இணை இயக்கி
[[File:Parallelitätsoperator.svg|thumb| &lt;math&gt;a \parallel b = c&lt;/math&gt; எனும் இணை இயக்கியை வரைபடத்தால் விளக்குதல்]]
 
 இணை இயக்கி ('''parallel operator''' &lt;math&gt;\|&lt;/math&gt;)(இது '''கொணர்கூட்டல்''' அல்லது '''இணை கூட்டுத்தொகை''' அல்லது '''இணை கூட்டல்''' எனவும் வழங்கும்)&amp;nbsp; இங்கு&lt;math&gt;\|&lt;/math&gt; ( &quot;இணை&quot; எனும் சொல்,&lt;ref name=&quot;Duffin_1971&quot;/&gt; வடிவியலின் இணைகோடுகள் குறியீட்டைப் பின்பற்றி வழங்கும் சொல்லாகும்)&lt;ref name=&quot;Kersey_1673&quot;/&gt;&lt;ref name=&quot;Cajori_1928&quot;/&gt;)  என்பது ஒரு [[சார்பு]] (கணிதம்) ஆகும்; இது குறிப்பாக [[மின்பொறியியல்|மின் பொறியியலிலும்]]&lt;ref name=&quot;TI_2004&quot;/&gt;&lt;ref name=&quot;Bober_2016&quot;/&gt;&lt;ref name=&quot;EECS_2018&quot;/&gt;&lt;ref group=&quot;nb&quot; name=&quot;NB_Origin&quot;/&gt;  இயக்கவியலிலும் பாய்ம இயக்கவியலிலும் நிதிசார் கணிதவியலிலும் சுருக்கெழுத்தாக  பயன்படுத்தப்படுகிறது. .&lt;ref name=&quot;Ellerman_1995&quot;/&gt;&lt;ref name=&quot;Ellerman_2004&quot;/&gt; இது [[பெருக்கல் நேர்மாறு|பெருக்கல் நேர்மாறைக்]] குறிப்பதோடு அதை வரையறுக்கிறது. 
 
 &lt;math&gt;\begin{array}{rlcl}
 \|:\ &amp; \overline{\mathbb{C}} \times \overline{\mathbb{C}} &amp;\to&amp; \overline{\mathbb{C}}\\
 &amp; (a,b) &amp;\mapsto&amp; a \| b = \frac{1}{\frac{1}{a} + \frac{1}{b}}
 \end{array}&lt;/math&gt;
 
 அதாவது, இது இரண்டு எண்களான a,  b  ஆகியவர்ரின் [[இசைச் சராசரி]]யைக் காட்டுகின்றது.
 
 ==குறிப்புகள்==
 {{Reflist|group=&quot;nb&quot;|refs=
 &lt;ref group=&quot;nb&quot; name=&quot;NB_Origin&quot;&gt;While the use of the symbol ∥ for &quot;parallel&quot; in [[வடிவவியல்]] reaches as far back as 1673 in John Kersey the elder's work,{{citeref|Kersey_(the_elder)|1673|A}} this came into more use only since about 1875.{{citeref|Cajori|1993|B}}&lt;!-- according to Florian Cajori in 1928 --&gt; The usage of a mathematical operator for [[தொடர் இணைப்பு மற்றும் பக்க இணைப்பு மின்வழிகள்]]s originates from network theory in [[மின்பொறியியல்]]. Sundaram Seshu introduced a ''reduced sum'' operator in 1956,{{citeref|Seshu|1956|C}} Kent E. Erickson proposed an [[உடுக்குறி (தமிழ் நடை)|உடுக்குறி]] (∗) to symbolize the operator in 1959,{{citeref|Erickson|1959|D}} whilst Richard James Duffin and William Niles Anderson, Jr. used a [[முக்கால்புள்ளி (தமிழ் நடை)|முக்கால்புள்ளி]] (:) for the ''parallel addition'' since 1966.{{citeref|Duffin|Hazony|Morrison|1966|E}} Sujit Kumar Mitra used a middot (∙) for it in 1970.{{citeref|Mitra|1970|F}} The first usage of the parallel symbol (∥) for this operator in applied electronics is unknown, but might have originated from {{ill|Stephen D. Senturia|d|Q94138984}} and Bruce D. Wedlock's 1974 book &quot;Electronic Circuits and Applications&quot;,{{citeref|Senturia|Wedlock|1975|G}} which evolved from their introductory electronics course at [[மாசாச்சூசெட்சு தொழில்நுட்பக் கழகம்]] (MIT) with concepts of teaching network theory and electronics derived from an earlier course taught by Campbell &quot;Cam&quot; Leach Searle. It was further popularized through John W. McWane's 1981 book &quot;Introduction to Electronics and Instrumentation&quot;,{{citeref|McWane|1981|H}} which grew out of an identically-named MIT course developed as part of the influential ''{{citeref|Wiesner|Johnson|Killian, Jr.|1978|Technical Curriculum Development Project|style=plain}}'' between 1974 and 1979. This symbol was probably also introduced because the other used symbols could be easily confused with signs commonly used for [[பெருக்கல் (கணிதம்)]] and [[வகுத்தல் (கணிதம்)|division]] in some contexts.&lt;/ref&gt;
 }}
 
