# Parallel (operator)

The parallel operator $\|$ (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in electrical engineering,[nb 1] but is also used in kinetics, fluid mechanics and financial mathematics.

It represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as "reciprocal formula") and is defined by:

${\begin{array}{rlcl}\|:\ &{\overline {\mathbb {C} }}\times {\overline {\mathbb {C} }}&\to &{\overline {\mathbb {C} }}\\&(a,b)&\mapsto &a\|b={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}={\frac {ab}{a+b}}\end{array}}$ with ${\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}$ being the extended complex numbers (with corresponding rules). The later form is sometimes also referred to as "product over sum".

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

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

$a\|a={\frac {a}{2}}$ .

Further, for all $a,b\in {\overline {\mathbb {C} }}$ :

$a\neq b\iff \left|a\|b\right|>{\tfrac {1}{2}}\min(|a|,|b|)$ with $\left|a\|b\right|$ representing the absolute value of $a\|b$ .

With $a$ and $b$ being positive real numbers follows $\left|a\|b\right|<\min(a,b)$ .

## Notation

The operator was originally introduced as reduced sum by Sundaram Seshu in 1956, studied as operator ∗ by Kent E. Erickson in 1959, and popularized by Richard James Duffin and William Niles Anderson, Jr. as parallel addition operator : in mathematics and network theory since 1966. In applied electronics, a ∥ sign became more common as the operator's symbol later on.[nb 1][nb 2] This was often written as doubled vertical line (||) available in most character sets, but now can be represented using Unicode character U+2225 (  ) for "parallel to". In LaTeX and related markup languages, the macros \| and \parallel are often used to denote the operator's symbol.

## Rules

For addition the parallel operator follows the commutative law:

$a\|b=b\|a$ and the associative law:

$\left(a\|b\right)\|c=a\|\left(b\|c\right)=a\|b\|c={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}}}={\frac {abc}{ab+ac+bc}}$ .

Multiplication is distributive over this operation.

Further, the parallel operator has $\infty$ as neutral element and for $a\in {\overline {\mathbb {C} }}$ the number $-a$ as inverse element. Hence $({\overline {\mathbb {C} }},\|)$ is an Abelian group.

In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction.

## Applications

In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits.[nb 2]

For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.

${\frac {1}{R_{\mathrm {eq} }}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}$ .

Likewise for the total capacitance of serial capacitors.[nb 2]

The same principle can be applied to various problems in other disciplines.

## Examples

Question:

Three resistors $R_{1}=270\,\mathrm {k\Omega }$ , $R_{2}=180\,\mathrm {k\Omega }$ and $R_{3}=120\,\mathrm {k\Omega }$ are connected in parallel. What is their resulting resistance?

$R_{1}\|R_{2}\|R_{3}=270\,\mathrm {k\Omega } \|180\,\mathrm {k\Omega } \|120\,\mathrm {k\Omega } ={\frac {1}{{\frac {1}{270\,\mathrm {k\Omega } }}+{\frac {1}{180\,\mathrm {k\Omega } }}+{\frac {1}{120\,\mathrm {k\Omega } }}}}\approx 56{,}84\,\mathrm {k\Omega }$ The effectively resulting resistance is ca. 57 kΩ.

Question:

A construction worker raises a wall in 5 hours. Another worker would need 7 hours for the same work. How long does it take to build the wall if both worker work in parallel?

$t_{1}\|t_{2}=5\,\mathrm {h} \|7\,\mathrm {h} ={\frac {1}{{\frac {1}{5\,\mathrm {h} }}+{\frac {1}{7\,\mathrm {h} }}}}\approx 2{,}92\,\mathrm {h}$ 