Talk:Parallel (operator)

Expert help needed
This unsourced article is translated from an unsourced German wikipedia article. Can an expert provide a sourcce? It's not easy to search via Google etc for "parallel". Thanks for any help. Pam D  14:04, 12 May 2016 (UTC)


 * Here's a source for this: Page 12 here https://inst.eecs.berkeley.edu/~ee16a/fa18/lectures/Note15.pdf. I haven't looked in detail at the article, so it could have errors, but it's in the right direction. The operator is in my experience (Studied physics in UK, and work with electronics) is rare, but does turn up in data sheets. ( I found this article researching the use of the operator in this data sheet http://www.ti.com/lit/ds/sbos222d/sbos222d.pdf )Russetrob (talk) 23:05, 27 December 2018 (UTC)


 * Ok went ahead and added references. The mathematics looks OK, but not certain on the arrow notation: While it definitely takes complex numbers, and results in complex numbers, I'm not sure if this is expressed correctly here.
 * Probably should be reviewed by a mathematician, and someone who teaches electrical engineeringRussetrob (talk) 02:31, 28 December 2018 (UTC)


 * Just added edit for the applications section.
 * Not 100% happy with wording, but it conveys 2 things:
 * Specifies that the operator can be applied to impedance in general.
 * Provides a backlink to the Series and parallel circuits :page.Russetrob (talk) 19:27, 13 July 2019 (UTC)

Computer algebra
this has nothing to do with the current article. Again, it should be “parallel (symbol)” because different sources ascribe unrelated meanings to the symbol. Incnis Mrsi (talk) 10:41, 14 July 2019 (UTC)
 * This usage in computer sciences is actually called "Paralleloperator" in the German source, so it belongs in the current article, although not as its main theme, which is about its usage in physics/network theory. This is why it is only mentioned under See also, one of the possible places for such links per WP:SEEALSO. Another possibility, but less desirable, would be a disambiguating hatnote, but I found that inadequate during the AfD.
 * I don't agree with you that Parallel (operator) should redirect to Parallel (symbol), but I could agree to having a disambiguation page under Parallel (symbol) listing the, so far, three different meanings: in geometry, in mathematics/physics/network theory, and in process algebra/computer sciences. But I think for our audience it is also important to at least briefly learn about the usages in other areas, so there should still be some crosslinking between the target articles or through a hatnote pointing to the disambiguation page.
 * Once we'd have a disambiguation page under Parallel (symbol) we should add a link to it to Parallel, but not remove the already existing links to the target articles.
 * --Matthiaspaul (talk) 11:04, 14 July 2019 (UTC)

Origin of ∥ as parallel operator
While we tracked down the origin of the "∥" symbol for "parallel" in geometry to John Kersey the elder in 1673, the exact origin of the usage of this symbol for parallel resistors in engineering is not yet known.

We have narrowed down the window in time to somewhen between 1956 (when the mathematical operator was first suggested in network theory, although the various authors still used other symbols like "∗", ":" or "∙" for it) and 1981 (when McWane's influential book used "∥" for it). However, McWane's book was based on the "MIT Technical Curriculum Development Project - Introduction to Electronics and Instrumentation" program, which took place between 1974 and 1979. It is quite likely that "∥" was already used during this program, but so far I could not find a source earlier than 1981 actually using it. Likewise, in the research on network theory most authors used the ":" symbol up into the early 1970s (and sometimes even up to the present). It is therefore likely that the usage of "∥" started somewhen in the mid-1970s in the USA. Given that the MIT program was partially sponsored by an Iranian organization (the Imperial Organization of Social Services (IOSS/IoSS)), the usage of the "∥" operator might have started to occur also in Middle Eastern publications since then. If you see it being used in this context in any sources older than 1981 (or in sources in other languages) please add them to the article or report this here, so that we can further track this down.

--Matthiaspaul (talk) 23:35, 9 August 2019 (UTC)


 * I found the "∥" operator being used in "Electronic Circuits and Applications" by Stephen D. Senturia and Bruce D. Wedlock written in 1974 and published in 1975. According to the authors, the book evolved from an introductory electronics course which was taught at MIT since about 1969, with the concepts of teaching network theory and electronics based on Campbell Searle's work.
 * It does not become clear if the "∥" notation was introduced in this book or the mentioned MIT course (or an earlier course by Campbell Searle), or elsewhere, therefore, if you find it being mentioned in any kind of publication prior to 1974, please report this here.
 * --Matthiaspaul (talk) 15:33, 3 October 2021 (UTC)

Answer to "What is $0 ‖ 0$ ?" clarification request
An alert and cautious editor inserted the following

