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0.9 (repeated) does NOT equal 1, with proof

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posted on Dec, 10 2006 @ 06:51 PM
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Originally posted by Akareyon
I always used this approach to prove for myself that .9r = 1
1 / 3 = .3r
3 x .3r = .9r
=> .9r = 1

But that has been said earlier already in other words :-)


If you try doing this in fractional form, that doesn't occur because .9r isn't a rational number. It just turns out 3(1/3) = 1. The only reason it appears to equal .9r is that you can't represent 3 dividing evenly into 1 in decimal form.

[edit on 10-12-2006 by Zurahn]




posted on Dec, 10 2006 @ 07:13 PM
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Zurahn, how did this argument start? If you don't mind my asking.

.9 (repeated) certainly does not equal 1. All can clearly see that.


But I am a simple thinker, and I don't know what compelled you to start this thread. So if you would, please explain a bit to me.

edit = clarity

[edit on 12/10/2006 by Mechanic 32]



posted on Dec, 10 2006 @ 09:57 PM
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It doesn't equal one but it approaches one as a limit when you carry this out to infinity. Literally you could probably say it never reached one, but as far as out measurements go, it's good enough. No one's that percise.



posted on Dec, 10 2006 @ 10:07 PM
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Originally posted by Mechanic 32
Zurahn, how did this argument start? If you don't mind my asking.

.9 (repeated) certainly does not equal 1. All can clearly see that.


But I am a simple thinker, and I don't know what compelled you to start this thread. So if you would, please explain a bit to me.

edit = clarity

[edit on 12/10/2006 by Mechanic 32]


I'm not exactly sure from where the concept arose, but it's now has spread fairly widely.

Here's the Wikipedia page on it:
en.wikipedia.org...


Ultimately I haven't seen argumentative proofs against it, only in favour. I want to see if there is a legidimate dispute with my argument in my original post. So far I haven't really gotten any feedback on that initial post specifically.

Basically, I made this thread because the common belief appears to be that in fact .9 repeated equals 1, and I feel as though I have disproven that. I would like feedback as to whether I am correct.

[edit on 10-12-2006 by Zurahn]



posted on Dec, 10 2006 @ 11:23 PM
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0=0.000000000000000000000000
1=1.000000000000000000000000
0.99999999999999999999999999999999999=0.99999999999999999999999999999999999

Of course .9999999999 dosn't equa one
The diferance is so small people just say it is one because they's little diferance.



posted on Dec, 10 2006 @ 11:32 PM
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Originally posted by Zurahn
But your saying 9 an infinite number of times, when infinity isn't constant. That's my concern there. For it to be a cardinality, the sets have to be the same size, which is infinity, which isn't constant, which means they aren't necessarily the same. Basically it's saying A = [9, 9, 9, ...] B = [9, 9, 9, ...]
|A| = infinity
|B| = infinity
|A| - |B| = 0
which isn't true. If I have interpreted you incorrectly, sorry, please clarify.

And I know that you can combine like terms, but that's not illustrating what that decimal proof is doing. When it moves the decimal point over by multiplying by 10, it's the equivalent of reducing 10/i down to 1/i.

...

Unless, that is, you can tell me what my mistake specifically was. If so, please point it out.


Perhaps your mistake was a typo, but you has 1/i where 10/i should have been in one of the equations. It isn't all that important, but I'm curious why you are so against the concept that .999... = 1. Why is that? Does it break something in your world or your mind?

Fine, we'll define * to be an infinitesimal. A positive infinitesimal is a number greater than zero AND less than all positive real numbers.

1 - * = .9r (hypothesis)

Now, .9r is defined as a value between 1 and the closest value to 1, but less than one. Since our desired value falls in-between an area that has no true value (i.e. it is an infinite sequence of 9s), we will try a proven method used to create modern calculus (although typically performed in 2+ dimensions).

Let's just do the calculations, shall we:

* > 0, * < R^+ (i.e. for all positive real numbers)
x = 1 - *
10x = 10(1 - *)
10x = 10 - 10*
10x - x = (10 - 10*) - (1 - *)
9x = 10 - 10* - 1 + *
9x = 9 - 9*
9x / 9 = (9 - 9*) / 9
x = 1 - *

So, the transformation follows the same algebraic rules. However, the final step of the infinitesimal needs to be completed. Since an infinitesimal is not a number, it must be eliminated. Here is the rule to eliminate an infinitesimal:


A number x is an infinitesimal if and only if for every integer n (ie. any multiple), |nx| is less than 1, no matter how large n is.


