Parallel (operator): Difference between revisions

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{{use dmy dates|date=July 2019|cs1-dates=y}}

{{use list-defined references|date=December 2021}}

[[File:Parallelitätsoperator.svg|thumb|Graphical interpretation of the parallel operator with <math>a \parallel b = c</math>.]]

The '''parallel operator''' <math>\|</math> (pronounced "parallel",<ref name="Duffin_1971"/> following the [[parallel (geometry)#Symbol|parallel lines notation from geometry]];<ref name="Kersey_1673"/><ref name="Cajori_1928"/> also known as '''reduced sum''', '''parallel sum''' or '''parallel addition''') is a [[binary functionoperation]] which is used as a shorthand in [[electrical engineering]],<ref name="TI_2004"/><ref name="Bober_2016"/><ref name="EECS_2018"/><ref group="nb" name="NB_Origin"/> but is also used in [[kinetics (physics)|kinetics]], [[fluid mechanics]] and [[financial mathematics]].<ref name="Ellerman_1995"/><ref name="Ellerman_2004"/> The name ''parallel'' comes from the use of the operator computing the combined resistance of [[Resistor#Series and parallel resistors|resistors in parallel]].

==Overview==

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</math>

Unlike for [[multiplication and repeated addition|repeated addition]], this does not commute: <math>a / b \neq b / a.</math>

:<math>\frac ab \neq \frac ba \quad \text{implies}\quad

\underbrace{a \parallel a \parallel \cdots \parallel a}_{b\text{ times}} \,\neq\, \underbrace{b \parallel b \parallel \cdots \parallel b}_{a\text{ times}}~\!.</math>

=== Binomial expansion ===

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\exp\left(\frac{1}{a\parallel b}\right) = \exp\left(\frac{1}{a}\right)\exp\left(\frac{1}{b}\right)

</math>

=== Parallel Functions ===

A ''parallel function'' is one which commutes with the parallel operation:{{cn|date=January 2023}}

:<math>

f\left(a\parallel b\right) = f(a)\parallel f(b)

</math>

For example, <math>f(x) = cx</math> is a parallel function, because <math>c(a\parallel b) = ca \parallel cb.</math>

=== Factoring parallel polynomials ===

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=== Quadratic formula ===

A [[linear equation]] can be easily solved via the parallel inverse:

:<math>\begin{align}

ax\parallel b &= \infty \\[3mu]

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\end{align}</math>

To solve a parallel [[quadratic equation]], [[Completing the square|complete the square]] to obtain an analog of the [[quadratic formula]]

:<math>

\begin{align}

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==Applications==

There are applications of the parallel operator in mechanics, electronics, optics, and study of periodicity:

===Reduced mass===

Given masses ''m'' and ''M'', the [[reduced mass]] <math>\mu = \frac{m M}{m + M} = m \parallel M</math> is frequently applied in mechanics. For instance, when the masses orbit each other, the [[moment of inertia]] is their reduced mass times the distance between them.

=== Circuit analysis ===

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Likewise for the total [[capacitance]] of serial [[capacitor]]s.<ref group="nb" name="NB_Application"/>

=== Coalescence of independent probability density functions. ===

The [[coalescence (statistics)|coalesced density function]] f_coalesced(x) of n independent probability density functions f₁(x), f₂(x), …, f_n(x), is equal to the reciprocal of the sum of the reciprocal densities.<ref>Van Droogenbroeck, Frans J., [https://www.academia.edu/127477986/ 'Coalescence, unlocking insights in the intricacies of merging independent probability density functions'] (2025). </ref>

:<math>\begin{align}

\frac{1}{f_{coalesced}(x)} &= \frac{1}{f_1(x)} + \frac{1}{f_2(x)} + \cdots + \frac{1}{f_n(x)} \\[5mu]

\end{align}</math>

=== Lens equation ===

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==Implementation==

[[File:Hp30bwp34s.jpg|thumb|150px|[[WP&nbsp;34S]] with parallel operator (<kbd>∥</kbd>) on the {{keypress|g|÷}} key.]]

Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959,<ref name="Erickson_1959"/> the parallel operator is implemented as a keyboard operator on the [[Reverse Polish Notation]] (RPN) scientific calculators [[WP&nbsp;34S]] since 2008<ref name="Bonin_2012"/><ref name="Bonin_2015"/><ref name="Bonin_2016"/> as well as on the [[WP&nbsp;34C]]<ref name="Dowrick_2015"/> and [[WP&nbsp;43S]] since 2015,<ref name="Bonin_2019_OG"/><ref name="Bonin_2019_RG"/> allowing to solve even cascaded problems with few keystrokes like {{keypress|270}}{{keypress|ENTER}}{{keypress|180}}{{keypress|∥}}{{keypress|120}}{{keypress|∥}}.

==Projective view==

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The two matrix products show that there are two subgroups of M(2,''F'') isomorphic to (''F'',+), the additive group of ''F''. Depending on which embedding is used, one operation is +, the other is <math>\parallel.</math>

==See also==

* [[Mediant (mathematics)]]

==Notes==