"In some circles, it’s decided that eventually 1.99999999999 gets SO close to 2 that it better be using contraception, so for all intents and purposes, 1.9999999999999 = 2."
In other, more mathematical circles, they post annoyingly precise proofs, such as the one below, which recasts your above observations in a more rigorous light.
Let x = 1.9999999... Then 10x = 19.999999...
10x = 19.9999... - x = 1.9999... ---------------- 9x = 18 x = 2
=> 1.99999... = 2 QED
In this case, the nines don't get there, they're just already there. There are limits where we approach something at infinity, but this is not an example. Normally, limits are applied to functions that change over some variable (say, as x changes). In the case of 1.9999..., however, nothing is changing. It's just a graphical representation of a (static) number. Instead of thinking about adding on 9's one after the other (which would be getting closer and closer to 2), you have to accept the mind blowing premise that all of the 9's are already there
( ... )
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In other, more mathematical circles, they post annoyingly precise proofs, such as the one below, which recasts your above observations in a more rigorous light.
Let x = 1.9999999...
Then 10x = 19.999999...
10x = 19.9999...
- x = 1.9999...
----------------
9x = 18
x = 2
=> 1.99999... = 2
QED
In this case, the nines don't get there, they're just already there. There are limits where we approach something at infinity, but this is not an example. Normally, limits are applied to functions that change over some variable (say, as x changes). In the case of 1.9999..., however, nothing is changing. It's just a graphical representation of a (static) number. Instead of thinking about adding on 9's one after the other (which would be getting closer and closer to 2), you have to accept the mind blowing premise that all of the 9's are already there ( ... )
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