There’s a simple algorithm I learned in 7th grade for converting, into a fraction, any decimal that infinitely repeats a finite group of digits.
Applied here: Let x = 0.6666666…
Then 10x = 6.6666666…
Now subtract x from 10x and the infinite strings go away:
9x = 6
x = 6/9 = ⅔
Bulletproof!
@Dutchess_III As you & @LuckyGuy pointed out the standard notation is either dots or a horizontal bar over the final digit(s). You could end the number with an ellipsis ( ... ) but then it may be ambiguous as to which digits are repeating. I’ve not seen any typographical conventions for rendering this at a keyboard—the situation doesn’t arise very often.
It’s customary in approximating the fraction ⅔ by a decimal string of 6’s to terminate the string with a 7, thus rounding off to the appropriate precision, since ⅔ closer to .7 than to .6.
On a related note, the value of 0.999… where an infinite string of 9’s follows the decimal point, exactly equals one; it’s just another representation of what we usually symbolize by “1”. Ref.…