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#1 Dark Discussions at Cafe Infinity » 1 does not equal .999999..... » 2013-04-08 14:29:47

ke
Replies: 20

The proof submitted:
x=0.999...
10x=9.999...
10x-x=9.999...-0.999...
9x=9
x=1
In the 4th line of the proof we see that from the RHS: 9.999...-0.999... = 9. This is not the case. While it is true that the number of 9s after the decimal point is infinite, it is neither sufficient or trivial to assume this step. For example:
Take an infinitely long integer consisting exclusively of 9s, call this number A. Take second infinitely long integer also consisting exclusively of 9s and call this number B.
A - B = 0
If and only if the number of digits in A and B are the same. Since x has been multiplied by 10, its infinite number of 9s after the decimal point is one less than in x by definition. (sorry to jump in here...but how is one set of INFINITE 9's smaller than another set of INFINITE 9's...by definition, they are the same) It is not sufficient either to say that there are an infinite number of 9s after the decimal place therefore it does not matter, irrespective of whether the repetition is finite or not, the number of digits does need to be the same, and as Mathematicians know, not all infinities are the same. (I laughed. What a joke. Go read some Cantor theorems. infinity +1 is ALWAYS equal to infinity). Therefore the 4th line should say:
That is only true for a finite series.
Say you have two sets, A and B. If A = {0, 1, 2...} and B = {1, 2, 3...) (both infinity), does that mean A has more digits than B simply because it has a zero? No, both sets are equal.
9x = 9 - d
Where d is a infinitesimally small number which is strictly non-zero but limited by zero.
0 is an infinitesimal, and the only one to exist in the real number system
(X=.99999 and X= 1?) how does that work?
You say the number of digits in A and B would not be the same if you multiply x by 10. I'm not entirely sure here, but wouldn't saying that multiplying a number by 10 (or 100, 1000 etc.) moves the decimal place one to the right be a simplified way of expressing the final answer? Doesn't 10 * x = x + x + x + x + x + x + x + x + x + x or x added together 10 times? Therefore, if you add .999... to .999... 10 times, shouldn't you maintain the same number of decimal places? I feel like I'm probably wrong but I'm not seeing where.