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''' (also known as '''reduced sum''', '''parallel sum''' or '''parallel addition''')&nbsp;<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 [[function (mathematics)|mathematicalbinary 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>

where {{mvar|a}}, {{mvar|b}}, and <math>a \parallel b</math> are elements of the [[extended complex numbers]] <math>\overline{\mathbb{C}} = \mathbb{C}\cup\{ \infty\}.</math><ref name="Georg_1999"/><ref name=ACAH>{{Wikibooks-inline|Associative Composition Algebra/Homographies}}</ref>

The operator gives half of the [[harmonic mean]] of two numbers ''a'' and ''b''.<ref name="Ellerman_1995"/><ref name="Ellerman_2004"/>

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Further, for all distinct numbers {{nobr|<math>a \neq b</math>:}}

:<math>\big| \,a \parallel b \,\big| > \tfrac12 \min\bigl(|a|, |b|\bigr),</math>

with <math>\big|\, a \parallel b \,\big|</math> representing the [[absolute value]] of <math>a \parallel b</math>, and <math>\min(x, y)</math> meaning the [[Maxima and minima#In relation to sets|minimum]] (least element) among {{mvar|x}} and {{mvar|y}}.

If <math>a</math> and <math>b</math> are distinct positive real numbers then <math>\tfrac12 \min(a, b) < \big|\, a \parallel b \,\big| < \min(a, b).</math>

The concept has been extended from a [[scalar (mathematics)|scalar]] operation to [[matrix (mathematics)|matrices]]<ref name="Mitra_1970"/><ref name="Duffin_1966"/><ref name="Anderson-Duffin_1969"/><ref name="Mitra-Puri_1973"/><ref name="Mitra-Bhimasankaram-Malik_2010"/> and further [[generalization (mathematics)|generalized]].<ref name="Eriksson-Bique-Leutwiler_1989"/>

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:<math>\varphi(z+t)=\varphi(z)\parallel \varphi(t)</math>

This implies immediately that <math>\widetilde{\C}</math> is a [[field (mathematics)|field]] where the parallel operator takes the place of the addition, and that isthis field is [[isomorphism|isomorphic]] to <math>\C.</math>

The following properties may be obtained by translating through <math>\varphi</math> the corresponding properties of the complex numbers.

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=== Field properties ===

As for any field, <math>(\widetilde{\C}, \,\parallel\,, \,\cdot\,)</math> satisfies a variety of basic identities.

It is [[commutative property|''commutative'']] under parallel and multiplication:

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k (a \parallel b) = \frac{k}{\dfrac1a + \dfrac1b} = \frac{1}{\dfrac1{ka} + \dfrac1{kb}} = ka \parallel kb.

</math>

=== Repeated parallel ===

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</math>

Or, multiplying both sides by {{mvar|n}},

:<math>

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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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The following identities hold:

:<math>

\log^frac{-1}{\log(ab)} = \log^frac{-1}{\log(a)}\parallel\log^frac{-1}{\log(b)},

</math>

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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==

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

===Reduced Circuit Analysis 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 ===

In [[electrical engineering]], the parallel operator can be used to calculate the total impedance of various [[Series and parallel circuits#Notation|serial and parallel]] electrical circuits.<ref group="nb" name="NB_Application" />

There is a [[duality (mathematics)|duality]] between the usual [[series sum|(series) sum]] and the parallel sum.<ref name="Ellerman_1995"/><ref name="Ellerman_2004"/>

For instance, the total [[electrical resistance|resistance]] of [[parallel resistors|resistors connected in parallel]] is the reciprocal of the sum of the reciprocals of the individual [[resistor]]s.

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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. ===

=== Lens Equation ===

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 Equationequation ===

In [[geometric optics]] the [[focal length#Thin lens approximation|thin lens approximation]] to the lens maker's equation.

:<math>f = \rho_{virtual}\parallel \rho_{object}</math>

===Synodic period===

There is a [[duality (mathematics)|duality]] between the usual [[series sum|(series) sum]] and the parallel sum.<ref name="Ellerman_1995"/><ref name="Ellerman_2004"/>

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

:<math>T_{syn} = T_1 \parallel (-T_2) .</math>

==Examples==

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Question:<ref name="Ellerman_1995"/><ref name="Ellerman_2004"/>

: 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 workerworkers work in parallel?

