Wait what? (1+2+3+4+...

[BAndrew]

First of all, I think it is absurd to suggest that you can have an endless series of 3s and yet not 9s. Furthermore, you can't use traditional arithmetic with nonreal numbers. You can multiply 0.333 by 3 and get 0.999, but who says that you get the same result with a nonterminating number? Regardless, 0.333 is a crude representation of the simple fraction 1/3. Certainly you aren't suggesting that 1/3 * 3 = 2.999…/3 is a valid answer. It makes unnecessary assumptions. But as I said, I will continue to reject your proofs as using faulty methods to evaluate numbers that it cannot process. There are plenty of examples of faulty systems that produce internally consistent but ultimately facile results, best summed up by the expression "garbage in, garbage out" (GIGO). In my opinion it's a loss of data and approximation - acceptable in most cases - but not robust proof. Clearly you disagree and there is no more discussion to be had.

So is 1/3 in fact the exact same as 0.34?

1/3 is not the same as 0,34, but 0,34=0,3399999999999...

Furthermore, you can't use traditional arithmetic with nonreal numbers.

What? 0,999... isn't a real number? Are you serious? 0,999... is a perfectly Real number.

Certainly you aren't suggesting that 1/3 * 3 = 2.999…/3 is a valid answer.

Yes it is. Why not?

But as I said, I will continue to reject your proofs as using faulty methods to evaluate numbers that it cannot process.

What numbers it cannot process exactly? They are real numbers.