I know that mathematics says that $0.99999....=1$. but some say that logic says that $0.99999....<1$. what I want to know, is weather or not there is logical proof (proof without formulas, and that relies solidly on common sense ) that $0.99999....$ must be equal to $1$.
I have heard great arguments for both sides, but they all seem to end with logic and math disagreeing. I want answers that are insightful, and make math and logic agree with each other.
OK a lot of people are asking what the logical proof is, well a few of the ones that I have heard of so far are: If you move a ninth of distance to your destination then a ninth of whats left, then a ninth of whats left after that, and so on, and so on, you will never get there. Another logical question that comes up is: How can an infinite sum be equal to any non fractional whole number, isn't it never ending? can you help me explain why these ideas are flawed without using math?
Also if you'r wondering what I mean when say logical proof, I mean proof by logical deduction with as little use of mathematical formulas as possible, in other words I want proof using common sense. all of the other proofs I have seen are great and fallow the laws of mathematics perfectly, but their are some people that agree with their own common sense more than math. I want an answer I can give to them. so the less formulas the better, try to rely on firm logic.