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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 :-)
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]
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.
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.
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.
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...
Originally posted by dbates
It doesn't equal one but it approaches one as a limit when you carry this out to infinity.
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
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.
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.
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.
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.
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.