MATH.22:5.3 - Introduce total reciprocal notation without retaining a false law

Suppose arithmetic uses field laws, including 0!=1 and distributivity, and the reciprocal law x*inv(x)=1 for x!=0. The proposed change makes inv a total operation and asks that same law to hold at zero.

The retained laws already imply 0*y=0. Indeed, (0+0)*y=0*y+0*y by distributivity, and cancelling one 0*y from 0*y=0*y+0*y leaves 0=0*y. The proposed new reciprocal law at zero would therefore imply 0=1. The retained assumptions and the new law conflict.

A different change works: over the rational numbers, define inv(0)=0 and inv(x)=1/x for x!=0. This gives a total operation, retains the nonzero reciprocal law, and leaves the field operations unchanged. It has not made cancellation of a zero factor valid.

For a use involving the equation x*y=x*z, cancellation still requires x!=0. At x=0, all rational y,z satisfy the equation. A calculation using total reciprocal notation must retain that condition or inspect the zero case.

The result is a consistent construction within rational arithmetic and a revised law with its condition, rather than the initially requested unconditional inverse law. Total notation and stronger algebraic permission have been separated.