I inserted the following comment, and commented out the clarification request:


 * The answer rests on a technicality: 0, $∞ ∊ ℂ$ have no phase: Any definition of phase has a discontinuity at both of those points, so being non-unique, does not exist. Checking limits of $δ  ‖  ε$ as $(δ,ε) ⟶ (0,0)$, both one after the other and simultaneously, is enough to convince one that $abs(δ ‖ ε) ⟶ 0 ,$ regardless of phase issues, and even though the phase of the limit is random (depends on direction of approach), that's immaterial, since no phase exists for the result, because zero and infinity have that exclusive property (no phase) out of all the other numbers in $$\, \overline{\, \mathbb{C} \,} ~.$$

Correct me if I'm wrong, but this question is the kind of thing that I don't think a general reader would really be bothered by, and I don't think its a particularly enlightening issue, because the (my) answer relies on an exotic property of the two numbers 0 and $∞$: Neither has a phase, so a phase discontinuity for any approach to either of them doesn't matter.

My take on this is that it's the sort of thing that if you know enough to recognize the problem and be bothered by it, then you should also be able to work out the answer with a few moments of thought and a little pencil work. So I suggest that this issue and it's easy-out answer not be added to the article. It isn't that it's too hard to follow, it's just that the whole thing is a let-down / time-waster for the reader (either general or technical). Astro-Tom-ical (talk) 09:17, 1 March 2022 (UTC)


 * $$0 \parallel 0 = 0.$$ This seems unambiguous. By the definition, $$0 \parallel 0 = \frac{1}{\infty + \infty} = \frac1\infty = 0.$$ –jacobolus (t) 06:23, 21 December 2022 (UTC)


 * Oh I see, the issue is that $$\infty + \infty$$ could have limit 0 if we take $$\lim_{a \to \infty} a + (-a).$$ I think it still probably usually makes sense to explicitly define $$0 \parallel 0 = 0.$$ –jacobolus (t) 06:56, 21 December 2022 (UTC)

Second, similar, worse issue
There's a similar, but unresolvable question about $a ‖ b$ for $a ⟶ 0$ and $b ⟶ ∞$, or vice versa. In that case the no-phase technicality doesn't help. If they go separately, then the limit of whichever one is taken first is the value of the function (so that alone gives two different limits). If they go simultaneously, then you can contrive to give them a constant, finite proportion (e.g. set $a = c/b$, or similar, with any constant $c ∊ ℂ\{0}$  ) and then for $b ⟶ ∞$ get $a  ‖  b ⟶ c ≠ 0$ , and no technicality will get you out of trouble; the only way out is to exclude $∞$ from the function domain.

So for that reason, the function has to be constrained to $$\, \mathbb{C} \otimes \mathbb{C} \,$$ only, where there isn't any $∞$, and not its closure $$\, \equiv \overline{\, \mathbb{C} \otimes \mathbb{C} \,} ~.$$ As a practical matter, this shouldn't bother anyone: The mathematicians are trained to dodge singularities; the electrical engineers will have already noticed that parts with infinite resistance, capacitance, or inductance, and signal generators that can provide infinite frequencies to drive them, are not available on the market, or always break before infinity is reached; and physicists should be confident that such items are not feasible, regardless of utility for hand-waving. In a word, no matter how big you can get it (resistance, capacitance, inductance, or frequencies) infinity is just as far away from what you've got as it ever was. Astro-Tom-ical (talk) 09:17, 1 March 2022 (UTC)


 * $$0 \parallel \infty = 0$$ seems reasonable to me. For any $x$, $$x \parallel \infty = \frac1{x^{-1} + 0} = x.$$ –jacobolus (t) 06:23, 21 December 2022 (UTC)

Product over sum
The phrase "product over sum" was removed as un-referenced. Please re-insert when reference is found. Clarification for complex projective line is available for this operation in rings. A new section "Projective view" has been appended and may be expanded, for example with 2x2 matrices. Rgdboer (talk) 21:35, 26 December 2022 (UTC) "phrase" Rgdboer (talk) 21:37, 26 December 2022 (UTC)
 * The mnemonic device "product over sum" doesn't seem all that important, but it's definitely a thing: Google Scholar search for «"product over sum" parallel». –jacobolus (t) 20:57, 27 December 2022 (UTC)
 * I know that original research is banned on Wikipedia, but I think it should differentiate between research that needs citing very much (such as claiming that someone has a certain tattoo or that something usually takes about 46 minutes and 3 seconds (some sources may consider this round or the variation may be small)) and short research that one can easily verify themselves (but it's still better to mark it as original if it is). If "product over sum" was original research *, it would be the latter. Also, there's an intermediate type, where it's easy to deduce from elsewhere on Wikipedia.
 * * Maybe it is — sometimes people come up with the same idea independently: for example, I thought of conic singularities and P2P messaging before finding out others also did (I have a lot of my own research about conic singularities (for which I've (randomly) chosen the name "portals"), but I've only thought of P2P messaging as a way to deliver messages faster and didn't name it then). (I've originally put this on a  tag, but it didn't work in the preview, so I changed it to this.) Orisphera2 (talk) 16:22, 12 May 2024 (UTC)
 * Whether a particular phrase is used as a mnemonic device is not something "one can easily verify themselves" without external input. But it can be easily verified by doing a literature search to find reliable sources. –jacobolus (t) 17:36, 12 May 2024 (UTC)

