Are all repeating decimals rational numbers

Short Yes. 

I can't think of any that are not. Some go out a pretty long ways before they repeat, but they are still rational. I use a rather limiting definition for rational numbers. If you know which digit you are on, you can tell what the next digit will be. For example, If you have a repeating decimal like
0.2334445523344455... and you are told what the 100th digit is, you will be able to claim you know what the 101st digit is. It may not be a simple calculation, but it can be done. 

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