0.999... is as much an approximation to 1 as 0.333... is to 1/3. That is, not at all, since we use these representations to stand in for the rational number tha

0.999... is as much an approximation to 1 as 0.333... is to 1/3. That is, not at all, since we use these representations to stand in for the rational number that is their limit. You will find that exactly the numbers that can be represented exactly in finite digits (a.k.a. ending in 000...) have two representations.