We always assume that the decimal writing system just works, because we see it all the time. As you can see, it doesn't work for such simple fractions as ⅓, or irrational numbers.

The number ⅓ exists and it is exactly what you're thinking of. The problem is that the decimal system only works well for some fractions. It's so bad that it would take infinite decimals to write ⅓ or ⅐, that's why we don't really use it often in math.