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The most gaping advantage can be very easilly explained through example:

Single core processors have a single thread, and can process a single set of instructions per clock cycle. This looks like this (Saying this processor can process 2 instructions a clock):

(Note this is in an optimal setting where data is perfectly threaded)

Clock 1: Instruction 1; Instruction 2;

Clock 2: Instruction 3; Instruction 4;

Clock 3: Instruction 5; Instruction 6;

Clock 4: Instruction 7; Instruction 8;

Dual-Core processing would do this same instruction set much quicker:

Clock 1: Instruction 1; Instruction 2; Instruction 3; Instruction 4

Clock 2: Instruction 5; Instruction 6; Instruction 7; Instruction 8

In a perfectly threaded application, two equivilent-performance cores on a dual core processor would power through the work twice as quickly as a single-core model. A quad-core with these specs would do the entire instruction set in a single clock.

Even if it isn't always a 2x increase, multiple-core procesors have a distinct advantage in a very large range of applications.

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