A basic question about bitcoin's anti double spending mechanism Note: much more basic than: Bitcoin and preventing double spending in decentralized digital currencies

I'm trying to understand the concern that bitcoin's chain of blocks technique tries to solve (see http://www.bitcoin.org/bitcoin.pdf). Warning: not a crypto-dude, be tolerant.

I understand that basically we're worried about "double spending". I'm able to grasp this idea when talking about discrete, indivisible coins. E.g. there are only 3 coins in the world, which cant be divided to cents, and we need to track the history of all 3, and to make sure that each payer really had the coin he claimed to have while paying with it.

While I'm able to see how this concept extends to millions of coins, I can't get how does it extend to fractions of coins. How does a "double spending" attack scenario actually looks like? how do the honest nodes scrutinize a nominated new block, after some node proposes it as the new head node? please provide me with some illustrative examples...
The correct answer was given to me here: http://forum.bitcoin.org/index.php?topic=27979.msg352498#msg352498.

The gist: The entire value of the output contributes to the subsequent transaction, so if you wish to transfer smaller amount, you must add an additional output on the second transaction to send the remainder back to yourself.

Note that the accounting is done for transactions outputs and inputs. The block concept is not necessary for the verification process and is only used for technical convenience.