Quote:
Originally Posted by seiki
But damn I can not believe .9999999 = 1 can be proven with a geometric sequence.
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You really should be writing that with "..." to show that you mean an infinitely repeating decimal, i.e. .999... = 1. When I first saw this thread I thought you were going to mention some example of really bad roundoff error.
But if you think about it a bit, it does seem intuitively reasonable that 0.999... = 1. Since you can approximate 0.999... as close to 1 as you like simply by listing enough 9s after the decimal, there's no reason to suspect that the infinitely repeating decimal is not the same as 1.
And here's another way to think about it, if you remember that 1/3 has the infinitely repeating decimal 0.333.... Well, obviously 3 * 1/3 must equal 1. But if you take the repeating decimal representation of 1/3 and multiply that by 3, you would get 0.999.... Therefore 0.999.... must be equal to 1.
I'm sure the geometric sequence is a much more rigorous way to demonstrate the equality, though.