Answer:

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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==

Given a [[field (mathematics)|field]] ''F'' there are two [[embedding]]s 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 [[homography|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 multiplication]]s in M(2,''A''):

Two charts make an [[atlas (topology)|atlas]] that covers the [[complex projective line]]. The first chart covers zero but not infinity: [''z'':1] → ''z'', the second chart covers infinity but not zero: [1:''z''] → ''z''. The projectivities of the projective line are represented by matrices in the [[general linear group]] GL(2,C):

:<math>\begin{align}

\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}

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\end{align}</math>

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

==See also==

* [[Mediant (mathematics)]]

==Notes==

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<ref name="Cajori_1928">{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations – Notations in Elementary Mathematics |chapter=§ 184, § 359, § 368 |volume=1 |orig-date=September 1928 |publisher=[[Open court publishing company]] |___location=Chicago, US |date=1993 |edition=two volumes in one unaltered reprint |pages=[https://archive.org/details/historyofmathema00cajo_0/page/193 193, 402–403, 411–412] |isbn=0-486-67766-4 |lccn=93-29211 |url=https://archive.org/details/historyofmathema00cajo_0/page/193 |access-date=2019-07-22 |quote-pages=402–403, 411–412 |quote=§359. […] ∥ for parallel occurs in [[William Oughtred|Oughtred]]'s ''Opuscula mathematica hactenus inedita'' (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when [[Robert Recorde|Recorde]]'s sign of equality won its way upon [[the Continent]], vertical lines came to be used for parallelism. We find ∥ for "parallel" in [[John Kersey the elder|Kersey]],{{citeref|A|ref=FC-A}} [[John Caswell|Caswell]], [[William Jones (mathematician)|Jones]],{{citeref|B|ref=FC-B}} Wilson,{{citeref|C|ref=FC-C}} [[William Emerson (mathematician)|Emerson]],{{citeref|D|ref=FC-D}} Kambly,{{citeref|E|ref=FC-E}} and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens{{citeref|F|ref=FC-F}} use "par{{citeref|F|ref=FC-F}} or ∥" for parallel […] {{anchor|FC-A}}[A] [[John Kersey the elder|John Kersey]], ''{{citeref|Kersey (the elder)|1673|Algebra|style=plain}}'' (London, 1673), Book IV, p. 177. {{anchor|FC-B}}[B] [[William Jones (mathematician)|W. Jones]], ''Synopsis palmarioum matheseos'' (London, 1706). {{anchor|FC-C}}[C] John Wilson, ''Trigonometry'' (Edinburgh, 1714), characters explained. {{anchor|FC-D}}[D] [[William Emerson (mathematician)|W. Emerson]], ''Elements of Geometry'' (London, 1763), p. 4. {{anchor|FC-E}}[E] {{ill|Ludwig Kambly{{!}}L.<!-- Ludwig --> Kambly|de|Ludwig Kambly}}, ''Die Elementar-Mathematik'', Part 2: ''Planimetrie'', 43. edition (Breslau, 1876), p. 8. […] {{anchor|FC-F}}[F] H. S.<!-- Henry Sinclair --> Hall and F. H.<!-- Frederick Haller --> Stevens, ''Euclid's Elements'', Parts I and II (London, 1889), p. 10. […]}} [https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf]</ref>

<ref name="Seshu_1956">{{cite journal |title=On Electrical Circuits and Switching Circuits |date=September 1956 |author-first=Sundaram |author-last=Seshu |journal=[[IRE Transactions on Circuit Theory]] |volume=CT-3 |issue=3 |publisher=[[Institute of Radio Engineers]] (IRE) |pages=172–178 |doi=10.1109/TCT.1956.1086310 |url=https://www.researchgate.net/publication/3440524_On_Electrical_Circuits_and_Switching_Circuits}} (7 pages) (NB. See {{citeref|Seshu|Gould|1957|errata|style=plain}}.)</ref>

<ref name="Seshu_1957">{{cite journal |title=Correction to 'On Electrical Circuits and Switching Circuits' |series=Correction |date=September 1957 |author-first1=Sundaram |author-last1=Seshu |author-first2=Roderick |author-last2=Gould |journal=[[IRE Transactions on Circuit Theory]] |volume=CT-4 |issue=3 |publisher=[[Institute of Radio Engineers]] (IRE) |page=284 |doi=10.1109/TCT.1957.1086390 |url=https://www.researchgate.net/publication/3440591_Correction_to_'On_Electric_Circuits_and_Switching_Circuits'|doi-access=free }} (1 page) (NB. Refers to {{citeref|Seshu|1956|previous|style=plain}} reference.)</ref>