To sum up, an infinitesimal is smaller than the smallest unit in a complete infinite set. So, we must technically "round" to the closest value. That value is, of course, 1, where x = 1 - * => x = 1

Now, it could be argued that if it were possible to have the largest sequence of .9999999999...99 WITHOUT being infinite AND THEN ADDING AN INFINITESIMAL, the result would either be that an infinite sequence was created, or the largest possible sequence of .999s resulted. However, performing this operation would only prove that there are two ways to get .9r or that our original assumption was correct (twice).

So what's the conclusion? It is completely reasonable to believe that the slightest possible reduction of 1, where 1 - * is that reduction, results in the value of 1 when replicating the procedures used to show that .9r = 1

Therefore, it is reasonable to assume that .9r = 1 because it is not possible for a lesser number (one of a finite sequence) to result from an infinitesimal reduction.

So: .9r = 1 - * = 1

Wow, that probably went way over everyone's head.

[edit on 10-12-2006 by Protector]



posted on Dec, 10 2006 @ 11:44 PM
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Egads!

Equations like that are almost like trying to find the absolute value of pi, without any remainder and without using fractions.



posted on Dec, 10 2006 @ 11:56 PM
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Originally posted by Mechanic 32
Egads!

Equations like that are almost like trying to find the absolute value of pi, without any remainder and without using fractions.


In computer science we sometimes use:
4 * arctan(1) = pi (in radians)

Because:
tan(pi/4) = 1 (in radians)

So using 4 * arctan(1) will get you the best precision pi value a computer can store (in theory). And, as you can see, it doesn't require fractions. In reality, an arctan breaks down into a fraction, but the illusion is PRETTY!!



posted on Dec, 11 2006 @ 06:45 AM
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It was not a typo in that 1/i and 10/i are both infinitesimals, and that it's similar to infinity+1 is still infinity and is how .9... * 10 is 9.9... since there's nowhere for the 0 to be placed.

From what you said, the inifitesimal is eliminated because in practical application it doesn't exist, but neither does .9 repeated. So considering that we start and end with the same equations, the infinitesimal would be eliminated from both, which means we're merely stating 1 = 1.

I'm not dead-set against the concept, but if I merely agree immediately, I have no basis as to why. To me right now, there is no "why" because it's not true.



posted on Dec, 11 2006 @ 05:01 PM
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After thinking about it and having a talk with an Analysis professor, I understand it fully and I hope I can make you understand.

It's important to note that none of these "proofs" are actual proofs - they're more of convention than anything else. This problem represents the fundamental flaw in our mathematics. Like I said before, 1/4 is commonly known to equal 0.25 but it also equals 0.249999999999999999999999 and so on. This is what I understood the problem to be initially but I confused myself.

This is just one of many examples of this flaw. It's not so much a proof - something like that could never be proven. I know of other ways to prove that 0.9r doesn't equal 1, but you have to change axioms to show this. It's mostly philosophical mumbo-jumbo...



posted on Dec, 11 2006 @ 06:50 PM
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Originally posted by T_Jesus
After thinking about it and having a talk with an Analysis professor, I understand it fully and I hope I can make you understand.

It's important to note that none of these "proofs" are actual proofs - they're more of convention than anything else. This problem represents the fundamental flaw in our mathematics. Like I said before, 1/4 is commonly known to equal 0.25 but it also equals 0.249999999999999999999999 and so on. This is what I understood the problem to be initially but I confused myself.

This is just one of many examples of this flaw. It's not so much a proof - something like that could never be proven. I know of other ways to prove that 0.9r doesn't equal 1, but you have to change axioms to show this. It's mostly philosophical mumbo-jumbo...


I wouldn't call it a fundamental flaw... I'd call it a feature.

.9r touches 1 from the left side.

Or you might say

Lim 1 - (1/x) = 1
x->oo

You could also ask:

Lim 1 + (1/x) = 1
x->oo

Now, in limit form, we say that both of those are equal to 1... but do they equal each other?

Lim 1 - (1/x) + (1/x) = 1
x->oo

But is:

Lim 1 - (1/x) ?= 1 + (1/x)
x->oo

In an equation, if two sides are equal, then subtracting them will result in zero:

1 - (1/x) - (1 + (1/x) = 1 - (1/x) - 1 - (1/x) = - 2/x != 0 (or does it?)
OR
1 + (1/x) - (1 - (1/x) = 1 + (1/x) - 1 + (1/x) = 2/x != 0 (or does it?)

