Parallel (operator)

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:

${\displaystyle a\parallel b\mathrel {:=} {\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{b}}}}={\frac {ab}{a+b}},}$

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 ${\displaystyle a\in {\overline {\mathbb {C} }}}$:

${\displaystyle a\parallel a={\frac {1}{2/a}}={\tfrac {1}{2}}a.}$

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

${\displaystyle {\big |}\,a\parallel b\,{\big |}>{\tfrac {1}{2}}\min {\bigl (}|a|,|b|{\bigr )},}$

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 ${\displaystyle {\tfrac {1}{2}}\min(a,b)<{\big |}\,a\parallel b\,{\big |}<\min(a,b).}$

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

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 around 1974. This was often written as doubled vertical line (||) available in most character sets (sometimes italicized as //), 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 (and rarely \smallparallel is used) to denote the operator's symbol.

Let ${\displaystyle {\widetilde {\mathbb {C} }}}$represent the extended complex plane excluding zero, ${\displaystyle {\widetilde {\mathbb {C} }}:=\mathbb {C} \cup \{\infty \}\smallsetminus \{0\},}$and ${\displaystyle \varphi }$the bijective function from ${\displaystyle \mathbb {C} }$to ${\displaystyle {\widetilde {\mathbb {C} }}}$such that ${\displaystyle \varphi (z)=1/z.}$One has identities

${\displaystyle \varphi (zt)=\varphi (z)\varphi (t),}$

and

${\displaystyle \varphi (z+t)=\varphi (z)\parallel \varphi (t)}$

This implies immediately that ${\displaystyle {\widetilde {\mathbb {C} }}}$is a field where the parallel operator takes the place of the addition, and that this field is isomorphic to ${\displaystyle \mathbb {C} .}$

The following properties may be obtained by translating through ${\displaystyle \varphi }$the corresponding properties of the complex numbers.

As for any field, ${\displaystyle ({\widetilde {\mathbb {C} }},\,\parallel \,,\,\cdot \,)}$satisfies a variety of basic identities.

It is commutative under parallel and multiplication:

${\displaystyle {\begin{aligned}a\parallel b&=b\parallel a\\[3mu]ab&=ba\end{aligned}}}$

It is associative under parallel and multiplication:

${\displaystyle {\begin{aligned}&(a\parallel b)\parallel c=a\parallel (b\parallel c)=a\parallel b\parallel c={\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{b}}+{\dfrac {1}{c}}}}={\frac {abc}{ab+ac+bc}},\\&(ab)c=a(bc)=abc.\end{aligned}}}$

Both operations have an identity element; for parallel the identity is ${\displaystyle \infty }$while for multiplication the identity is 1:

${\displaystyle {\begin{aligned}&a\parallel \infty =\infty \parallel a={\frac {1}{{\dfrac {1}{a}}+0}}=a,\\&1\cdot a=a\cdot 1=a.\end{aligned}}}$

Every element ${\displaystyle a}$of ${\displaystyle {\widetilde {\mathbb {C} }}}$has an inverse under parallel, equal to ${\displaystyle -a,}$the additive inverse under addition. (But 0 has no inverse under parallel.)

${\displaystyle a\parallel (-a)={\frac {1}{{\dfrac {1}{a}}-{\dfrac {1}{a}}}}=\infty .}$

The identity element ${\displaystyle \infty }$is its own inverse, ${\displaystyle \infty \parallel \infty =\infty .}$

Every element ${\displaystyle a\neq \infty }$of ${\displaystyle {\widetilde {\mathbb {C} }}}$has a multiplicative inverse ${\displaystyle a^{-1}=1/a}$:

${\displaystyle a\cdot {\frac {1}{a}}=1.}$

Multiplication is distributive over parallel:

${\displaystyle k(a\parallel b)={\frac {k}{{\dfrac {1}{a}}+{\dfrac {1}{b}}}}={\frac {1}{{\dfrac {1}{ka}}+{\dfrac {1}{kb}}}}=ka\parallel kb.}$

