Registers are fast memory, almost always connected to circuitry that allows various arithmetic, logical, control, and other manipulations, as well as possibly setting internal flags.
Most early computers had only one data register that could be used for arithmetic and logic instructions. Often there would be additional special purpose registers set aside either for temporary fast internal storage or assigned to logic circuits to implement certain instructions. Some early computers had one or two address registers that pointed to a memory location for memory accesses (a pair of address registers typically would act as source and destination pointers for memory operations). Computers soon had multiple data registers, address registers, and sometimes other special purpose registers. Some computers have general purpose registers that can be used for both data and address operations. Every digital computer using a von Neumann architecture has a register (called the program counter) that points to the next executable instruction. Many computers have additional control registers for implementing various control capabilities. Often some or all of the internal flags are combined into a flag or status register.
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