1. The problem statement, all variables and given/known data How many decimal expansions terminate in an infinite string of 9's? How many dont? 3. The attempt at a solution If we have a number terminate with an infinite amount of 9's then it will be a rational number. So there would be countably many of these. And since irrational numbers do not end with all 9's then their would be uncountably many of them that do not.