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Author Topic: 1 = .999… ??!
QuiGonValjean
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Icon 4 posted January 07, 2003 16:47      Profile for QuiGonValjean   Author's Homepage     Send New Private Message       Edit/Delete Post 
Okay, a friend and I have this very heated disagreement over one thing: he says he can prove to me by using correct math that .9999 to infinity can equal the number 1.

The first method he tried on me was this:

1/3 = .333…

Therefore:

1/3 * 3 = .333… * 3
3/3 = 1 = .999…

I didn't buy that one. (I believe that the Substitution Axiom is what would disprove that.)

Okay, so he gives me another proof:

x = .999…
x * 10 = .999… * 10
10x = 9.999…

Subtract the left column in the first equation from the left column in the third, and likewise with the right column…

9x = 9
9x / 9 = 9 / 9
x = 1

Initially, x was said to equal .999…, so…

.999… = 1

[Eek!]

Okay, how is this supposed to work? I'm thinking that his math is wrong somewhere, but so far I haven't found it yet. [Confused]

C'mon, can anyone help me out here, please!?

-QGV

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"I hope that by having this open dialogue we can learn to live in harmony. Or failing in that, I'll live in harmony by myself." - Dogbert

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rw

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Icon 1 posted January 07, 2003 21:16      Profile for rw   Author's Homepage     Send New Private Message       Edit/Delete Post 
I saw those proofs many years ago, and didn't want to believe them either -- but the person who showed them to me was my 12th grade math teacher, and she did know a thing or two about math.
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joliet_jane
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Icon 1 posted January 08, 2003 00:34      Profile for joliet_jane     Send New Private Message       Edit/Delete Post 
I too reject that. .9999 to infinity is .99999 forever, not 1. One is one!

Think of it as on the number line. Each end of the line goes to infinity. Between each number increment on the line, there can be fractions... 1/5... 6/40... or rather, y/x, where X can be any number as long as it's not smaller than y, and then you have to reduce and all that too. Of course, though, x can be anything; any number to infinity. That why I think that only 1 can be one and not any other number.


...then again, I am a moron. [Smile]

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QuiGonValjean
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Icon 1 posted January 08, 2003 04:14      Profile for QuiGonValjean   Author's Homepage     Send New Private Message       Edit/Delete Post 
See, the problem is that the guy who gave him these heretic ideas is HIS 12th grade math teacher too! Ulgh. So, my friend's pretty willing to believe anything this guy says now...

quote:
I too reject that. .9999 to infinity is .99999 forever, not 1. One is one!
Exactly what I told him!! But he tells me to look at the proof... it proves mathematically that what he says is true! Grrrr... [Mad]

C'mon, one number cannot equal another unequal number! ......Right?

-QGV

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"I hope that by having this open dialogue we can learn to live in harmony. Or failing in that, I'll live in harmony by myself." - Dogbert

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GameMaster
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Icon 1 posted January 08, 2003 07:51      Profile for GameMaster   Author's Homepage     Send New Private Message       Edit/Delete Post 
I've heard of this proof, from a great math minds, but if you think about.....


1) 2*1 = 2 by Identity Pr.
2) Let's assume 1 = .9999...
3) 2 = 2 * .9999... = 1.99999999999 by 1 and 2.
4) 1.999... = 1.999... * 1 by Identity.
5) 1.999... = 1.999... * .999... = 1.999...96
which is where I see the problem... if the 9 continue infinatly, the trailing 6 would go??? Which might indeed mean that the proof is valid.
[Eek!] [Frown] [Roll Eyes] [Frown] [Eek!]

But it might be a play off this abuse of infinity...
Infinity * 2 = Infinity by definition
Infinity * 10,000 = Infinity by definition
Infinity = Infinity by Identity
ergo: 2 = 10,000
[Beard of Peter Gabriel!] [Roll Eyes] [Beard of Peter Gabriel!]