Yes: c = ab/(a+b); addition of fractions and reciprocation. Details at an elementary level of algebraic geometry have been supplied for the projective view. Rgdboer (talk) 22:58, 27 December 2022 (UTC)

Can we substitute the name "operation" for "operator"?
The term operation is relatively unambiguous, see operation (mathematics), while the term operator often means something else, see operator (mathematics). (The name "operator" in this sense is much more common in computer programming; see operator (computer programming).)

Also, I would recommend we just move this article to parallel operation or parallel addition or the like, instead of using the current parenthesized parallel (operator). –jacobolus (t) 21:14, 27 December 2022 (UTC)

Structure transfer
In section, the field $$\widetilde \C$$ is defined by a “transfer of structure”. It seems that the subsection may be damatically reduced, by removing the proofs, and removing the formulas that are not used in applications. D.Lazard (talk) 11:10, 6 March 2023 (UTC)


 * I don’t understand the complaint. What’s the problem with spelling out the consequences? The section is not inordinately long and seems easy for anyone to skim past. –jacobolus (t) 19:32, 6 March 2023 (UTC)
 * To elaborate a bit, one of my chief complaints with many Wikipedia articles is that readers are assumed to be extremely knowledgeable and sophisticated able to read books worth of subtle consequences into abstract mathematical statements, and fundamental relations are skipped or compressed to the point that non-expert readers are left in the dark. –jacobolus (t) 19:37, 6 March 2023 (UTC)
 * In response to this, I started looking at our existing references to see what parts we can directly cite. This was amusing from Ellerman (1995) and seems relevant to this complaint: """There is a whole "parallel mathematics" which can be developed with the parallel sum replacing the series sum. Since the parallel sum can be defined in terms of the series sum (or viceversa), "parallel mathematics" is essentially a new way of looking at certain known parts of mathematics. Exclusive promotion of the series sum is "series chauvinism" or "serialism." Before venturing further into the parallel universe, we might suggest some exercises to help the reader combat the heritage of series chauvinism. Anytime the series sum seems to occur naturally in mathematics with the parallel sum nowhere in sight, it is an illusion. The parallel sum lurks in a "parallel" role that has been unfairly neglected.""" –jacobolus (t) 22:40, 6 March 2023 (UTC)
 * Hm, so are there interesting things to be said about, for example, Fourier parallels? Or is that a whoosh-bird I hear? —Tamfang (talk) 01:20, 7 March 2023 (UTC)
 * You are making a pun about two different meanings of the word series. "Series" in this context means a string of resistors in a row, though of course that is loosely related to the sense of infinite series. But if you hunted around I am sure you could find plenty of examples of sums like $S = 1\Big/{\sum\limits_{k\in\N} 1/f_k} = \mathop{\mathrm{P}}\limits_{k\in\N}f_k $ where $P$ means "parallel summation". If you do a literature search for "reciprocal Fourier series" I am guessing you could find at least dozens if not hundreds of relevant results. If there were a commonly used notation for this it might even see more use than it does. (Disclaimer: I don’t really have a dog in this fight. From my perspective the parallel operator seems cute but not mind blowing.) –jacobolus (t) 03:49, 7 March 2023 (UTC)

"Further reading" references seem inappropriate as-is
Many (most? all?) of the references in the "Further reading section are about generalizations of this concept to matrices, operators, etc., and seem substantially irrelevant to the content of the article as it is. A new section could perhaps be added called something like 'Generalizations' and these could be added in footnote citations, but it seems inappropriate to me to include them in a "Further reading" section with the implication that they are meaningful general references for typical readers of this article. If no new section is to be added, I would recommend removing these references. (Or perhaps they could be copied onto this talk page for the future event that someone may want to add that section.) –jacobolus (t) 22:17, 6 March 2023 (UTC)