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:
\[ a \parallel b + c = (a \parallel b) + c \qquad a \parallel bc = a \parallel (bc) \]
Theorem: Associativity of Parallel Sum
The parallel sum is associative:
\[ (a \parallel b) \parallel c = a \parallel (b \parallel c) \]
Proof
TODO