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TMBWITW,PB

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Icon 1 posted January 08, 2003 08:22      Profile for TMBWITW,PB     Send New Private Message       Edit/Delete Post 
The trouble here is that we are treating infinity as though it were a number. Which it isn't. It is the concept of something continuing out forever. That is why

Infinity + 2 = Infinity + 10,000 = Infinity

So looking at that .99999999 ad infinitum may not actually equal 1, but it is so close that there isn't any real point in distinguishing between them.

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greycat

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Icon 1 posted January 08, 2003 11:05      Profile for greycat   Author's Homepage     Send New Private Message       Edit/Delete Post 
My god, this is painful. rw, your math teacher was right (as is Quigon's friend). The written notation ".99999..." is really a short-hand for:

code:
(infinity)
(Sigma) 9/(10^i)
i=1

(Which would look a whole lot better if it were written with the correct symbols. I hope you know what Sigma means.)

This is what we call an infinite series. I can't recall off the top of my head how to prove that the sum of this infinite series is 1, but perhaps your math teacher does. Or perhaps you can work it out yourself -- it would certainly be better for you than self-delusion!

quote:
C'mon, one number cannot equal another unequal number! ......Right?
You can write a number in many different ways without changing its value. You could write one as 1 or 1.0 or 2/2 or 4/(2^2) or 0.9 with a bar over the 9 (curse you ASCII+UBB for your pitifully small expression capability!) or 0.999... or, in fact, an infinite number of other ways. They're all equal to one, the multiplicative identity element.
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GameMaster
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Icon 1 posted January 08, 2003 11:35      Profile for GameMaster   Author's Homepage     Send New Private Message       Edit/Delete Post 
TB, I know that... I was making fun of when I was a kid arguing with my brother as we'd yell at eachother "Oh, yeah, well 'uh-huh times infinty times 2".... "NO. NO,no,no." I thought that perhaps this might be an extention from that falicy. Which would make the proof valid, but only because the premises are assumed correct.

Right, but while it is indeed an infinate series, that does mean that there is an infintly small part, just keep doing the sumation that you wrote above...

code:
  infinity
--------
\ |
\ 9/10^1
/
/_______|
1 = 0

iteration 1:
.9
iteration 2:
.99

...

iteration (lazy 8)
.999...

Which would result in the tenths place to always be a nine, no matter how many billions and billions and billions of 9s follow. There will still be an (literally)infintesimal fraction left over. I'm not saying that I don't believe that this is the case, I'm just saying I don't see logically how this can be true. I suffice to say, it's time to figgure out how to type the sumation of 9/10^i, for i from 1 to infinity, into a goole search and not get a mess of junk. I'll post what I find, QuiGon.
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GameMaster
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Icon 1 posted January 08, 2003 11:41      Profile for GameMaster   Author's Homepage     Send New Private Message       Edit/Delete Post 
Not bad, just typed in the thread name... and found:
http://www.math.lsa.umich.edu/~mathsch/courses/Infinity/Equals1.shtml

I can't find the proof, but there is again an example of the sumation, and some reasoning.... looks rather familiar to me... I suppose I'll just have to eat that .000000.......0.....0......1

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Lex
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Icon 1 posted January 08, 2003 13:20      Profile for Lex   Author's Homepage     Send New Private Message       Edit/Delete Post 
I think this is basically Xeno's (Zeno?) Paradox in a different suit. If you start at 0 and want to walk to 2, you have to halve the distance first. So then you are at 1. Half again is 1.5, 1.75, and so on, until you get to 1.999..., and then it just gets progressively longer. It's another one of them sigma thingies, and the proof that you can get to 2 is taught in Calc 2 here. If I could remember any calculus, I'd give it, but I don't for some reason. Odd considering that I'm in Calc 3 this semester. *shrug*.
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quantumfluff
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Icon 1 posted January 08, 2003 14:21      Profile for quantumfluff     Send New Private Message       Edit/Delete Post 
The proof is a clever slight of hand. The math is flawed.