Throw that back in the limit:

Lim -2/x = 0
x->oo
OR
Lim 2/x = 0
x->oo

Now the answer IS zero, but I'm sure you're going to ask a fundamental question now. Is the result zero, or is it approaching zero? This subject could be argued for a long time, so obvious limits don't work, but they are yet another way to show that .9r can reasonably be called 1 (like a limit form of an infinitesimal). The difference in this approach is that I have shown that is reasonable to deduce that:
.9r = 1 = 1.0r1

What does that mean? Other than the fact that we don't have defined rules (logically) for values after an infinite sequence, it is probably assumed that any values that are placed after an infinite sequence are either impossible, illogical, or simply descriptive (because mathematically, there is no space for numbers AFTER an infinite sequence).

As a question, you should ask, if 1 is made of the sum of all of its parts, is .9r one of those parts or a representation of the whole summed? Let's face it, .9r is either a unit derived from 1 or 1 itself. By definition it is not greater than 1, nor less than or equal to zero. So it is either a piece or the whole.

So you might just want to look at .9r as a special case of 1. Another description of 1, if you will.

.9r is a safe description of 1 if you are approaching from the left. It might not be a safe description if you approach from the right. However, our equation above proves that .9r IS a reasonable description for 1 because, as a limit, it is equal to a value approaching from the right. The only way that is possible is if this statement is NOT TRUE:
.9r < 1 < 1.0r1

But we just saw that:
.9r = 1.0r1 (in limit form)

THEREFORE, I believe it is safe to assume that .9r = 1

If you can PROVE to me that every standard approach fails using another method or other axioms, then I might believe otherwise, but I have never been shown proof of this.

There is probably one reason why. Higher dimensions are made up of (defined as) an infinite number of the lower dimensions. It is a loose definition, but it is what exists. The only reason there are 3 mathematical dimensions is because there are two other stable dimensions below it to support its structure. Three dimensions underneath if you count the concept of the point.

So someone PLEASE PROVE ME WRONG. I haven't seen ONE SINGLE PROOF for a counter-argument. Never in my life have I seen one. Someone step up to the plate.



posted on Dec, 11 2006 @ 07:24 PM
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Originally posted by dbates
It doesn't equal one but it approaches one as a limit when you carry this out to infinity.


The problem with this thinking is it's not a limit equation. When you say .9 repeating there are implicit in that already an infinite number of nines following the decimal, therefore it is exactly equal to 1.



posted on Dec, 11 2006 @ 09:05 PM
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Lol, it's not what you want to call it - it is a fundamental flaw in mathematics. I don't know if you're ever taken a course in Advanced Calculus or Real Analysis, but this is stuff you should've learned as the fundamental flaw in our mathematical system due to specific axioms.

I know some people who do some research in fiddling with axioms, but I'm not a theoretical math guy. I stick to the practical stuff.



posted on Dec, 11 2006 @ 09:32 PM
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Originally posted by Protector
Lim -2/x = 0
x->oo
OR
Lim 2/x = 0
x->oo

Now the answer IS zero, but I'm sure you're going to ask a fundamental question now. Is the result zero, or is it approaching zero? This subject could be argued for a long time, so obvious limits don't work, but they are yet another way to show that .9r can reasonably be called 1 (like a limit form of an infinitesimal). The difference in this approach is that I have shown that is reasonable to deduce that:
.9r = 1 = 1.0r1


How have you shown anything? You say it could be argued either way, then jump to the conclusion of .9r = 1 = 1.0r1.



posted on Dec, 11 2006 @ 10:48 PM
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Originally posted by Zurahn

Originally posted by Protector
Lim -2/x = 0
x->oo
OR
Lim 2/x = 0
x->oo

Now the answer IS zero, but I'm sure you're going to ask a fundamental question now. Is the result zero, or is it approaching zero? This subject could be argued for a long time, so obvious limits don't work, but they are yet another way to show that .9r can reasonably be called 1 (like a limit form of an infinitesimal). The difference in this approach is that I have shown that is reasonable to deduce that:
.9r = 1 = 1.0r1


How have you shown anything? You say it could be argued either way, then jump to the conclusion of .9r = 1 = 1.0r1.


It shows that approaching 1 from the right is equal (via limits) to approaching from the left. In limits, if the values converge from both sides, then the result is the value converged upon. In this case, both sides converge on 1, so .9r = 1. The problem is that limits are sometimes rejected as a choice method for solving certain problems. They are, in my opinion, a great tool for this kind of problem. You end up in a philosophical discussion without facts as to whether or not the value is approaching 1 or at 1. THAT DOES NOT MEAN THEIR IS NOT EVIDENCE. Merely that most people won't go research it... I'm quite positive you won't. However, my proof was showing that you get the same results when approaching from the right, which isn't possible if:
.9r < 1 < 1.0r1

It doesn't matter if I try to show you a thousand ways. If you don't understand the math and its advantages and limitations, you won't know where to question it and where it is strictly defined as true. Limits are used to define reasonable solutions to problems like these. If you can't give me counter evidence, I'll go on knowing that such methods result in .9r = 1

At this point, everyone is just arguing without showing me proof.



posted on Dec, 11 2006 @ 11:10 PM
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Some people have argued it for 24 pages:
www.volconvo.com...

