# Parallel (operator) Graphical interpretation of the parallel operator with $a\parallel b=c$ .

The parallel operator (also known as reduced sum, parallel sum or parallel addition$\|$ (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.

## Overview

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

{\begin{aligned}{}\parallel {}:&&{\overline {\mathbb {C} }}\times {\overline {\mathbb {C} }}&\to {\overline {\mathbb {C} }}\\&&(a,b)&\mapsto a\parallel b={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}={\frac {ab}{a+b}},\end{aligned}} with ${\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}$ being the extended complex numbers (with corresponding rules).[clarification needed] The latter 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 $a\in {\overline {\mathbb {C} }}$ :

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

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

$a\neq b\iff {\big |}a\parallel b{\big |}>{\frac {1}{2}}\min(|a|,|b|)$ with ${\big |}a\parallel b{\big |}$ representing the absolute value of $a\parallel b$ .

With $a$ and $b$ being positive real numbers follows ${\big |}a\parallel b{\big |}<\min(a,b)$ .

The concept has been extended from a scalar operation to matrices and further generalized.

## 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 or parallel sum operator : in mathematics and network theory since 1966. While some authors continue to use this symbol up to the present, for example, Sujit Kumar Mitra used ∙ as a symbol in 1970. 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:

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

$(a\parallel b)\parallel c=a\parallel (b\parallel c)=a\parallel b\parallel c={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}}}={\frac {abc}{ab+ac+bc}}$ Multiplication is distributive over this operation.

Further, the parallel operator has $\infty$ as neutral element and, for $a\in {\overline {\mathbb {C} }}\setminus \{0\},$ the number $-a$ as inverse element. However, $\left({\overline {\mathbb {C} }},\parallel \right)$ is not an abelian group, as $a\parallel 0=0$ for every nonzero a, and $0\parallel 0$ is not well defined (indeterminate form).

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

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

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

There is a duality between the usual (series) sum and the parallel sum.

## 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?

$R_{1}\parallel R_{2}\parallel R_{3}=270\,\mathrm {k\Omega } \parallel 180\,\mathrm {k\Omega } \parallel 120\,\mathrm {k\Omega } ={\frac {1}{{\frac {1}{270\,\mathrm {k\Omega } }}+{\frac {1}{180\,\mathrm {k\Omega } }}+{\frac {1}{120\,\mathrm {k\Omega } }}}}\approx 56.84\,\mathrm {k\Omega }$ 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?

$t_{1}\parallel t_{2}=5\,\mathrm {h} \parallel 7\,\mathrm {h} ={\frac {1}{{\frac {1}{5\,\mathrm {h} }}+{\frac {1}{7\,\mathrm {h} }}}}\approx 2.92\,\mathrm {h}$ 