Repeated parallel is equivalent to division,

${\displaystyle \underbrace {a\parallel a\parallel \cdots \parallel a} _{n{\text{ times}}}={\frac {1}{\underbrace {{\dfrac {1}{a}}+{\dfrac {1}{a}}+\cdots +{\dfrac {1}{a}}} _{n{\text{ times}}}}}={\frac {a}{n}}.}$

Or, multiplying both sides by n,

${\displaystyle n(\underbrace {a\parallel a\parallel \cdots \parallel a} _{n{\text{ times}}})=a.}$

Unlike for repeated addition, this does not commute:

${\displaystyle {\frac {a}{b}}\neq {\frac {b}{a}}\quad {\text{implies}}\quad \underbrace {a\parallel a\parallel \cdots \parallel a} _{b{\text{ times}}}\,\neq \,\underbrace {b\parallel b\parallel \cdots \parallel b} _{a{\text{ times}}}~\!.}$

Using the distributive property twice, the product of two parallel binomials can be expanded as

${\displaystyle {\begin{aligned}(a\parallel b)(c\parallel d)&=a(c\parallel d)\parallel b(c\parallel d)\\[3mu]&=ac\parallel ad\parallel bc\parallel bd.\end{aligned}}}$

The square of a binomial is

${\displaystyle {\begin{aligned}(a\parallel b)^{2}&=a^{2}\parallel ab\parallel ba\parallel b^{2}\\[3mu]&=a^{2}\parallel {\tfrac {1}{2}}ab\parallel b^{2}.\end{aligned}}}$

The cube of a binomial is

${\displaystyle (a\parallel b)^{3}=a^{3}\parallel {\tfrac {1}{3}}a^{2}b\parallel {\tfrac {1}{3}}ab^{2}\parallel b^{3}.}$

In general, the nth power of a binomial can be expanded using binomial coefficients which are the reciprocal of those under addition, resulting in an analog of the binomial formula:

${\displaystyle (a\parallel b)^{n}={\frac {a^{n}}{\binom {n}{0}}}\parallel {\frac {a^{n-1}b}{\binom {n}{1}}}\parallel \cdots \parallel {\frac {a^{n-k}b^{k}}{\binom {n}{k}}}\parallel \cdots \parallel {\frac {b^{n}}{\binom {n}{n}}}.}$

The following identities hold:

${\displaystyle {\frac {1}{\log(ab)}}={\frac {1}{\log(a)}}\parallel {\frac {1}{\log(b)}},}$

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

As with a polynomial under addition, a parallel polynomial with coefficients ${\displaystyle a_{k}}$in ${\textstyle {\widetilde {\mathbb {C} }}}$(with ${\displaystyle a_{0}\neq \infty }$) can be factored into a product of monomials:

${\displaystyle {\begin{aligned}&a_{0}x^{n}\parallel a_{1}x^{n-1}\parallel \cdots \parallel a_{n}=a_{0}(x\parallel -r_{1})(x\parallel -r_{2})\cdots (x\parallel -r_{n})\end{aligned}}}$

for some roots ${\displaystyle r_{k}}$(possibly repeated) in ${\textstyle {\widetilde {\mathbb {C} }}.}$

Analogous to polynomials under addition, the polynomial equation

${\displaystyle (x\parallel -r_{1})(x\parallel -r_{2})\cdots (x\parallel -r_{n})=\infty }$

implies that ${\textstyle x=r_{k}}$for some k.

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

${\displaystyle {\begin{aligned}ax\parallel b&=\infty \\[3mu]\implies x&=-{\frac {b}{a}}.\end{aligned}}}$

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

${\displaystyle {\begin{aligned}ax^{2}\parallel bx\parallel c&=\infty \\[5mu]x^{2}\parallel {\frac {b}{a}}x&=-{\frac {c}{a}}\\[5mu]x^{2}\parallel {\frac {b}{a}}x\parallel {\frac {4b^{2}}{a^{2}}}&=\left(-{\frac {c}{a}}\right)\parallel {\frac {4b^{2}}{a^{2}}}\\[5mu]\left(x\parallel {\frac {2b}{a}}\right)^{2}&={\frac {b^{2}\parallel -{\tfrac {1}{4}}ac}{{\tfrac {1}{4}}a^{2}}}\\[5mu]\implies x&={\frac {(-b)\parallel \pm {\sqrt {b^{2}\parallel -{\tfrac {1}{4}}ac}}}{{\tfrac {1}{2}}a}}.\end{aligned}}}$

