Harmonic operations

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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:

$O_{-1}(a_{1},a_{2},...,a_{n})\equiv \left ( \sum_{i=1}^{n}a_{i}^{-1} \right )^{-1} ,\,\, a_{i} \neq 0$

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:

$x \boxplus y = \left ( \frac{1}{x} + \frac{1}{y} \right )^{-1} \equiv \frac{xy}{x+y}; \,\, x, y \neq 0$

The harmonic additive unit is infinity instead of zero: x ⊞ ∞ = ∞. Harmonic increment is:

$x \boxplus y = \left ( \frac{1}{x} + 1 \right )^{-1} \equiv \frac{x}{x+1}; \,\, x \neq 0$

Harmonic addition is commutative, associative, and distributes with multiplication.

Harmonic subtraction is defined as:

$x \boxminus y = x \boxplus (-y) = \left ( \frac{1}{x} - \frac{1}{y} \right )^{-1} \equiv \frac{xy}{y-x}; \; x,y\neq 0$

$x\boxminus y >0 \Leftrightarrow y>x;\; x\boxminus y <0 \Leftrightarrow y<x$

Harmonic multiplication is defined as:

$x \boxtimes y = \left ( \frac{1}{x} \times \frac{1}{y} \right )^{-1} \equiv xy; \,\, x, y \neq 0$

Harmonic division is defined then as:

$x \: \Box \: y = \left ( \frac{1}{x} \div \frac{1}{y} \right )^{-1} \equiv \frac{x}{y}; \,\, x, y \neq 0$

Harmonic exponentiation is defined as:

$x \curlywedge y = \left(\frac{1}{x} \right)^{(-(\frac{1}{y})^{-1})} \equiv \left( \frac{1}{x} \right) ^{-y}; \,\, x, y \neq 0$

The harmonic mean is defined as:

H(x, y) := (2x) ⊞ (2y) = 2xy(x + y)⁻¹

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