 == மேற்கோள்கள் ==
 {{Reflist|refs=
 
 &lt;ref name=&quot;Duffin_1971&quot;&gt;{{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&amp;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 [[பரிமாற்றுத்தன்மை|commutative]] and [[சேர்ப்புப் பண்பு|associative]]. Moreover, multiplication is [[பங்கீட்டுப் பண்பு|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&amp;nbsp;+ B'' as &quot;A series B&quot; and ''A&amp;nbsp;: B'' as &quot;A parallel B&quot; 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)&lt;/ref&gt;
 
 &lt;ref name=&quot;Kersey_1673&quot;&gt;{{cite book |author-first=John |author-last=Kersey (the elder) |title=The Elements of that Mathematical Art, commonly called Algebra&lt;!-- , Expounded in Four Books --&gt; |location=London |date=1673 |volume=Book IV - The Elements of the Algebraical Arts |chapter=Chapter I: Concerning the Scope of this fourth Book and the Signification of Characters, Abbreviations and Citations used therein |publisher=Thomas Passinger, Three-Bibles, London-Bridge |pages=177–178 |url=https://archive.org/details/b30333726_0002 |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20200805135008/https://archive.org/details/b30333726_0002 |archive-date=2020-08-05}}&lt;/ref&gt;
 
 &lt;ref name=&quot;Cajori_1928&quot;&gt;{{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 Oughtred's ''Opuscula mathematica hactenus inedita'' (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for &quot;parallel&quot; in Kersey,{{citeref|A|ref=FC-A}} Caswell, Jones,{{citeref|B|ref=FC-B}} Wilson,{{citeref|C|ref=FC-C}} 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 &quot;par{{citeref|F|ref=FC-F}} or ∥&quot; for parallel […] {{anchor|FC-A}}[A] John Kersey, ''{{citeref|Kersey (the elder)|1673|Algebra|style=plain}}'' (London, 1673), Book IV, p. 177. {{anchor|FC-B}}[B] W. Jones, ''Synopsis palmarioum matheseos'' (London, 1706). {{anchor|FC-C}}[C] John Wilson, ''Trigonometry'' (Edinburgh, 1714), characters explained. {{anchor|FC-D}}[D] W. Emerson, ''Elements of Geometry'' (London, 1763), p. 4. {{anchor|FC-E}}[E] {{ill|Ludwig Kambly{{!}}L.&lt;!-- Ludwig --&gt; Kambly|de|Ludwig Kambly}}, ''Die Elementar-Mathematik'', Part 2: ''Planimetrie'', 43. edition (Breslau, 1876), p. 8. […] {{anchor|FC-F}}[F] H. S.&lt;!-- Henry Sinclair --&gt; Hall and F. H.&lt;!-- Frederick Haller --&gt; 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]&lt;/ref&gt;
 
 &lt;ref name=&quot;Ellerman_1995&quot;&gt;{{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 |edition=illustrated |publisher=Rowman &amp; 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&amp;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 ''[[கூட்டல் (கணிதம்)|கூட்டல்]]'') of the resistances a&amp;nbsp;+ b. If the resistances are placed in parallel, their compound resistance is the ''parallel sum'' of the resistances, which is denoted by the [[முக்கால்புள்ளி (தமிழ் நடை)|முக்கால்புள்ளி]] […]}} [https://web.archive.org/web/20150917191423/http://www.ellerman.org/Davids-Stuff/Maths/sp_math.doc] (271 pages)&lt;/ref&gt;
 
