the arithmetic logic unit
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It is called a formula.
Yes, that is correct.
A PET scan would be most useful for detecting brain areas that are most active as a person performs mathematical calculations.
In programming: They let you do generic calculations, with numbers that you don't know in advance. In algebra: They let you do calculations with unknown values - either generic calculations, or do calculations to find such unknown values in a specific case.
The Romans did their calculations on an abacus counting device which was the equivalent to a primitive calculator.