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One, does that mean it's as fast as a single core processor running at 4Ghz?
No, a dual-core processor is not the same speed as a single-core processor times 2. That's not how it works. You need a better understanding of how single-threaded and multi-threaded applications are handled by Windows. It's a complicated comparison that doesn't necessarily mean 2 processors is double the performance of one. Also, the multi-core processors of today are much different from the single-core processors from 5 years ago.
Intel's first attempt at the consumer level was to release the Pentium D. This was essentially two Pentium 4's slapped together on one die. Neither could communicate directly with the other without first going through the FSB and accessing main memory (RAM). That among other things was a very poor design in dual-core architecture and one that was easily beaten by the fastest single-core CPU's in most benchmarks.
Enter the Core 2 duo based on the Core microarchitecture in 2006. Intel decided to return to its roots ditching NetBurst and going with a "new" microarchitecture that was actually a successor to the Pentium III. Each CPU core was more efficient at a lower clock speed than the Pentium 4 ever was. A Core 2 E4300, for example, rated at 1.8GHz per core would beat a Pentium 4 3.0GHz in practically all single-threaded benchmarks. It would even give a 3.4GHz model a run for its money. It was clear after this that the Core microarchitecture was a step in the right direction in solving the speed limitations that single-core CPUs were beginning to run into.
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Which would be faster, my dual core processor, or two single cores running on a motherboard with two cpu sockets?
As explained above, this question isn't one you would ask today. Maybe 5-6 years ago but not today. There aren't any single-core solutions still being developed by either AMD or Intel in which you would need to ask that. Keep in mind that two processors running in separate sockets on the mobo are not as efficient as being on the same die. The main reasoning behind that is the amount of distance between cores. If you can fit it on one die, you're better off.
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How can you tell if you have a 32 bit processor or a 64 bit processor?
Intel 64 (formerly known as EM64T) is supported in all models since the Pentium 4 Prescott models designated with a 500-level model number. For example:
~cdogg
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