In the text TAOCP
A real number is a quantity x that has a decimal expansion $$x = n + 0.d_1d_2d_3...$$ where n is an integer, each $d_i$ is a digit between 0 and 9, and the sequence of digits doesn't end with infinitely many 9s
Why is such a case ? Isn't all the numbers between 0 and 1 are all real ? Then the numbers which ends with infinitely many 9s should be real too, Isn't it ?