 &lt;ref name=&quot;Ellerman_2004&quot;&gt;{{cite web |title=Introduction to Series-Parallel Duality |author-first=David Patterson |author-last=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&amp;rep=rep1&amp;type=pdf&lt;!-- https://archive.today/20190810080659/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&amp;rep=rep1&amp;type=pdf --&gt; |archive-date=2019-08-10 |quote=The parallel sum of two positive real numbers x:y&amp;nbsp;= [(1/x)&amp;nbsp;+ (1/y)]&lt;sup&gt;−1&lt;/sup&gt; arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a duality between the usual [[கூட்டல் (கணிதம்)|(series) sum]] and the parallel sum. […]}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&amp;rep=rep1&amp;type=pdf] (24 pages)&lt;/ref&gt;
 
 &lt;ref name=&quot;TI_2004&quot;&gt;{{cite web |title=INA 326/INA 327 – Precision, Rail-to-Rail I/O Instrumentation Amplifier |publisher=Burr-Brown / Texas Instruments |date=2018 |id=SBOS222D |orig-date=November 2004, November 2001 |pages=3, 9, 13 |url=http://www.ti.com/lit/ds/sbos222d/sbos222d.pdf |access-date=2019-07-13 |url-status=live |archive-url=https://web.archive.org/web/20190713200507/http://www.ti.com/lit/ds/sbos222d/sbos222d.pdf |archive-date=2019-07-13}}&lt;/ref&gt;
 
 &lt;ref name=&quot;Bober_2016&quot;&gt;{{cite book |title=Numerical and Analytical Methods with MATLAB for Electrical Engineers |author-first1=William |author-last1=Bober |author-first2=Andrew |author-last2=Stevens |chapter=Chapter 7.6. Laplace Transforms Applied to Circuits |series=Applied and Computational Mechanics |edition=1 |publisher=CRC Press |date=2016 |isbn=978-1-46657607-0 |id={{ISBN|1-46657607-3}} |page=224 |url=https://books.google.com/books?id=O2POBQAAQBAJ |chapter-url=https://books.google.com/books?id=O2POBQAAQBAJ&amp;pg=PA224}} (388 pages)&lt;/ref&gt;
 &lt;ref name=&quot;EECS_2018&quot;&gt;{{cite book |title=EECS 16A Designing Information Devices and Systems I |chapter=Chapter 15.7.2 Parallel Resistors |editor-first1=Gireeja |editor-last1=Ranade |editor-first2=Vladimir |editor-last2=Stojanovic |date=Fall 2018 |type=lecture notes |id=Note 15 |page=12 |publisher=[[கலிபோர்னியா பல்கலைக்கழகம் (பெர்க்லி)]] |url=https://inst.eecs.berkeley.edu/~ee16a/fa18/lectures/Note15.pdf |chapter-url=https://inst.eecs.berkeley.edu/~ee16a/fa18/lectures/Note15.pdf#page=12 |access-date=2018-12-28 |url-status=live |archive-url=https://web.archive.org/web/20181227224413/https://inst.eecs.berkeley.edu/~ee16a/fa18/lectures/Note15.pdf |archive-date=2018-12-27 |quote-page=12 |quote=[…] This mathematical relationship comes up often enough that it actually has a name: the &quot;parallel operator&quot;, denoted ∥. When we say x∥y, it means &lt;math&gt;\frac{xy}{x+y}&lt;/math&gt;. Note that this is a mathematical operator and does not say anything about the actual configuration. In the case of resistors the parallel operator is used for parallel resistors, but for other components (like capacitors) this is ''not'' the case. […]}} (16 pages)&lt;/ref&gt;
 }}
 
 ==வெளி இணைப்புகள்==
 * https://github.com/microsoftarchive/edx-platform-1/blob/master/common/lib/calc/calc/calc.py&lt;!-- 2013 --&gt;
 
 [[பகுப்பு:நுண்புல இயற்கணிதம்]]
 [[பகுப்பு:அடிப்படை இயற்கணிதம்]]