Comment by tsimionescu

The way I was taught decimals in school (in Romania) always made 0.99... seem like an absurdity to me: we were always taught that fractions are the "real" representation of rational numbers, and decimal notation is just a shorthand. Doing arithmetic with decimal numbers was seen as suspect, and never allowed for decimals with infinite expansions. So, for example, if a test asked you to calculate 2 × 0.2222... [which we notated as 2 × 0,(2)], then the right solution was to expand it:

  2 × 0.2222...
   = 2 × 2/9 
   = 4/9 
   = 0.444...

Once you're taught that this is how the numbers work, it's easy(ish) to accept that 0.999... is just a notational trick. At the very least, you're "immune" to certain legit-looking operations, like

  0.33... + 0.66...
    = 1/3 + 2/3
    = 3/3
    = 1

Instead of

  0.33... + 0.66...
    = 0.99...

So, in this view, 0.3 or 0.333... are not numbers in the proper sense, they're just a convenient notation for 3/10 and 1/3 respectively. And there simply is no number whose notation would be 0.999..., it's just an abuse of the decimal notation.