ambiguity from powers of 10 in denominator I am reading the book Introduction to Calculus and Analysis by Richard Courant, and he is discussing decimal fractions (infinite and otherwise). In this discussion he discusses the ambiguity between say $3.999\ldots$ and $4.000\ldots$ since they are same number. Either representation could have been found from the same process of nested sequences of intervals.
Then he says:

Hence the only case in which an ambiguity can arise is for a rational
  number x which can be written as a fraction having a power of ten
  for its denominator. We can eliminate even this ambiguity by excluding
  decimal representations in which all digits from a certain point on are
  nines.

I don't quite get this. What is special about numbers which can be represented with a power of 10 in the denominator? How/why are these the only ones which can have ambiguous representations?
 A: This is because there are two provably equivalent decimal (base-$10$) representations of $1$ viz. $0.9999... = 0.\overline 9$ and $1$
Note that $1$ is also a rational with a denominator that is a power of $10$ because $1 = \frac 1{10^0}$.
In fact, all whole numbers (integers) and all rational numbers with a terminating decimal representation can be represented as a fraction with a denominator that is a power of $10$, e.g. $0.375 = \frac 38 = \frac{375}{10^3}$.
No such representation can be made for non-terminating decimal numbers, even if they are rational. e.g. $0.333... = 0.\overline 3 = \frac 13$ (no representation with a power of $10$ as denominator can be done).
Every terminating decimal can be represented in two ways. The first way is the "usual" way, the second way is with the trailing digit decremented by one and an infinite string of nines appended to its right.
e.g. $0.375 = 0.374999... = 0.374\overline 9$
This creates an ambiguity in representation. It is not usually an issue (except in naive understanding issues such as people not "getting" that $0.\overline 9 = 1$ and wondering what lies between), except when you are considering particular questions like the cardinality of certain sets such as the rationals. Then the issue of two possible representations of a subset of such numbers may become an issue.
But the issue is easy to eliminate, simply by defining a convention that you only allow one of these representations. So we can say that $\frac 38$ is only "allowed" to be represented as $0.375$ but not as $0.374\overline 9$
That, I believe, is what the author is trying to say here.
Note that there is nothing special about decimal base (base-$10$), the same issue crops up in any natural number base, e.g. in binary (base-$2$), $0.1111..._2 = 0.\overline 1_2 = 1_2$. Here, note that the "ambiguous" representation involves an infinite trailing string of the bit $1$. You should be able to deduce that this is one less than the base used in representation ($2 - 1 = 1$), exactly analogous to what happened in decimal base ($10 - 1 = 9$).
