MATH.7:5.1 - Shifted numbers with a shifted addition law

Start with real numbers under addition. Let h(x)=x+1 and r(y)=y-1. Both inverse equations hold on all real numbers. The transported addition is:

u ⊕ v=h(r(u)+r(v))=u+v-1.

The source zero becomes h(0)=1, and source negation becomes neg_Y(u)=h(-r(u))=2-u. Thus u⊕1=u and u⊕(2-u)=1. Associativity follows directly because both groupings of three inputs give u+v+w-2; it also follows from the general transport argument.

To solve x+3=7, translate it as y⊕4=8. The receiving equation gives y=5, and decoding gives x=4. Using ordinary addition on the new labels would instead solve y+4=8 and return x=3, which fails the original equation.

The transported operation is useful as a representation of the source addition. If ordinary addition on Y is part of the receiving requirement, this h does not preserve that requirement; the construction has exposed a different operation.

The same map can transport real division, defined when the source denominator is nonzero. Its receiving formula is u⊘v=(u-1)/(v-1)+1, with domain v≠1. The label 1 decodes to the forbidden denominator zero; the label 0 decodes to the allowed denominator −1. For example, 3⊘0=-1 decodes to −2, the result of the source calculation 2/(-1). Using the familiar restriction v≠0 would exclude an allowed input and admit a forbidden one.