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To oversimplify it the "operational amplifier" was originally designed to perform mathematical operations in electronic analog computers. The designer set the mathematical operation of each amplifier by designing its feedback network. Some operations that could be done were: addition, subtraction, constant multiplication/division, logarithm, exponentiation, variable multiplication/division by combining logarithm-addition/subtraction-exponentiation, integration, differentiation, absolute value, clipping, etc.
A "normal" amplifier was just designed to produce a certain amount of voltage or current gain.
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