I disagree that 0.9999999999... is equal to the integer 1. All integers are understood to be exactly 1 or exactly 2, etc, as if they had an infinite number of zeros after a decimel point. Of course, they are practically the same in any conceivable everyday context, but they differ by that tiny amount. For example, 1/(1-1) is not defined because one cannot divide by zero. 1/(1-0.9999999.....) approaches infinity asymptotically the more nines you tack on at the end.

This is basic calculus. You'd express the above as the limit of 1/(1-x) as x approaches 1 from zero.