphere" (see my post "Z^n and the Riemann Sphere" in the complex analysis forum to find out what the Riemann Sphere is).
 
towr: It is actually very common to define a single infinity which serves as both limx[to]0+1/x and limx[to]0-1/x. This infinity wraps up the Real line to make a circle. [infty] is the antipode of 0 on this circle. Again, see my post on the Reimann Sphere in the complex analysis forum to see the same trick done to the complex plane.
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Re: 0.999.  
« Reply #149 on: Dec 29th, 2002, 4:56am »

x=1
y=0.99999 (using the underline to represent an infinite amount)
 
2x=2
2y=1.99998
 
3x=3
3y=2.9997
 
this shows a difference between the values of x and y, but i dunno if thats mathematically legal
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