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1-0.999.... = 0.0000......1
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Incorrect, there is no termination to the result, you cant do "00...1" on the end.
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Even imaginary numbers are bigger than 1-.9r and they don't exist (or so we've been told).
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Imaginary numbers "exist" to the same extent that Real numbers "exist". Further more there is no orderedness in complex (imaginary) numbers compared to purely real numbers, so you cant really say they are "bigger" without using a different metric |a+b
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It doesn't matter if it is infinitely small or infinitely large, it still isn't equal to zero. It's like if you have to move an object 1 foot. You start by moving it half a foot. Then 1/4 ft, then 1/8 ft..... You will never move it the full 1ft, even if there is no way to measure the distance between the object and the 1ft barrier, there is still distance to be covered.
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Firstly, you are getting confused with the partial sum and the sum itself, its true to say that
for any N one chooses, but the fact is that the sum to infinity does equal 1.
So if you mean by "infinitely large" you mean the sum to infinity then you are wrong and the sum does equal 1. However if you mean by "infinitely large" "for any choice of N, the sum is less than 1 then you are correct. Its not clear which one you ment.
The problem occurs where you say "..it still isn't equal to zero." i presume you mean the result of 1-0.9r. Well this does equal zero, as it is the sum to infinity :
not a partial sum to N.