This revised post follows up on harmonic (or reciprocal or parallel) addition mentioned in a previous post here. See Grossman & Katz, Non-Newtonian Calculus (Lee Press, Pigeon Cove, MA: 1972) p.59f; and 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.
Harmonic 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 := (x−1 + y−1)−1 = xy(x + y)−1
x ⊡ y := (x−1 ∙ y−1)−1 = xy
etc., where zero replaces any division by zero.
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.
Harmonic 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.
Harmonic 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 harmonic addition is thus defined as:
The harmonic additive unit is infinity instead of zero: x ⊞ ∞ = ∞. Harmonic increment is:
Harmonic addition is commutative, associative, and distributes with multiplication.
Harmonic subtraction is defined as:
Harmonic multiplication is defined as:
Harmonic division is defined then as:
Harmonic exponentiation is defined as:
The harmonic mean is defined as:
H(x, y) := (2x) ⊞ (2y) = 2xy(x + y)−1