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October 14, 2009 at 2:13 pm (This post was last modified: October 14, 2009 at 2:27 pm by Violet.)
To skip to the most important part (The part I think you will understand), and not argue the same thing continuously [and mistaken as a strawman?]: please jump to the next bold. Other text hidden for convenience of reader.
Quote:Why are you bringing infinity into it? This is not a discussion of infinity as a value (and I am well aware of the arguments that infinity is not a value...I agree wholeheartedly with them). Infinity is completely different from infinitely long numbers though. An infinitely long number doesn't equal infinity, as it could be 1.999999... (infinitely long string of 9's), making the number equal to 2 (or for sake of your argument, "just less" than 2).
Infinity is a concept tied very much to 1.9^. Infinity is an uncertain value... because of its endless nature. Until you remove the uncertainty from 1.9^... you cannot calculate with certainty. Infinity is not a number (value/whatever), it is a concept... and it is a concept present within 1.9^ that makes the number itself incalculable (because you have a value under the effects of an infinite concept, which makes the number undefined.).
If one has tied an infinite concept into a value, as one can see with .9^: then one has made the value of this number dependent upon infinity. The value of .9^ is based within something without calculable value. I thought I should bring this up before I disproved this 'equality'.
Quote:An infinitesimal is a number that cannot be measured because it is "so" small. It cannot be distinguished from 0. Your number 0.0^1 is not an infinitesimal number for two reasons:
1) You cannot logically or mathematically have an infinite string of 0's (as denoted by your 0^) which is then followed by a 1. We have been over this. An infinite string has no end, so you have no place to put the 1 on.
2) Even if this number were to exist, it is easily distinguishable from 0, since it is 0.0^1 away from 0. Ergo it does not have the attributes of an infinitesimal number.
An infinitesimal is 'the infinith' number of infinity. By the very definition of it, 0.0^1 is the definition of the infinitesimal. It is no easier to distinguish from 0 than .9^ is from 1. They are an equivalent distance away from 0 and 1. The reason infinitesimals exist is because the infinith number exists. Infinity is like an endless gap... between two pieces of land infinitely far away from each other (And that land exists, just so much as that gap exists). .9^12343 is no different in this concept. If you remove the infinite from the value: you can calculate that 12343. That is why such numbers are able to exist, assuming infinites can be calculated at all.
Quote:If you don't like Wikipedia, I suggest you don't use it to try and argue about infinitesimals. If you want other sources, I suggest you use Google:
I didn't say that I don't like wikipedia... I said it is not always a reliable source. If mathematicians truly think this, then they are incorrect. So far as I can tell, and I have seen no reason to think otherwise: these 'proofs' are nothing but errors of mathematics.
Quote:There was no infinity in the calculation. There was an infinitely long string. There is a difference; I have been over this time and time again. I subtracted an infinitely long string of 9's from an infinitely long string of 9's. Both infinitely long strings are exactly the same thing, therefore subtracting one from the other leaves nothing.
You can think about it like this:
9.9 - 0.9 = 9
9.99 - 0.99 = 9
9.999 - 0.999 = 9
etc, etc.
No matter how many 9's you stick on the end, as long as the same amount goes onto the end of the other number, the answer will be the same.
This is exactly your error in proving this inequality is equal. You have removed the infinite from the equation. An infinitely long string of numbers is infinite. It is undefinable, and it is endlessly long (thus endlessly gargantuan). Is a hot fire not fire? Then is something infinitely long not infinite? (The Identity's Equality). Not at all. The concept is exactly the same. In 'infinitely long' you are describing the length as being infinite. And here again you are tying an infinite into the definition (and thus value) of something. That's what is being done here .0^1 & .9^
This is precisely why these 'proofs' fail to prove what they are trying to do: they are removing the infinite from the equation to prove it. That is fallacious reasoning to me. And some people do not believe that one even can remove an infinite.
Quote:Yes, I'm canceling out the *infinitely long number* (not infinity...again) because subtracting it from itself returns 0. This is basic math, basic algebra. Of course it's a new value. If you subtract something from itself, you get 0, which is a new number.
