Parallel (operator)

The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as "reciprocal formula") and is defined by:[9][6][10][11]

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

with ${\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$ being the extended complex numbers (with corresponding rules).[12] 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.[7][8]

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

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

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

${\displaystyle a\neq b\iff {\big |}a\parallel b{\big |}>{\frac {1}{2}}\min(|a|,|b|)}$

with ${\displaystyle {\big |}a\parallel b{\big |}}$ representing the absolute value of ${\displaystyle a\parallel b}$.

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

The concept has been extended from a scalar operation to matrices[13][14][15][16][17] and further generalized.[18]

The operator was originally introduced as reduced sum by Sundaram Seshu in 1956,[19][20][13] studied as operator ∗ by Kent E. Erickson in 1959,[21][22][13] 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.[14][15][1] While some authors continue to use this symbol up to the present,[7][8] for example, Sujit Kumar Mitra used ∙ as a symbol in 1970.[13] In applied electronics, a ∥ sign became more common as the operator's symbol later on.[23][24][25][26][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.

For addition, the parallel operator follows the commutative law:

${\displaystyle a\parallel b=b\parallel a}$

and the associative law:[12][7][8]

${\displaystyle (a\parallel b)\parallel c=a\parallel (b\parallel c)=a\parallel b\parallel c={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}}}={\frac {abc}{ab+ac+bc}}}$

Multiplication is distributive over this operation.[1][7][8]

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

In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction.[1][27][9][10]

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.

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

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.[7][8]

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:

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

The effectively resulting resistance is ca. 57 kΩ.

Question:[7][8]

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?

Answer:

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

They will finish in close to 3 hours.

WP 34S with parallel operator (∥) on the g+÷ key.

Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959,[21] the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008[28][29][30] as well as on the WP 34C[31] and WP 43S since 2015,[32][33] allowing to solve even cascaded problems with few keystrokes like 270↵ Enter180∥120∥.

• Pekarev, Èdward L.; Šmul'jan, Ju. L. (1976-04-30). "Parallel Addition and Parallel Subtraction of Operators". Mathematics of the USSR-Izvestiya. American Mathematical Society. 10 (2): 351–370. Bibcode:1976IzMat..10..351P. doi:10.1070/IM1976v010n02ABEH001694.
• Duffin, Richard James; Morley, Tom D. (July 1978). "Almost Definite Operators and Electro-mechanical Systems". SIAM Journal on Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM). 35 (1): 21–30. doi:10.1137/0135003. JSTOR 2101028. (10 pages)
• Morley, Tom D. (July 1979). "Parallel Summation, Maxwell's Principle and the Infimum of Projections" (PDF). Journal of Mathematical Analysis and Applications. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA. 70 (1): 33–41. doi:10.1016/0022-247X(79)90073-8. Archived from the original on 2020-08-20. Retrieved 2020-08-20.
• Seeger, Alberto (May 1990) [1988-03-22]. "Direct and Inverse Addition in Convex Analysis and Applications" (PDF). Journal of Mathematical Analysis and Applications. Department of Mathematics, University of Washington, Seattle, Washington, USA: Academic Press, Inc. 148 (2): 317–349. doi:10.1016/0022-247X(90)90004-Y. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20. (33 pages)
• Bryant, Randal E.; Tygar, J. Doug; Huang, Lawrence P. (1994). "Geometric characterization of series-parallel variable resistor networks" (PDF). IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 41 (11): 686–698. doi:10.1109/81.331520. Archived from the original (PDF) on 2017-08-14.
• Antezana, Jorge; Corach, Gustavo; Stojanoff, Demetrio (April 2006) [2005-09-14]. "Bilateral Shorted Operators and Parallel Sums" (PDF). Linear Algebra and Its Applications. La Plata, Argentina & Buenos Aires, Argentina. 414 (2–3): 570–588. arXiv:math/0509327. doi:10.1016/j.laa.2005.10.039. Archived (PDF) from the original on 2017-08-09. Retrieved 2020-08-20. (19 pages)
• Chansangiam, Pattrawut (February 2016) [August 2015, July 2015]. "Mathematical aspects for electrical network connections". KKU Engineering Journal. 43 (1): 47–54. doi:10.14456/kkuenj.2016.8. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20.
• Besenyei, Ádám (2016-09-01). "The irresistible inequality of Milne" (PDF). Budapest: Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University. CIA2016. Archived (PDF) from the original on 2019-08-08. Retrieved 2019-08-11.

• https://github.com/microsoftarchive/edx-platform-1/blob/master/common/lib/calc/calc/calc.py