I'm sorry leo, but 0.3r is equal to 1/3. You can prove it easily with long division and induction.
So either long division is wrong, induction is wrong, or your assertion is wrong. Unfortunately for you, both these methods are provably correct, leaving only your assertion being wrong as the answer.
You say you aren't convinced that a value less than 1 equals 1. Well obviously you aren't, and neither are we. 0.9r isn't less than 1, it is the same as 1. You have been brought up to think that any number with a prefix of 0. must be less than 1. The point is, that is a linguistic mistake, not a mathematical one. The fact is, you can represent the same value in multiple ways in the decimal system. That is all.
So either long division is wrong, induction is wrong, or your assertion is wrong. Unfortunately for you, both these methods are provably correct, leaving only your assertion being wrong as the answer.
You say you aren't convinced that a value less than 1 equals 1. Well obviously you aren't, and neither are we. 0.9r isn't less than 1, it is the same as 1. You have been brought up to think that any number with a prefix of 0. must be less than 1. The point is, that is a linguistic mistake, not a mathematical one. The fact is, you can represent the same value in multiple ways in the decimal system. That is all.
