The vast majority of users will likely only run a 'lightweight' client, which can verify transactions made sent to itself by downloading only the relevant transactions via the network in real time, keeping only the block headers (80 bytes each) on local storage. The lightweight client neither needs the entire blockchain, nor all of the transactions, in order to function well. Another way to look at is that the blockchain is a massive, distributed ledger of the entire history of the system; and generation is the automatic equivalent of a notary public (or a bank) that substitutes for the trusted third party normally required in everyday credit based transactions. You don't need either a notary or a bank to by a hamburger at McD's with cash, the cashier only needs to look at the paper and judge that it's *very likely* that said paper is actually cash. The same is true with Bitcoin, if someone sends you bitcoins by creating a transaction and sending that to your client, the client can verify that the sender legitimately owned those coins at one time; and usually that is enough (depending on how well you trust the counterparty) but if it isn't then the blockchain will let you know with ever increasing confidence every ten minutes. With each new block, it becomes increasing unlikely that a 'double spend' or other fraud was committed against you as a seller, until that likelyhood crosses the "astronomicly unlikely" point and "matures" in the client. Bitcoins that are not yet mature can still be sent, but the receiver has a lower mathmatical certainty that they will mature for him, and so the client usually sends the most mature coins first. The maturity level of transactions/coins also positively affects the priority system for block inclusion (with regard to free transactions, a volutary fee can be added to a transaction to bump up the priority) , so there is an incentive for users to favor using the most mature coins that they possess.
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