If you go to college and take a Discrete Math course, you're likely to be introduced to this proof. I had seen it in high school and didn't really believe it because of the "logical" problems with it. Try to turn your intuition off for a moment; if you can't do that, you really don't have much hope in higher level math anyway.
My professor explained it after he showed the proof in a way I hadn't heard before, but helped me understand a lot better.
.(9) does, in fact = 1, here's the thing
No one would ever dispute the fact that 1/3 = .(3) however, you're talking about two numbers living in totally different worlds. 1/3 lives in Q (rational numbers), .(3) lives in R (real numbers). These are two ways, in different number sets to express the same idea (after all, numbers are simply ideas). Likewise, we have 1 (which lives in N (natural numbers), or Z, Q, or R for that matter) and .(9) which lives in R (real numbers).
1/3 + 1/3 + 1/3 does, in fact, = 1 You aren't going to dispute that, because it is fact.
.(3) + .(3) + .(3) does, in fact, =.(9) You aren't going to dispute that, because it is fact.
You can always substitute x for x. 1/3 + 1/3 + 1/3 = 1/3 + 1/3 + 1/3 = .(3) + .(3) + .(3)... 1 = .(9)
You're thinking about this in absolute terms when numbers are not absolute. We have different ways of representing many numbers. Because they are in different number sets we perceive them differently; they are not different in the slightest.
