This revised post follows up on reciprocal (or harmonic or parallel) addition mentioned in a previous post here.
Reciprocal arithmetic is based on an automorphism that interchanges the zero with the infinite and the greater-than-one with the less-than-one: 0 ↔ ∞ and x ↔ 1/x. So zero becomes the new inaccessible number and infinity becomes the new additive unit. That is,
x ⊞ y := g−1(g(x) + g(y))
x ⊡ y := g−1(g(x) ∙ g(y))
etc., where g(x) = 1/x with x ≠ 0.
The identity element is ∞ if it is included, so that x ⊡ ∞ = x and x ⊡ ∞ = ∞. The additive inverse of x is −x, and the multiplicative inverse is x−1.
Reciprocal arithmetic is isomorphic to the ordinary numerator operations exchanged with their denominator counterparts. It is like counting down from a maximum, in which an increment of one reduces the amount slightly.
Reciprocal addition (also known as parallel addition) is defined as a power operation:
with the understanding that (1/0) → ∞ and (1/∞) → 0. An intermediate value may be zero but not a final value.
Simple reciprocal addition is thus defined as:
The reciprocal additive unit is infinity instead of zero: x ⊞ ∞ = ∞. Reciprocal increment is:
Reciprocal addition is commutative, associative, and distributes with multiplication. See Kent E. Erickson, “A New Operation for Analyzing Series-Paralled Networks,” IRE Trans. on Circuit Theory, March 1959, pp.124-126, See also here and here.
Reciprocal subtraction is defined as:
Reciprocal multiplication is defined as:
Reciprocal division is defined then as:
Reciprocal exponentiation is defined as: