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I understand how whole Bitcoins are traded (each coin has its own identifier, this identifier is associated with a wallet), though I do not understand how it is possible to trade parts of a Bitcoin (e.g. I can buy 0.7 BTC from Mt. Gox). How are the Bitcoins tracked once they are split into parts, and associated with different wallets?
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It is not true that individual bitcoins have an identifier. In fact, individual bitcoins do not exist.
What your wallet holds are addresses. These addresses can occur in the Bitcoin block chain, which can - at its simplest - be seen as a big database of balances for each Bitcoin address. Your wallet also holds a private key for each address, which can be seen as the password needed to spend the balance that is accredited to the corresponding address.
When you spend some bitcoins, you send them from one of your addresses to another address. Only the person that owns the private key corresponding to the address, can spend the bitcoins on its balance.
Now, that still not explains how bitcoins can be divided into smaller fractions. In fact, the Bitcoin protocol does not really work with the unit bitcoin, but with a smaller unit, called satoshi. 1 bitcoin equals 100,000,000 (= 10^8) satoshi. So 1 satoshi (=10^-8 BTC) is the smallest amount you can send.
This means that when you send 0.7 BTC, you actually send 70,000,000 satoshis.
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Your understanding of bitcoins is incorrect. A bitcoin is only a number in a ledger, called the blockchain. When someone receives bitcoins, a transaction is added to their ledger which shows them as having received the bitcoins. When they send out bitcoins, a transaction is added to their ledger which shows them as having given away bitcoins.
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Each coin does not have it's own identifier. You create a bitcoin "address" which is just a public key (you hold the private side which is proof that you own the address). You can have an unlimited amount of public keys. At all times, the blockchain contains information that defines how many BTC each address owns. The blockchain is like an accounting journal. Everyone knows how many BTC every address owns at any one time.
Expanding on this, if you want to send BTC to someone, you send ALL of the BTC that address owns. However, you send to more than one recipient, one of those recipients can be yourself.
This is demonstrated in this transaction: http://blockchain.info/tx/b2391755af7382c6643822a52e25620899396cc0dceab0a399033aeafa516310
The address at the left sent 1.6628596 BTC to the first address and 10.57632087 BTC to the second. This is also how you make "change". You send a portion of the owned BTC to the recipient, and the rest to yourself (and this also serves to further anonymize transactions).
Knowing this, you can probably see that you can send a small portion (It's claimed that one BTC is infinitely divisible) out of one address and to another.
Bitcoins are not infinitely divisible. – Steven Roose – 2013-05-14T16:34:22.340
You defined the lower limit in your post, thank you. I was just assuming that there was a practical limit even though it is claimed that there is not: https://en.bitcoin.it/wiki/FAQ. "Bitcoin, however, offers a simple and stylish solution: infinite divisibility. Bitcoins can be divided up and trade into as small of pieces as one wants, so no matter how valuable Bitcoins become, one can trade them in practical quantities."
– abommarito – 2013-05-15T19:46:26.340It's true that the Bitcoin concept supports infinite divisibility. But the current implementation of the protocol does not. The protocol could be changed to allow for higher divisibility, and perhaps that will happen whenever the price becomes to high and a higher divisibility is desired. – Steven Roose – 2013-05-15T20:05:47.197