Originally Posted by Amenity
Uhm...wow. 
*deep breath*
What's 0.[0]?
Think about that.
1. No.
2. Ok, you can say that, yeah. What's the difference here? The repeating portion is zero. You're saying 0.[0] is the same as 0.[1], 0.[2], 0.[3], etc etc...
This isn't that hard, guys. You can try to convince yourself in every way imaginable...but 0.[9] is NOT one, and 0.0000......1 is NOT 0. |
0.[0]
is the same general idea as 0.[1] and 0.[2], etc. Without the ability to use infinite repeating sequences after the decimal point, we limit the decimal numbering system to only being able to accurately represent numbers that evenly divide into whatever number we choose as the base (10, in this case). At some point, mathematicians decided that having to deal with infinite repeating sections after the decimal point was worth the hassle—as a result, we have a decimal numbering system that is well defined for all rational numbers. What's even better is that absolutely nothing breaks if we consider 0.[3] exactly equal to 1/3, or 0.[9] exactly equal to 1.
Sure, we could just as easily define the decimal numbering system in such a way as to only be exact when we're talking about nice, neat numbers like 1/2 or 1/5. But then we've have to start throwing around the ≈ symbol around a lot more.
There are multiple potential ways to define the decimal numbering system, and the math world in general happened to pick one that has a weird consequence. We could've defined it differently, but it's still a definitional issue, not an absolute rule of the universe.