The parallelism operator ${\displaystyle {\|}}$ (Read " parallel") is a mathematical function that is used primarily as a shorthand notation in electrical engineering. It computes the reciprocal of a sum of reciprocal values, ​​and is defined as

${\displaystyle {\begin{matrix}\|:\ &{\overline {\mathbb {C} }}\times {\overline {\mathbb {C} }}&\to &{\overline {\mathbb {C} }}\\&(a,b)&\mapsto &a\|b={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}\end{matrix}}}$

where ${\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$ is the set of extended complex numbers (with its usual rules of operation).

Properties

1. ${\displaystyle a\|a={\frac {a}{2}}}$.
2. ${\displaystyle a\neq b\iff \left|a\|b\right|>{\tfrac {1}{2}}\min(|a|,|b|)}$, where ${\displaystyle \left|a\|b\right|}$ denotes the absolute value of ${\displaystyle a\|b}$.
3. When ${\displaystyle a}$ and ${\displaystyle b}$ are positive real numbers, ${\displaystyle \left|a\|b\right|<\min(a,b)}$.
4. The parallelism operator is commutative: ${\displaystyle a\|b=b\|a}$.
5. The parallelism operator is associative: ${\displaystyle \left(a\|b\right)\|c=a\|\left(b\|c\right)=a\|b\|c}$.
6. The parallelism operator has ${\displaystyle \infty }$ as its identity operator, and, for ${\displaystyle a\in {\overline {\mathbb {C} }}}$, ${\displaystyle -a}$ is the inverse element. Thus, ${\displaystyle ({\overline {\mathbb {C} }},\|)}$ is an Abelian group.

Examples

Example 1

Problem:

A bricklayer can build a brick wall in 5 hours. A second bricklayer can build the same wall in 7 hours. How long does it take if both bricklayers work on the wall simultaneously?

Solution:

${\displaystyle t_{1}\|t_{2}=5\,\mathrm {h} \|7\,\mathrm {h} ={\frac {1}{{\frac {1}{5\,\mathrm {h} }}+{\frac {1}{7\,\mathrm {h} }}}}\approx 2.917\,\mathrm {h} }$

Thus, it takes just under 3 hours.

Example 2

Problem:

Three resistors of resistances ${\displaystyle {R_{1}=270\,\mathrm {k\Omega } }}$, ${\displaystyle {R_{2}=180\,\mathrm {k\Omega } }}$, and ${\displaystyle {R_{3}=120\,\mathrm {k\Omega } }}$ are connected in parallel. What is the total resistance of the circuit?

Solution:

${\displaystyle R_{1}\|R_{2}\|R_{3}=270\,\mathrm {k\Omega } \|180\,\mathrm {k\Omega } \|120\,\mathrm {k\Omega } ={\frac {1}{{\frac {1}{270\,\mathrm {k\Omega } }}+{\frac {1}{180\,\mathrm {k\Omega } }}+{\frac {1}{120\,\mathrm {k\Omega } }}}}\approx 56.842\,\mathrm {k\Omega } }$

Thus, the circuit has a total resistance of about 57 k Ω.