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The Control Unit
The control unit is responsible for managing the flow of a program. It's the component that retrieves the next instruction to be acted upon or the data to be processed.
The ALU (Arithmetic Logic Unit) calculates and compares numbers. The ALU does most of the work of the processor, but it's best suited to working with operations that act on whole numbers (not fractions)
The FPU (Floating Point Unit) is designed specifically to work with real numbers ( numbers with fractional components, and very large, very small, or very precise numbers). It's faster and more efficient at performing mathematical manipulations than the ALU is.
Registers are very small, very fast, memory locations for holding instructions or units of data. Registers operate at the same speed as the CPU, whereas normal system memory can be many times slower.
During operations, CPU's store data and instructions in registers. That information is then transferred back to main system memory. To speed operations, the control unit can "prefetch" instructions and data from system memory and store it in the CPU's registers.
CPU's can have many registers, with groups of registers devoted to a specific purpose (and thus unavailable for other uses). Some modern processors can use registers as needed for the task at hand, rather than being limted by a limted quantity of special-purpose registers.