[Zeke]

Originally Posted by ijdgaf

that kosher?

Yep -- in fact, it's my favourite way of making sense of the seeming paradox. Mathematically, though, the simplest proof is to observe that an infinite decimal expansion is really just shorthand for an infinite series -- in this case 0.9 + 0.09 + 0.009 + ... This has initial term 0.9 and constant ratio 0.1; when you plug them into the formula a/(1-r) for the sum of an infinite geometric series, you get 1.

A looser argument is that given any two numbers which don't have the same value, we can always find a third which is strictly between them. But there's no number bigger than 0.9999... and less than 1.0000... If there were, it would have to either start out like 0.9999... but then differ from it in some decimal place (which would make it less, not greater) or start out like 1.0000... and then differ somewhere (which would make it greater, not less). Thus they can't be different numbers, so they can only be equal.

Gatac: The notation (bar) in the posts above means infinite recurrence.