<ref name="Erickson_1959">{{cite journal |author-first=Kent E. |author-last=Erickson |title=A New Operation for Analyzing Series-Parallel Networks |journal=[[IRE Transactions on Circuit Theory]] |volume=CT-6 |issue=1 |date=March 1959 |publisher=[[Institute of Radio Engineers]] (IRE) |doi=10.1109/TCT.1959.1086519 |pages=124–126 |url=https://www.researchgate.net/publication/3440722_A_New_Operation_for_Analyzing_Series-Parallel_Networks |quote-page=124 |quote=[…] The operation&nbsp;∗ is defined as A&nbsp;∗ B = AB/A&nbsp;+ B. The symbol&nbsp;∗ has algebraic properties which simplify the formal solution of many series-parallel network problems. If the operation&nbsp;∗ were included as a subroutine in a [[digital computer]], it could simplify the programming of certain network calculations. […]}} (3 pages) (NB. See {{citeref|Kaufman|1963|comment|style=plain}}.)</ref>

<ref name="Kaufman_1963">{{cite journal |author-first=Howard |author-last=Kaufman |title=Remark on a New Operation for Analyzing Series-Parallel Networks |journal=[[IEEE Transactions on Circuit Theory]] |volume=CT-10 |issue=2 |date=June 1963 |publisher=[[Institute of Electrical and Electronics Engineers]] (IEEE) |doi=10.1109/TCT.1963.1082126 |page=283 |quote-page=283 |quote=[…] Comments on the operation&nbsp;∗ […] a∗b&nbsp;= ab/(a+b) […]}} (1 page) (NB. Refers to {{citeref|Erickson|1959|previous|style=plain}} reference.)</ref>

<ref name="Duffin_1966">{{cite journal |title=Network synthesis through hybrid matrices |author-last1=Duffin |author-first1=Richard James |author-link1=Richard James Duffin |author-first2=Dov |author-last2=Hazony |author-first3=Norman Alexander |author-last3=Morrison |journal=[[SIAM Journal on Applied Mathematics]] |publisher=[[Society for Industrial and Applied Mathematics]] (SIAM) |volume=14 |number=2 |date=March 1966 |orig-date=1965-04-12, 1964-08-25 |pages=390–413 |jstor=2946272 |doi=10.1137/0114032}} (24 pages)</ref>

<ref name="Anderson-Duffin_1969">{{cite journal |author-last1=Anderson, Jr. |author-first1=William Niles<!-- https://web.archive.org/web/20110723004431/http://www.tomstrong.org/kappa/03/b_ind_0399.html --> |author-last2=Duffin |author-first2=Richard James |author-link2=Richard James Duffin |title=Series and parallel addition of matrices |journal=[[Journal of Mathematical Analysis and Applications]] |publisher=[[Academic Press, Inc.]] |volume=26 |issue=3 |pages=576–594 |date=1969 |orig-date=1968-05-27 |doi=10.1016/0022-247X(69)90200-5 |quote-page=576 |quote=[…] we define the parallel sum of A and B by the formula A(A&nbsp;+ B)⁺B and denote it by A&nbsp;: B. If A and B are nonsingular this reduces to A&nbsp;: B&nbsp;= (A⁻¹&nbsp;+ B⁻¹)⁻¹ which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the [[Hermitian matrix|Hermitian]] semi-definite matrices form a [[commutative law|commutative]] partially ordered semigroup under the parallel sum operation. […]|doi-access=free }} [https://kilthub.cmu.edu/articles/Series_and_parallel_addition_of_matrices/6479477/1]</ref>

