Parallel Sum

Definition: Parallel Sum

The parallel sum of two extended complex numbers $a$ and $b$ is defined as

$$\frac{1}{\frac{1}{a}+\frac{1}{b}}=\frac{ab}{a+b}$$

NOTATION

$$a\parallel b$$

The parallel sum is stronger than addition but weaker than multiplication:

$$a\parallel b+c=(a\parallel b)+ca\parallel bc=a\parallel (bc)$$

Theorem: Associativity of Parallel Sum

The parallel sum is associative:

$$(a\parallel b)\parallel c=a\parallel (b\parallel c)$$

PROOF

TODO