And thus you are no longer proving that the infinitely long number is equal to a finite number... because you have cancelled out specifically that which you are trying to prove. I do not deny that this is basic math... that it is simple basic math that 1 ≠ .9^ is my point entirely in fact.
Quote:Erm...2*9 is equal to 18. I'm not sure why you think it isn't...
Exactly, it is not any different from the other answer (I was asking a question of you, not stating a fact [which I thought was obvious, but must have been mistaken]). However, this is?:
Quote:I'm sorry, but if you set x to 2.2^, then 10x - x does indeed equal 9x.
x = 2.2^
10x = 22.2^
10x - x = 22.2^ - 2.2^
9x = 20
x = 2.2^ (20/9)
As I've said before, this type of calculation works for infinitely long strings of 9s, since there are no gaps between this number and the decimal value "above" it.
Umm... 2.2^x9 = 19.9^8. which is ≠ to 20 (even further from equality than .9^ is from 1). In one of the equations, the infinitely long number remains. In the other: it has been neutralized. This discrepancy between otherwise equivalent methods is because an infinite is an undefined, and until you remove the undefined: you cannot have a defined. In one method it was removed, and the finite answer of 20 is available. In the other: it was not removed, and 19.9^8 is the closest approximate we can get. It remains undefined.
In none of these infinite proofs, will you be able to prove that the undefined = a defined: because it is not possible... by definition. If you found them to be equal, then you would not have an undefined in the first place.
Quote:Once again, the number 8.9^1 does not exist. It is a logical impossibility, a mathematical impossibility, and a verbal impossibility. You cannot have an infinitely long string of 9s (or any number for that matter) and then put a 1 (or any number for that matter) on the end, simply because there is no end for that number to go.
Citing my gapped land example above: yes, you can. You can have an infinitely long value separating other values. There is no impossibility about it. An infinitesimal is infinitely small number, which can easily be represented by 0.0^1... infinite zeros to show that it is infinitely small... and a 1 for the infinith digit to show that it is different than zero. Once again: nothing impossible about it.
Of course, an infinite tied into a number usually makes the number irreparably uncertain... simply because of the nature of the concept. This is once again because the number is nonfinite, and impossible to define because of such. Arguing that .9^=1 is an infinitesimal (Representable as 0.0^1) distance away from one... but still the distance is there. .9^ is infinite, and is not a finite. You cannot write a finite number infinitely... because to do so immediately revokes the finiteness of the finite.
Quote:I don't expect you to accept the proof of an infinite equaling a finite, but that's because nobody here is arguing that. We are not talking about infinitely large numbers, we are talking about infinitely long numbers. You continually make this strawman (whether intentionally or not), and it does nothing to help your argument.
Good that you don't then Infinite is infinite. Wether in length, width, height, general size, or any other attribute. It is a concept applied to another attribute, often to denote that attribute's endlessness and often it's indefiniteness. This is not a strawman, it is simply The Identity's Equality, that something is itself, and is not something else.
(October 14, 2009 at 1:38 pm)Tiberius Wrote:
(October 14, 2009 at 12:40 pm)Ephrium Wrote: I have the BEST answer yet. It steps side all the argument about infinity here.
It is no secret that you cannot represent all fractions as rational numbers. What this means is that you cannot represent accurately by our base ten system that which can easily be represented by a fraction.
Simply, for some numbers (ie 1/7, 1/3, 1/574365972843.3), one can only attain a degree of accuracy... in other words: cannot write the number in a base ten form as any more than an approximation.
The decimal .3^ is our best estimate of the value of 1/3, simply because we cannot perfectly split our base ten system into 3s, or 7s, or a lot of things really. But it would be to presume too far that .3^ is equal to either .34 or .3... and in either case one is rounding.
October 14, 2009 at 2:47 pm (This post was last modified: October 14, 2009 at 2:48 pm by Retorth.)
