>>12911619Algebraic manipulation isn't a valid proof given the circumstances. You could show stuff like 1-1+1-1+1-1...=0.5 using the same approach. Moreover, and I know this will trigger alot of yall, but mathematics isn't objective fact, it's a set of arbitrary rules that we define for preforming logical operations. It's purely abstract, it has no basis in reality. We could define 0.999... to not equal 1 if we wanted to. Of course, whether that's a useful definition is another question, but these kinds of semantics games get us nowhere.
The truth is that we have two symbols 1 an 0.999... that represents two distict ideas. Whether we choose to define these symbols as being interchangable depends on context. If you ask me, 0.999... as a symbol refers to the largest possible number smaller than one. In other words, it by definition is not equal to one. Though that concept doesn't make much sense within the confines of regular number theory, you would have to pull up something like surreal numbers for that distinction to even make sense.