The extended complex numbers including zero, ${\displaystyle {\overline {\mathbb {C} }}:=\mathbb {C} \cup \infty ,}$is no longer a field under parallel and multiplication, because 0 has no inverse under parallel. (This is analogous to the way ${\displaystyle {\bigl (}{\overline {\mathbb {C} }},{+},{\cdot }{\bigr )}}$is not a field because ${\displaystyle \infty }$has no additive inverse.)

For every non-zero a,

${\displaystyle a\parallel 0={\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{0}}}}=0}$

The quantity ${\displaystyle 0\parallel (-0)=0\parallel 0}$can either be left undefined (see indeterminate form) or defined to equal 0.

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

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

Given masses m and M, the reduced mass ${\displaystyle \mu ={\frac {mM}{m+M}}=m\parallel M}$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.

In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits. There is a duality between the usual (series) sum and the parallel sum.

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

${\displaystyle {\begin{aligned}{\frac {1}{R_{\text{eq}}}}&={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}\\[5mu]R_{\text{eq}}&=R_{1}\parallel R_{2}\parallel \cdots \parallel R_{n}.\end{aligned}}}$

Likewise for the total capacitance of serial capacitors.

The 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.

${\displaystyle {\begin{aligned}{\frac {1}{f_{coalesced}(x)}}&={\frac {1}{f_{1}(x)}}+{\frac {1}{f_{2}(x)}}+\cdots +{\frac {1}{f_{n}(x)}}\\[5mu]\end{aligned}}}$

In geometric optics the thin lens approximation to the lens maker's equation.

${\displaystyle f=\rho _{virtual}\parallel \rho _{object}}$

The time between conjunctions of two orbiting bodies is called the synodic period. If the period of the slower body is T₂, and the period of the faster is T₁, then the synodic period is

${\displaystyle T_{syn}=T_{1}\parallel (-T_{2}).}$

Question:

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

Answer:

${\displaystyle {\begin{aligned}R_{1}\parallel R_{2}\parallel R_{3}&=270\,\mathrm {k\Omega } \parallel 180\,\mathrm {k\Omega } \parallel 120\,\mathrm {k\Omega } \\[5mu]&={\frac {1}{{\dfrac {1}{270\,\mathrm {k\Omega } }}+{\dfrac {1}{180\,\mathrm {k\Omega } }}+{\dfrac {1}{120\,\mathrm {k\Omega } }}}}\\[5mu]&\approx 56.84\,\mathrm {k\Omega } \end{aligned}}}$

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 workers work in parallel?

Answer:

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

They will finish in close to 3 hours.

Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 as well as on the WP 34C and WP 43S since 2015, allowing to solve even cascaded problems with few keystrokes like270↵ Enter180∥120∥.

Given a field F there are two embeddings of F into the projective line P(F): z → [z : 1] and z → [1 : z]. These embeddings overlap except for [0:1] and [1:0]. The parallel operator relates the addition operation between the embeddings. In fact, the homographies on the projective line are represented by 2 x 2 matrices M(2,F), and the field operations (+ and ×) are extended to homographies. Each embedding has its addition a + b represented by the following matrix multiplications in M(2,A):

${\displaystyle {\begin{aligned}{\begin{pmatrix}1&0\\a&1\end{pmatrix}}{\begin{pmatrix}1&0\\b&1\end{pmatrix}}&={\begin{pmatrix}1&0\\a+b&1\end{pmatrix}},\\[10mu]{\begin{pmatrix}1&a\\0&1\end{pmatrix}}{\begin{pmatrix}1&b\\0&1\end{pmatrix}}&={\begin{pmatrix}1&a+b\\0&1\end{pmatrix}}.\end{aligned}}}$

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 ${\displaystyle \parallel .}$

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