iSoul Time has three dimensions

# Harmonic arithmetic

This post follows up on harmonic addition mentioned in the previous post here.

Harmonic arithmetic is an inverse arithmetic. It 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.

Harmonic addition was defined as a power operation:

$\textup{Harmonic&space;addition}\;&space;O_{-1}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\sum_{i=1}^{n}a_{i}^{-1}&space;\right&space;)^{-1}$

Simple harmonic addition is thus defined as:

$x\oplus&space;y=\left&space;(&space;\frac{1}{x}+\frac{1}{y}&space;\right&space;)^{-1}&space;=&space;\frac{xy}{x+y}.$

The harmonic additive unit is infinity instead of zero. Harmonic adding one to infinity equals one. Incrementing one leads to one-half, and in general each increment leads to a smaller number:

$x\oplus&space;1=\left&space;(&space;\frac{1}{x}+1&space;\right&space;)^{-1}&space;=&space;\frac{x}{x+1}.$

Harmonic subtraction is defined as:

$x\ominus&space;y&space;=x&space;\oplus&space;(-y)&space;=&space;\left&space;(&space;\frac{1}{x}-\frac{1}{y}&space;\right&space;)^{-1}&space;=&space;\frac{xy}{y-x}.$

Note the reverse of x and y in the denominator. Harmonic multiplication is defined from multiple harmonic additions as:

$x&space;\odot&space;y&space;=&space;\frac{x}{y}$

which is surprisingly non-commutative. Harmonic division is defined then as:

$x&space;\oslash&space;y=xy$

which is surprisingly commutative. Harmonic exponentiation is defined from multiple harmonic multiplications as:

$x\,&space;(\wedge&space;)\,&space;y=x^{^{1-y}}.$

The harmonic square is the inverse:

$x\,&space;(\wedge&space;)\,&space;2=x^{-1}.$

Harmonic arithmetic is like counting down from infinity, in which an increment of one reduces the amount slightly.