Am I in Wonderland??
Man. Grab any advanced calculus or abstract algebra book before explain anything.
0.999.... is not a number. It's just an infinite geometric series representation.
0.9999... means Sum[from n = 1 to infinity] (9/power(10,n))
(sorry, lack of proper tool to show a formula)
For example, 0.999... = 9/10 + 9/100 + 9/1000 + 9/10000 + ... and keep going. And if you know infinite sum of geometric
series, you can fiqure out, [initial term] = 9/10, [common ratio] = 1/10
So, Sum = (9/10) / (1 - 1/10) = (9/10) / (9/10) = ***1***.
It's something like asking WHY "2+1 = 3"? because "2+1" is not "3" (literally! 2+1 has THREE characters, 3 has ONE character).
By asking 0.9999.... = ?, what you are really seeking is an ANSWER for that series, which is ONE.
Any *infinitely repeating* decimal point representation is just a series representation, which could have an answer (like 0.999...) or not. (like Phi = 3.1415.....???, e = 2.71...??? we just assign symbol Phi or e to REPRESENT them.) When I say "answer", I mean in "RATIONAL" numbers (at least in this decimal point case).
It's merely the way we represent Numbers. No significant meaning. PLEASE DO NOT SPEND your life on such questions. I've seen many so called "amature" mathematicians (yes, I'm a professonal mathematician), who are spending so much time in meaningless questions. (like "how to divide arbitrary angle into three equal angles, using only rulers, etc, etc... which have been solved (disproved) hundreds years ago, but hard to explain to somebody who has no professional math knowledge. Have to use Galois Theory, which is very last part of abstract algebra courses). Do not confuse "disproof" with "misunderstanding".
Actually, whole confussion is from the concept of "limit" which most of ppl do not understand properly. Even mathematician confuses sometimes. If you ever learn advanced calculus II or Real Analysis in college, you would remember epsilon-delta definition of limit. Which is defined by double negations (not-not) which does not have any easier way to understand. 0.999.... is not just bunch of numbers. It's a "processing" which keeps going into 1.
Wanna go further? Yes. When people *invented* number system, the first one was Whole (natural) Numbers (obviously).
0, 1, 2, 3, 4.... (ok. Zero is not in the first place but...)
Then when we were trying to solve x + 3 = 0. Oops, no answer. So they invented whole system of negative numbers.
-2, -1, 0, 1, 2,... which make INTEGERs together with whole numbers.
Now, we have bunch of them but still we cannot solve some a*x - b = 0 (1st degree equation) when a, b is interger,
i.e. 2x + 3 = 0. So there goes RATIONAL NUMBERs. (so that x = b / a, obviously a != 0), which means "RATIO" between two integer numbers. (So, "RATIO"nal)
But we still may not have any answer in higher degree polynomials. For example, x^2 = 3... what's x?? no such rational number.
So, we considered *REAL NUMBERs*.
Real Number is not easy thing. There are several way to define it, and one thing is that real numbers are a family of limit points of rational sequences. 0.9999.... is one of such expression (representation).
Mathematically, integers are called "group" ('cause they have an identity (0) and inverse for plus), rational numbers and real numbers are "field" (commutative division ring, which means have two operations (+,*), two identities (0,1), and inverse for both operations and commutative.)
Even though Real Numbers form a field, it's not closed one in the sense that we cannot solve arbitrary polynomial whose coefficients are in real numbers. For instance, there's no answer for this simple quadratic x^2 + 1 = 0 in real field. That's why we are using whole concept of "COMPLEX" numbers. (I saw somebody's mentioning "NON-REAL Numbers". I've studied a math like 20 years and got a PhD in Math, but never seen such term as "NON-Real". Maybe "complex"?)
Complex is the first closed field (algebraic closure) in number system (oh well at least for infinite field). Any degree of polynomials in complex has an answer in it. But still we could have series of complex numbers, in which point we could introduce new number system... but it's beyond scope, I guess.
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