[split] 0.999... equals 1

[Tiberius]

Edit: I should note before i begin, that there are some people who do not think you can add or subtract an infinite a few math teachers I know personally [and discussed the concept of infinity with] among them. Even the concept of infinity is not accepted by every mathematician as having the same definition or even in being possible... so how could all mathematicians be on board with the idea of .9^=1? Be aware that not everyone even agrees on infinity's concept in math... let alone on a specific concept. For the full context of the wikiquote: http://en.wikipedia.org/wiki/Infinity

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).

I did read it... and I do not see how it is equal to 1.8. The point of the article is simply this: .9^ approaches 1... but never reaches it. It is http://en.wikipedia.org/wiki/Infintesimal far away from 1. As you can see, infinitesimals exist as much as infinity... therefore my use of 0.0^1 (An infinitesimal number)... which you stated does not exist: essentially exists as much as .9^, which is an infinitesimal distance from one. Please do not insult me by suggesting I did not read the articles I used as evidence... esp. since I found them to be rather fascinating (therefore insulting what I find fascinating).

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.

Also, I do not think wikipedia is an entirely trustable source... http://en.wikipedia.org/wiki/Wikipedia#Reliability. So yes, I think it is possible that a disproof might not be mentioned. I was not convinced by these pages of proofs, and so I will actively hunt down each one and disprove it. In previous posts, i just suggested that the notion of this inequality being equal was laughable... now I am giving evidence to back up my previous laughter.

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:

http://www.google.co.uk/search?hl=en&q=0...rt=10&sa=N

Or you could expand the search so only universities show up:

http://www.google.co.uk/search?hl=en&q=0...earch&meta=
http://www.google.co.uk/search?hl=en&q=0...earch&meta=

If you want to argue this with mathematicians, then be my guest, but this is taught in all higher level mathematics courses, before and during university level. None of the proofs have ever been disproven, since to do so would be to toss algebra, calculus, and other elements of mathematics in the bin. Be my guest though...

I'll start with the first one.

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No it doesn't. 1 is provably 0.999... (reoccuring)

Here is my favourite proof:

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

So x = 0.999... = 1

Fine the error in my math if you want, but you won't since this thing has been known by mathematicians for years

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You removed the infinite from the equation when you subtracted it from itself. Essentially, you have cancelled out the effect of infinity.

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.

Essentially... this 'proof' is invalid, because you have changed the value of X by canceling out the effect of infinity upon the number. As in, you are no longer proving that the first value of X (.9^)is equal to one... but that the new value (1) is. You changed the value of X when you canceled out part of the equation. Simply, 1-1=0... just as infinity - infinity=0. You negated its effect in your calculation, so that you could solve it without infinity's effect.

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.

You accept that 2*10=20, 20-2=18, no? Then how come when we take 2*9, it is ≠ 18? Essentially, that is what is being done in this 'proof'. Registering different answers for 20-2 and 2*9 is a fallacy under finite circumstance... and is only (and according to some people, not even then) possible when eliminating an infinite or infinitesimal or other undefined number.

Erm...2*9 is equal to 18. I'm not sure why you think it isn't...

For example, 10x2.2^-2.2^ would not be equal to 9x2.2^.

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.

x=.9^
9x=8.9^1
8.9^1/9= .9^

Once you nullify the infinite part of x, then you are no longer proving .9^. You are proving a finite by canceling out the infinite. By doing so, you get a different answer for 10x-x, and for simply 9x. If this were a finite number to begin with: you would not get different results from those two normally equivalent expressions. I do not accept this 'proof' of an infinite equalling a finite... for it is only by the canceling of the infinite concept that the inequality can 'solved'.

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.

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.