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*) := (2*x*) ⊞ (2*y*) = 2*xy*(*x* + *y*)^{−1}