.333_ * 3 is not .999_. It is 1.

The thing is that ".333_" is not an actual number, but a notation for representing infinite expansion. When you multiply it by a scaler, the results are an aproximation. Greycat approached it the right way by looking at the sum of the series. Each of them will converge to 1 if you can actually add the series up infinitly.

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Charisma
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Icon 1 posted January 08, 2003 17:27      Profile for Charisma   Author's Homepage     Send New Private Message       Edit/Delete Post 
tricky!

I can see how you could prove it mathematically... but it doesn't make sense!

I think it only makes sense... if... wait... hmmmmm.

Like, if it ends? Because, since it's infinite, it'll never end and 1.000000000 isn't equal to 0.999999999.

Just my take on it.

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The Famous Druid

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Icon 1 posted January 08, 2003 21:56      Profile for The Famous Druid     Send New Private Message       Edit/Delete Post 
You can produce a similaryl wrong proof that

1 ** infinity = 2.718281828

Which is based on the fact that

e = Limit ((1 + 1/n) ** n) as n approaches infinity

so if you (wrongly) state that (1 + 1/infinity) = 1

this becomes e = 1 ** infinity

As someone said earlier, infinity is not a number, it's a concept. Treating it like a number in equations is a mistake.

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The Famous Druid

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Icon 1 posted January 08, 2003 22:06      Profile for The Famous Druid     Send New Private Message       Edit/Delete Post 
There is a fine distinction between these two statements

limit ( 1 - 1/n) as n approaches infinity = 1 (correct)

and

(1-1/n) = 1 when n is infinity (wrong)

The 'limit' statement is basically saying
'You can decrease the error to practically nothing by increasing n'

the key point to remember is that 'practically nothing' is not the same as 'actually zero'

--------------------
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Hikaru
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Icon 1 posted January 08, 2003 22:17      Profile for Hikaru   Author's Homepage     Send New Private Message       Edit/Delete Post 
even more simply stated,
using Windows user logic
it just doesnt there, oh, and I dont like your vastly superior OS *
(*= Mac OS, Mac OSX, Lindows, Lunix, Linux, Lisa OS, Amiga OS, Basic ect)

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ModularMan
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Icon 1 posted January 09, 2003 01:00      Profile for ModularMan     Send New Private Message       Edit/Delete Post 
In standard analysis (which is taught in Calculus) 1 equals 0.999999.... There is no doubt about this.

But in Non-standard Analysis this is not necessarily true. Two useful links:

Another View of Nonstandard Analysis

Introduction to Unified Analysis

For more information simply google for "non standard analysis".

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technosquirrel 101
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Icon 1 posted January 10, 2003 08:14      Profile for technosquirrel 101     Send New Private Message       Edit/Delete Post 
i think there's a mistake to take

1/3 = .333…

as a precise statement. like many things in math it's a shortcut to explain something and to make it easily imaginable.

just like you can express PI with a standard 10-digit calculator as 3.141592653 and get good results. after the 10th digit the calculator rounds it and only gives back better results 'cause it rounds other values, too.

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Evilbunny
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Icon 1 posted January 13, 2003 11:15      Profile for Evilbunny   Author's Homepage     Send New Private Message       Edit/Delete Post 
Hello everyone! I've been gone for quite a while but somehow this thread has sparked my interest.

This is a simple (and true) Calculus argument. Basically what its saying is that .99999999 repeating equals one because as the number of nines approaches infinity, the number comes closer and closer to becoming one. Thus, at infinity (and heance the repeating decimal) .999999 repeating equals one.

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QuiGonValjean
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Icon 1 posted January 15, 2003 10:22      Profile for QuiGonValjean   Author's Homepage     Send New Private Message       Edit/Delete Post 
Ah yes, I was pretty sure you'd show up sooner or later, Evilbunny. [Smile]

So, by following your same logic, Evilbunny, could you also say that .7777… would also equal 1 at infinity?

