Sixty-four-bit processors have been with us since 1992, and in the 21st century they have started to become mainstream. Both Intel and AMD have introduced 64-bit chips, and the Mac G5 sports a 64-bit processor. Sixty-four-bit processors have 64-bit ALUs, 64-bit registers, 64-bit buses and so on.
One reason why the world needs 64-bit processors is because of their enlarged address spaces. Thirty-two-bit chips are often constrained to a maximum of 2 GB or 4 GB of RAM access. That sounds like a lot, given that most home computers currently use only 256 MB to 512 MB of RAM. However, a 4-GB limit can be a severe problem for server machines and machines running large databases. And even home machines will start bumping up against the 2 GB or 4 GB limit pretty soon if current trends continue. A 64-bit chip has none of these constraints because a 64-bit RAM address space is essentially infinite for the foreseeable future -- 2^64 bytes of RAM is something on the order of a billion gigabytes of RAM.
With a 64-bit address bus and wide, high-speed data buses on the motherboard, 64-bit machines also offer faster I/O (input/output) speeds to things like hard disk drives and video cards. These features can greatly increase system performance.
Servers can definitely benefit from 64 bits, but what about normal users? Beyond the RAM solution, it is not clear that a 64-bit chip offers "normal users" any real, tangible benefits at the moment. They can process data (very complex data features lots of real numbers) faster. People doing video editing and people doing photographic editing on very large images benefit from this kind of computing power. High-end games will also benefit, once they are re-coded to take advantage of 64-bit features. But the average user who is reading e-mail, browsing the Web and editing Word documents is not really using the processor in that way.
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