Sae,
Your issue:
0.99999999999999999999999999999999999999(infinite9) cannot = 1 because there should be a 0.0000000000000000000000000000000000000000000000000000000000001 somewhere at the end.
I believe this is accurately your issue yes?
While it makes sense that 0.9 can never equal 1 because 0.1 is needed, what Adrian, and I think pretty much everyone else on this thread, is trying to show you is that the string of .999999999(infinite9)'s can be subtracted off by itself. Instead of looking solely at your issue above that 0.00001 is needed to fill the "gap", look at the string subtracting itself. This effectively makes your case for argument redundant if I may say so.
As Adrian pointed out earlier:
9.9 - 0.9 = 9
9.99 - 0.99 = 9
9.999 - 0.999 = 9
and so on and so forth.
If its 9.999999(infinite9) - 0.9999999(infinite9), it will still be 9 because the decimal placing is infinite which makes it both the same length so it subtracts itself off making it a non-issue and leaving you with a simple, sexy 9.
The dark side awaits YOU...AngryAtheism "Only the dead have seen the end of war..." - Plato “Those who wish to base their morality literally on the Bible have either not read it or not understood it...” - Richard Dawkins
And thus you have removed the infinite from the equation, and are left with a simple (sexy? lol ^_^) finite 9. I explained that (I thought?) in the hidden part of my post
However, looking at this same conceptual understanding of infinity:
.9=1
.99=1
.999=1
And on and so forth?
When I used that way to envision infinite .9's ≠ 1 a few pages ago, others called it an inaccurate definition because I was cutting the number off at some point.
and so on and so forth but these are finite numbers. Thats the beauty of "infinite"
.99999(inifinite9) = 1
Only because the (infinite) decimal place can be successfully subtracted from itself. This will never work for those examples above no matter how long the string of 9's.
The dark side awaits YOU...AngryAtheism "Only the dead have seen the end of war..." - Plato “Those who wish to base their morality literally on the Bible have either not read it or not understood it...” - Richard Dawkins
Quote:Infinite is infinite. Wether in length, width, height, general size, or any other attribute. It is a concept applied to another attribute, often to denote that attribute's endlessness and often it's indefiniteness. This is not a strawman, it is simply The Identity's Equality, that something is itself, and is not something else.
Calling an infinite number = to a finite number is ridiculous. .9^=1 is to declare an infinite equal to a finite, which it is not. In all of these 'proofs', the infinite must be canceled out in some way before the number has any possibility of equaling a finite value.
Also, oo - oo = 0, there is no disagreement from me (though there is opposition from at least some individuals to the idea that you can subtract or add infinites [Which to me makes no sense that they can multiply them and divide them instead? Though it would seem there is opposition to that as well {IE, my math teacher, who is currently getting her masters}]).
Anyway, by canceling out the undefined, you can solve otherwise impossible problems. However, it should be noted that you have in the process changed what you were solving.
I don't quite understand what you mean. In what is being used to prove the calculation, the infinite is being removed by itself (another infinite).
9.9999999999(infite9) - 0.99999999999(infinite9)
The .9999999999999(infinite9) cancels itself out to leave you with 9. I don't get whats so hard to understand lol If its a finite number then, again, doing !0x - x won't work. However, with 0.9(infinite9), no matter how many times you multiply it, the decimal still remains 0.9(infinite9). That is, as I said, the beauty of infinite. We aren't using a finite number in correlation to the infinite number.
The dark side awaits YOU...AngryAtheism "Only the dead have seen the end of war..." - Plato “Those who wish to base their morality literally on the Bible have either not read it or not understood it...” - Richard Dawkins
I don't disagree... but now you've completely confused me as to what you're trying to say ^_^ lol ^_^
Yes, we are canceling out the infinite. If you do not do that: then .9^ will never equal 1. I don't understand what's not to get, and now I'm all confuzzeled >_^
Infinite is infinite. Wether in length, width, height, general size, or any other attribute. It is a concept applied to another attribute, often to denote that attribute's endlessness and often it's indefiniteness. This is not a strawman, it is simply The Identity's Equality, that something is itself, and is not something else.
