Majority consensus in bitcoin is represented by the longest chain, which required the greatest amount of effort to produce. If a majority of computing power is controlled by honest nodes, the honest chain will grow fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of that block and all blocks after it and then surpass the work of the honest nodes. The probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added.
The overwhelming majority of bitcoin transactions take place on a cryptocurrency exchange, rather than being used in transactions with merchants. Delays processing payments through the blockchain of about ten minutes make bitcoin use very difficult in a retail setting. Prices are not usually quoted in units of bitcoin and many trades involve one, or sometimes two, conversions into conventional currencies. Merchants that do accept bitcoin payments may use payment service providers to perform the conversions.
Early Bitcoin client versions allowed users to use their CPUs to mine. The advent of GPU mining made CPU mining financially unwise as the hashrate of the network grew to such a degree that the amount of bitcoins produced by CPU mining became lower than the cost of power to operate a CPU. The option was therefore removed from the core Bitcoin client's user interface.
How do they find this number? By guessing at random. The hash function makes it impossible to predict what the output will be. So, miners guess the mystery number and apply the hash function to the combination of that guessed number and the data in the block. The resulting hash has to start with a pre-established number of zeroes. There's no way of knowing which number will work, because two consecutive integers will give wildly varying results. What's more, there may be several nonces that produce the desired result, or there may be none (in which case the miners keep trying, but with a different block configuration).
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Computing power is often bundled together or "pooled" to reduce variance in miner income. Individual mining rigs often have to wait for long periods to confirm a block of transactions and receive payment. In a pool, all participating miners get paid every time a participating server solves a block. This payment depends on the amount of work an individual miner contributed to help find that block.