<ref name="Mitra_1970">{{cite journal |title=A Matrix Operation for Analyzing Series-parallel Multiports |journal=[[Journal of the Franklin Institute]] |publisher=[[Franklin Institute]] |series=Brief Communication |volume=289 |issue=2 |pages=167–169 |date=February 1970 |doi=10.1016/0016-0032(70)90302-9 |author-first=Sujit Kumar |author-last=Mitra<!-- 1932–2004 http://www.iisc.ernet.in/currsci/aug102004/395.pdf --> |url=https://www.researchgate.net/publication/239368282_A_matrix_operation_for_analyzing_series-parallel_multiports |quote-page=167 |quote=The purpose of this communication is to extend the concept of the [[scalar (mathematics)|scalar]] operation Reduced Sum introduced by {{citeref|Seshu|1956|Seshu|style=plain}} […] and later elaborated by {{citeref|Erickson|1959|Erickson|style=plain}} […] to matrices, to outline some interesting properties of this new matrix operation, and to apply the matrix operation in the analysis of series and parallel [[n-port network|''n''-port network]]s. Let A and B be two non-singular [[square matrices]] having [[inverse matrix|inverses]], A⁻¹ and B⁻¹ respectively. We define the operation&nbsp;∙ as A&nbsp;∙ B&nbsp;= (A⁻¹&nbsp;+ B⁻¹)⁻¹ and the operation&nbsp;⊙ as A&nbsp;⊙ B&nbsp;= A&nbsp;∙ (−B). The operation&nbsp;∙ is [[commutative law|commutative]] and [[associative law|associative]] and is also [[distributive law|distributive]] with respect to multiplication. […]}} (3 pages)</ref>

<ref name="Duffin_1971">{{cite book |title=Mathematical Aspects of Electrical Network Analysis |series=Proceedings of a Symposium in Applied Mathematics of the American Mathematical Society and the Society for Industrial and Applied Mathematics held in New York City, 1969-04-02/03 |chapter=Network Models |pages=65–92 [68] |volume=III of SIAM-AMS Proceedings |date=1971 |orig-date=1970, 1969 |publisher=[[American Mathematical Society]] (AMS) / [[Society for Industrial and Applied Mathematics]] (SIAM) |edition=illustrated |publication-place=Providence, Rhode Island |editor-first1=Herbert Saul |editor-last1=Wilf |editor-link1=Herbert Saul Wilf |editor-first2=Frank |editor-last2=Hararay |editor-link2=Frank Harary |author-first=Richard James |author-last=Duffin |author-link=Richard James Duffin |___location=Durham, North Carolina, USA |issn=0080-5084 |isbn=0-8218-1322-6 |id={{ISBN|978-0-8218-1322-5}}. Report 69-21 |lccn=79-167683 |chapter-url=https://books.google.com/books?id=j2vFhxA5K-UC&pg=PA65 |url=https://books.google.com/books?id=j2vFhxA5K-UC |access-date=2019-08-05 |quote-pages=68–69 |quote=[…] To have a convenient short notation for the joint resistance of resistors connected in parallel let […] A:B = AB/(A+B) […] A:B may be regarded as a new operation termed ''parallel addition'' […] Parallel addition is defined for any nonnegative numbers. The network model shows that parallel addition is [[commutative law|commutative]] and [[associative law|associative]]. Moreover, multiplication is [[distributive law|distributive]] over this operation. Consider now an algebraic expression in the operations (+) and (:) operating on positive numbers A, B, C, etc. […] To give a network interpretation of such a polynomial read ''A&nbsp;+ B'' as "A series B" and ''A&nbsp;: B'' as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […]}} [https://kilthub.cmu.edu/articles/Network_models/6478514] [https://kilthub.cmu.edu/ndownloader/files/11915081] (206 pages)</ref>

