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Negatives and rationals are generated from natural numbers in a similar way by taking a binary operator, its inverse, and its identity:
a 0 - b
to create a set closed under the inverse of addition, and
a * 1 / b
to get a set closed under the inverse of multiplication.
So with the reals, we have a set we want to close under the inverse of exponentiation—but instead of something analogous to the above, we use
a i * b.
Why do we use addition and multiplication in place of the operation whose inverse we want to close, and why do we use i instead of the identity element?
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