Question

Why do negative reciprocals always have a product of -1?

Answer 1

For any number x, when we multiply by -1/x, we get -x/x, or -1.  As we haven't made any assumptions about x (other than x is not equal to zero, in which case x does not have a defined reciprocal), all negative reciprocals have a product of -1.

Answer 2

All real numbers except 0 have reciprocals. The reciprocal of a negative number must itself be a negative number so that the number and its reciprocal multiply.