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Harmonic algebra

This revised post follows up on harmonic (or parallel) addition mentioned in a previous post here.

Harmonic algebra 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,

xy := g−1(g(x) + g(y))

xy := g−1(g(x) ∙ g(y))

etc., where g(x) = 1/x with x ≠ 0.

Harmonic algebra is isomorphic to the ordinary numerator operations exchanged with their denominator counterparts. It is like counting down from infinity, 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 \oplus 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 \oplus 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. 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.

Harmonic subtraction is defined as:

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

x\ominus y >0 \Leftrightarrow y>x;\; x\ominus y <0 \Leftrightarrow y<x

Harmonic multiplication is defined as:

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

Harmonic division is defined then as:

x \oslash 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

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