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<ref name="MIT_1978">{{cite book |title=Report of the President and the Chancellor 1977–78 – Massachusetts Institute of Technology |chapter=School of Engineering – Center for Advanced Engineering Study (C.A.E.S.) – Research and Development – Technical Curriculum Research and Development Project |editor-first1=Jerome Bert |editor-last1=Wiesner |editor-link1=Jerome Bert Wiesner |editor-first2=Howard Wesley |editor-last2=Johnson |editor-link2=Howard Wesley Johnson |editor-first3=James Rhyne |editor-last3=Killian, Jr. |editor-link3=James Rhyne Killian |date=1978-04-11 |publisher=[[Massachusetts Institute of Technology]] (MIT) |pages=249, 252–253 |url=https://libraries.mit.edu/archives/mithistory/presidents-reports/1978.pdf |access-date=2019-08-08 |url-status=live |archive-url=https://web.archive.org/web/20150910145633/http://libraries.mit.edu/archives/mithistory/presidents-reports/1978.pdf |archive-date=2015-09-10 |quote-pages=249, 252–253 |quote=[…] The Technical Curriculum Research and Development Program, sponsored by the {{ill|Imperial Organization of Social Services|fa|سازمان شاهنشاهی خدمات اجتماعی}} of [[Iran]], is entering the fourth year of a five-year contract.<!-- 1974–1979 --> Curriculum development in electronics and mechanical engineering continues. […] Administered jointly by [[C.A.E.S. (MIT)|C.A.E.S.]] and the [[MIT Department of Materials Science and Engineering|Department of Materials Science and Engineering]], the Project is under the supervision of Professor Merton C. Flemings. It is directed by Dr. John W. McWane. […] Curriculum Materials Development. This is the principal activity of the project and is concerned with the development of innovative, state-of-the-art course materials in needed areas of engineering technology […] new introductory course in electronics […] is entitled Introduction to Electronics and Instrumentation and consists of eight […] modules […] dc Current, Voltage, and Resistance; Basic Circuit Networks; Time Varying Signals; Operational Amplifiers; Power Supplies; ac Current, Voltage, and Impedance; Digital Circuits; and Electronic Measurement and Control. This course represents a major change and updating of the way in which electronics is introduced, and should be of great value to [[Shiraz Technical Institute|STI]] as well as to many US programs. […]}}</ref>

<ref name="Senturia-Wedlock_1974">{{cite book |title=Electronic Circuits and Applications |chapter=Part A. Learning the Language, Chapter 3. Linear Resistive Networks, 3.2 Basic Network Configurations, 3.2.3. Resistors in Parallel |date=1975 |orig-date=August 1974 |author-first=Stephen D. |author-last=Senturia |author-link=:d:Q94138984 |author-first2=Bruce D. |author-last2=Wedlock |___location=Massachusetts Institute of Technology, Cambridge, Massachusetts, USA |publisher=[[John Wiley & Sons, Inc.]] |publication-place=New York, London, Sydney, Toronto |edition=1 |isbn=0-471-77630-0 |lccn=74-7404 |s2cid=61070327 |pages=viii–ix, 44–46 [45] |quote-pages=viii, ix, 45 |quote=This textbook evolved from a one-semester introductory electronics course taught by the authors at the [[Massachusetts Institute of Technology]]. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor [[Campbell Searle]], who taught the introductory electronics course when one of us ([[:d:Q94138984|S.D.S.]]) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor [[Louis Dijour Smullin|Louis D. Smullin]], has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R₁ and R₂ is the same as that which occurs across a single resistance of value R₁&nbsp;R₂&nbsp;(R₁&nbsp;+&nbsp;R₂). Because this expression for parallel resistance occurs so often, it is given a special notation (R₁∥R₂). That is, when R₁ and R₂ are in parallel, the equivalent resistance is <math>(R_1 \parallel R_2) = \frac{R_1 R_2}{R_1+R_2}</math> […]}} (xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 {{citeref|Wedlock|1978|book|style=plain}}.) [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4321201]</ref>

<ref name="Wedlock_1978">{{cite book |title=Basic circuit networks |series=Introduction to electronics and instrumentation |work=Technical Curriculum Research and Development Project |date=1978 |author-first=Bruce D. |author-last=Wedlock |publisher=[[Massachusetts Institute of Technology]] (MIT), Technical Curriculum Research and Development Project }} (81 pages) (NB. This formed the basis for Part I of McWane's 1981 {{citeref|McWane|1981|book|style=plain}}. See also: Senturia's and Wedlock's 1975 {{citeref|Senturia|Wedlock|1975|book|style=plain}}.)</ref>