Or here's an article from Wikipedia:
en.wikipedia.org...

And then their is always Dr. Math:
mathforum.org...



Skepticism in education

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

* Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[12]
* Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[13]
* Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.[14]
* Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount.
* Some students believe that the value of a convergent series is an approximation, not the actual value.


[edit on 11-12-2006 by Protector]



posted on Dec, 12 2006 @ 12:55 AM
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Repeated .9 never equals 1 even though it progressively draws near it. If it did equal 1, it would be so indicated at the beginning. Mathematical rules can be bent and shaped so that it equals 1 or only repeated .9. Linguistic rules state that it never equaled 1 unless these rules are bent and shaped after the first numeral, 1 or .9.

As always, there are two or more proofs based on which assumptions are made.



posted on Dec, 12 2006 @ 02:03 AM
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Originally posted by GreatTech
Repeated .9 never equals 1 even though it progressively draws near it. If it did equal 1, it would be so indicated at the beginning.


Again, .9repeating is exactly equal to 1 just like 1/4 is exactly equal to 2/8, 4/16, etc.

Just like 10/5 is an "improper" or at least awkward to say 2, .9repeating is an awkward way to say 1. There is absolutely not even an infinitessimally small difference between them -- they are one and the same.

If you refuse to believe that, then you have some kind of mental block against a fact that as said above may not be intuitive initially, but is totally true.



posted on Dec, 12 2006 @ 08:52 AM
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Originally posted by Protector

Originally posted by Zurahn

Originally posted by Protector
Lim -2/x = 0
x->oo
OR
Lim 2/x = 0
x->oo

Now the answer IS zero, but I'm sure you're going to ask a fundamental question now. Is the result zero, or is it approaching zero? This subject could be argued for a long time, so obvious limits don't work, but they are yet another way to show that .9r can reasonably be called 1 (like a limit form of an infinitesimal). The difference in this approach is that I have shown that is reasonable to deduce that:
.9r = 1 = 1.0r1


How have you shown anything? You say it could be argued either way, then jump to the conclusion of .9r = 1 = 1.0r1.


It shows that approaching 1 from the right is equal (via limits) to approaching from the left. In limits, if the values converge from both sides, then the result is the value converged upon. In this case, both sides converge on 1, so .9r = 1. The problem is that limits are sometimes rejected as a choice method for solving certain problems. They are, in my opinion, a great tool for this kind of problem. You end up in a philosophical discussion without facts as to whether or not the value is approaching 1 or at 1. THAT DOES NOT MEAN THEIR IS NOT EVIDENCE. Merely that most people won't go research it... I'm quite positive you won't. However, my proof was showing that you get the same results when approaching from the right, which isn't possible if:
.9r < 1 < 1.0r1

It doesn't matter if I try to show you a thousand ways. If you don't understand the math and its advantages and limitations, you won't know where to question it and where it is strictly defined as true. Limits are used to define reasonable solutions to problems like these. If you can't give me counter evidence, I'll go on knowing that such methods result in .9r = 1

At this point, everyone is just arguing without showing me proof.


What your proof came down to was that one side equalled a negative infinitesimal, the other a positive. If you assume them to be zero, then you have proven it. If you don't, you haven't.

[edit on 12-12-2006 by Zurahn]



posted on Dec, 12 2006 @ 08:57 AM
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Originally posted by djohnsto77

Originally posted by GreatTech
Repeated .9 never equals 1 even though it progressively draws near it. If it did equal 1, it would be so indicated at the beginning.


Again, .9repeating is exactly equal to 1 just like 1/4 is exactly equal to 2/8, 4/16, etc.

Just like 10/5 is an "improper" or at least awkward to say 2, .9repeating is an awkward way to say 1. There is absolutely not even an infinitessimally small difference between them -- they are one and the same.

If you refuse to believe that, then you have some kind of mental block against a fact that as said above may not be intuitive initially, but is totally true.


Which is what my original post was for. If there's no infinitesimal difference between the two values, tell me where I've gone wrong in my original post where I showed .9 repeating is equal to 1 subtract an infinitesimal. I'm not insistent that my original post is correct, but if you're going to state something directly to contrary, tell me why mine is contradictory.



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