I agree with the idea that infinity is a concept, not a number. As finite creatures, we cannot comprehend infinity, so attempting to argue it is rather pointless anyway. But it's fun! [Big Grin] Keeps me out of trouble... Okay, so maybe not, but anyway...

My friend and I finally called a truce, on the grounds that we will never determine an answer anyway, since such a problem cannot phyically be proved. And besides, trying to comprehend infinity hurts my head. [Smile]

-QGV

--------------------
"I hope that by having this open dialogue we can learn to live in harmony. Or failing in that, I'll live in harmony by myself." - Dogbert

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greycat

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Icon 1 posted January 15, 2003 12:46      Profile for greycat   Author's Homepage     Send New Private Message       Edit/Delete Post 
No, .77777... equals 7/9 (seven divided by nine).

Evilbunny was trying to simplify it for a nontechnical audience; unfortunately, she simplified it so far it became useless.

What she was trying to say, I believe, was:

code:
.99999... = lim (n -> inf) sigma (i=1 .. n)(9/10^n)

which is the fully formal expression of what I wrote a few days ago. This limit is 1, so we define the meaning of .99999... to be 1.
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Evilbunny
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Icon 1 posted January 29, 2003 18:02      Profile for Evilbunny   Author's Homepage     Send New Private Message       Edit/Delete Post 
Thanks greycat! That's exactaly what I meant.
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neo
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Icon 1 posted February 26, 2003 21:30      Profile for neo   Author's Homepage     Send New Private Message       Edit/Delete Post 
one number is always one number. aka, 1=1 and never 1=\= 9. infinity = infinity.
but infinity could equal to 9. let me tell you why.
we use numerology to do this. for example.
9+9=18
1+8=9
get it?
the highest number is 9 and infinity=9 not 9=infnity. although infinity is not a number, 9 represents it because we can look at the nine times table for example.
9*
1)9 =9
2)18 =1+8=9
3)27 =2+7=9
4)36 =3+6=9
5)45 =4+5=9
and from here, it's reversed!
6)54(45) =5+4=9
7)63(36) =3+6=9
and so on.
upon further investigations, i found that 9 is similar to infinity. the end number is 9. if you square 9, you will get 81(18) and it will give you 9.
so this is the trick i give those kids if they want to know what comes after 9.
get it? i don't think so. hope you do. just keep adding. even the 0. infinity enough?

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neotatsu
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Icon 1 posted February 26, 2003 22:30      Profile for neotatsu     Send New Private Message       Edit/Delete Post 
quote:
Originally posted by neo:

*poke* *poke*... *POKE!* (O.O) [crazy] [crazy] [Embarrassed]
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Twinkle Toes
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Icon 3 posted February 27, 2003 02:21      Profile for Twinkle Toes   Author's Homepage     Send New Private Message       Edit/Delete Post 
quote:
Originally posted by neo:

1)9 =9
2)18 =1+8=9
3)27 =2+7=9
4)36 =3+6=9
5)45 =4+5=9
and from here, it's reversed!
6)54(45) =5+4=9
7)63(36) =3+6=9
and so on.
upon further investigations, i found that 9 is similar to infinity. the end number is 9. if you square 9, you will get 81(18) and it will give you 9.

All this reminds me of this habit of mine with group numbers... I add the [separated] numbers in the group and try and come up with a certain main special number they all equal. I know it's a terrible explanation, but it was similar to what greycat posted in 'this one thread', in which Lumina (-I think she actually started it, but my memory fails too often) talked about how she doesn't step on cracks and so forth and so on.

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neo
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Icon 1 posted March 02, 2003 20:58      Profile for neo   Author's Homepage     Send New Private Message       Edit/Delete Post 
wow, i have a stalker!

--------------------
~What lies do you believe?~

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