<ref name="McWane_1981">{{cite book |title=Introduction to Electronics and Instrumentation |author-first=John W. |author-last=McWane <!-- |contribution=Illustrations |contributor-first1=Barry I. |contributor-last1=Levine |contributor-first2=Julie |contributor-last2=Gecha --> |date=1981-05-01 |edition=illustrated |publisher=[[Breton Publishers]], [[Wadsworth, Inc.]] |___location=North Scituate, Massachusetts, USA |isbn=0-53400938-7 |id={{ISBN|978-0-53400938-0}} |pages=78, 96–98, 100, 104 |url=https://books.google.com/books?id=NKy1nQ8Y_V0C |access-date=2019-08-04 |quote-page=xiii, 96–98, 100 |quote=[…] Bruce D. Wedlock […] was the principle contributing author to Part I, {{citeref|Wedlock|1978|BASIC CIRCUIT NETWORKS|style=plain}} including the design of the companion examples. […] Most of the development of the IEI program was undertaken as part of the {{citeref|Wiesner|Johnson|Killian, Jr.|1978|Technical Curriculum Research and Development Project|style=plain}} of the [[MIT Center of Advanced Engineering Study]]. […] shorthand notation […] shorthand symbol ∥ […]}} (xiii+545 pages) (NB. In 1981, a 216-pages laboratory manual accompanying this book existed as well. The work grew out of an [[Massachusetts Institute of Technology|MIT]] course program "{{citeref|Wiesner|Johnson|Killian, Jr.|1978|The MIT Technical Curriculum Development Project - Introduction to Electronics and Instrumentation|style=plain}}" developed between 1974 and 1979. In 1986, a second edition of this book was published under the title "Introduction to Electronics Technology".)</ref>

<ref name="Eriksson-Bique-Leutwiler_1989">{{cite journal |author-last1=Eriksson-Bique |author-first1=Sirkka-Liisa Anneli |author-link1=:fi:Sirkka-Liisa Eriksson |author-last2=Leutwiler |author-first2=Heinz |title=A generalization of parallel addition |journal=[[Aequationes Mathematicae]] |publisher=[[Birkhäuser Verlag]] |date=February 1989 |orig-date=1989-01-10 |volume=38 |issue=1 |pages=99–110 |doi=10.1007/BF01839498 |url=https://link.springer.com/content/pdf/10.1007/BF01839498.pdf |access-date=2020-08-20 |url-status=live |archive-url=https://web.archive.org/web/20200820155352/https://link.springer.com/content/pdf/10.1007/BF01839498.pdf |archive-date=2020-08-20}}</ref>

<ref name="Ellerman_1995">{{cite book |title=Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics |chapter=Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics |series=G – Reference, Information and Interdisciplinary Subjects Series |work=The worldly philosophy: studies in intersection of philosophy and economics |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |edition=illustrated |publisher=[[Rowman & Littlefield Publishers, Inc.]] |date=1995-03-21 |isbn=0-8476-7932-2 |pages=237–268 |url=http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |chapter-url=https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237 |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20160305012729/http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |archive-date=2016-03-05 |quote-page=237 |quote=[…] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the ''[[series sum]]'') of the resistances a&nbsp;+ b. If the resistances are placed in parallel, their compound resistance is the ''parallel sum'' of the resistances, which is denoted by the [[full colon]] […]}} [https://web.archive.org/web/20150917191423/http://www.ellerman.org/Davids-Stuff/Maths/sp_math.doc] (271 pages)</ref>

<ref name="Georg_1999">{{cite book |title=Elektromagnetische Felder und Netzwerke: Anwendungen in Mathcad und PSpice |chapter=Chapter 2.11.4.3: Aufstellen der Differentialgleichung aus der komplexen Darstellung - MATHCAD Anwendung 2.11-6: Benutzerdefinierte Operatoren |chapter-url=https://books.google.com/books?id=wBAjBgAAQBAJ&pg=PA246 |language=de |author-first=Otfried |author-last=Georg |date=2013 |orig-date=1999 |edition=1 |series=Springer-Lehrbuch |publisher=[[Springer-Verlag]] |isbn=978-3-642-58420-6 |id={{ISBN|3-642-58420-9}} |doi=10.1007/978-3-642-58420-6 |pages=246–248 |url=https://books.google.com/books?id=wBAjBgAAQBAJ |access-date=2019-08-04}} (728 pages)</ref>

<ref name="Ellerman_2004">{{cite web |title=Introduction to Series-Parallel Duality |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |publisher=[[University of California at Riverside]] |date=May 2004 |orig-date=1995-03-21 |citeseerx=10.1.1.90.3666 |url=http://www.ellerman.org/wp-content/uploads/2012/12/Series-Parallel-Duality.CV_.pdf |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20190810011716/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf<!-- https://archive.today/20190810080659/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf --> |archive-date=2019-08-10 |quote=The parallel sum of two positive real numbers x:y&nbsp;= [(1/x)&nbsp;+ (1/y)]⁻¹ arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a [[duality (mathematics)|duality]] between the usual [[series sum|(series) sum]] and the parallel sum. […]}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf] (24 pages)</ref>