The parallel operator ${\displaystyle \|}$ (pronounced "parallel", following the parallel lines notation from geometry; also known as reduced sum, parallel sum or parallel addition) is a binary operation which is used as a shorthand in electrical engineering, but is also used in kinetics, fluid mechanics and financial mathematics. The name parallel comes from the use of the operator computing the combined resistance of resistors in parallel.

The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as the "reciprocal formula" or "harmonic sum") and is defined by:

where a, b, and ${\displaystyle a\parallel b}$ are elements of the extended complex numbers ${\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.}$

The operator gives half of the harmonic mean of two numbers a and b.

As a special case, for any number ${\displaystyle a\in {\overline {\mathbb {C} }}}$:

Further, for all distinct numbers ${\displaystyle a\neq b}$:

with ${\displaystyle {\big |}\,a\parallel b\,{\big |}}$ representing the absolute value of ${\displaystyle a\parallel b}$, and ${\displaystyle \min(x,y)}$ meaning the minimum (least element) among x and y.

If ${\displaystyle a}$ and ${\displaystyle b}$ are distinct positive real numbers then ${\displaystyle {\tfrac {1}{2}}\min(a,b)<{\big |}\,a\parallel b\,{\big |}<\min(a,b).}$