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:

Simple harmonic addition is thus defined as:

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:

Harmonic subtraction is defined as:

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

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

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

The harmonic square is the inverse:

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