A strange decimal expansion
Call a number strange if and only if its decimal expansion ends in the infinite string : .
So, for example, the number : , is strange.
Is the set of strange numbers empty,finite, countably infinite or uncountable?
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1 solution
There are clearly infinite, but the goal is to have a well-defined list of them. This can be done by considering the digits before the infinite string begins. We can then have finite lists of strange numbers in increasing order that start at some digit, like the tens or hundredth digit, and then have the infinite string start at some other defined place. Thus, the trick is to find some well-defined way to combine these lists using those two numbers, which define each list. This is analagous to finding a way to list rational numbers, which are also defined by 2 integers. Since rational numbers are countably infinite, strange numbers must be as well.