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What I was trying to say is that if you use basic mathematical operations with infinite numbers, you get impossible results.

Operations with infinite numbers do indeed cause problems - if you extend the ordinary numbers (reals) to include infinity, you break a lot. The point is that 0.99... does not require infinitely large numbers at all.

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If what you are saying is that portions of rational numbers are not possible without infinity concepts, then I see your point.

What I was saying is that the same concepts used to define 0.99... are used to define the set of real numbers. 0.99... is well defined if and only if real numbers in general are well defined.

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However, the algebraic proof I was attempting to refute cannot work without rational numbers. Whenever you use an "X = Y" statement, for the rest of that proof, X can never equal anything other than Y. What I believe the proof is trying to do is say that "X = Y, and Y = Z, therefore, X = Z". I would go with this more than I would go with the way it was formatted.

A statement "x = y" means that the object on the left is the same as the object on the right. The initial step states that x is being used to denote the same object denoted by 0.99...

What the proof shows, in essence, is that x = 0.99... implies x = 1. From this it follows that 0.99... = 1, by the transitivity of equality. The only implicit assumption here is that 0.99... is a real number (whatever real number it may be), as that's required to use ordinary operations on it.

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If the definitions of X and Y were the same, then I would agree with this whole thing. But as far as I know, the definitions of .999~ and 1 are NOT the same... Am I wrong?

The definitions don't need to be the same. The definitions of 2/2 and 1 are different, yet they denote the same object. You can define the same things in many ways.

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My understanding of a need for .999~ was because 1 is not evenly divisible by 3.

There's actually no need for .99... at all. In fact, to avoid confusion, some textbooks explicitly restrict decimal expansions to those not involving repeating 9s, since otherwise you can get two decimal expansions that denote the same number (e.g. 1.00... and 0.99...).

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It's like a language...there's no TRUE translation for 1/3 in the decimal number system, since the number isn't able to be represented by decimals for its true value without using infinity.

There is, in fact, a true translation in decimal form for 1/3. A decimal expansion is defined by assigning a digit to each slot in the decimal expansion. There is a slot for every natural number.

i.e. 1 corresponds to the first decimal place in the expansion.

n corresponds to the nth decimal place.

By assigning 3 to every natural number n, and correspondingly to every decimal place, you've defined a number in the decimal system.

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That being said, I assume that for all intents and purposes, when someone says, ".999~ is the same as 1" it's no different than someone saying, "agape love is the same as unconditional love", even though the meanings are always going to have some infinitesimal difference, but the overwhelming majority of the meanings are so close that it's pointless to discuss the difference between the two.

You can't do that in mathematics. If you say x = y when there is a difference, no matter how trivial or insignificant that difference is, you'll blow your mathematics up. You'll literally be able to derive anything, such as 1 = 2, or jam = square root of your mom.

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I think the definitions of the numbers is the key...if it's what I think it is, the problem with people's opinions is that they're trying to represent the same number in two different numeric systems. Is this the point everyone has been trying to make?

The proof shows that the two definitions (of 0.99... and 1) denote the same number. It's not just different systems though. After all, you can define 1 in decimal notation in another way: 1.00...

These are both infinite decimal expansions.

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612 people, and counting, are retarded.

612 + every single mathematician in the world.