The question of whether 0.9999…. is equal to 1 is, and probably always will be, a source of heated disagreement among people who know a certain amount of math, but not too much. High school math teacher Polymathematics delivered a magisterial series of posts on this question last year, which covers with admirable thoroughness every one of the many, many strange trails this argument likes to wander down. So I’ll leave that to him, and just use the question as an excuse to copy in one of my favorite quotes from G.H. Hardy, from his 1948 book Divergent Series:
“…it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition: it was not natural to them to say, in so many words, `by X we mean Y.’ … it is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define
1 – 1 + 1 – 1 + …
‘What is 1 – 1 + 1 – 1 + …
and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.”
The connection to 0.9999… is quite direct. To ask what 0.9999… is is to miss the point. Rather, you ought to ask “How can we choose a real number which deserves to be called ‘the value of 0.9999…?'” And once you have done this, you realize you are not at all sure what the definition of a real number is … and before long, you’ve learned the first few weeks’ material of a course in real analysis and all confusion has departed.