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When we want to perform division, we write e.g. $8/2$ (this is what we already learn at school). But when we want to express that $2$ is a divisor of $8$, we write: $2\mid 8$. What the heck?? I do find this very counterintuitive, I would have expected $8\mid 2$ instead.

So, is there a good reason to write $2\mid 8$ instead of $8\mid 2$, and who invented that notation?

Michael Hardy
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SearchSpace
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2 Answers2

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In mathematics, we often write relations between $a$ and $b$ in the form $aRb$. I mean this both in the sense that we write that string to represent an abstract relation, as well as using that form to write expressions with particular relations. In almost every case, these are read as "$a$ [relation] $b$." For a few examples, we have

  1. $a:=b$, "is defined to be"
  2. $a\geq b$, "is greater than or equal to"
  3. $a\in b$, "in / is an element of"
  4. $a\subseteq$ "is a subset of"
  5. $a\to b$, "maps to / is mapped to"
  6. $a=O(b)$, "is big-O of"

Notably, every relation on this list is antisymmetric, so the ordering of $a$ first and then $b$ is important. This list is extremely incomplete, and there are dozens more.

The correct reading of the symbol $|$ is "divides / is a divisor of." When interpreted in this way, $a|b$ aka "$a$ divides $b$" fits this very well established pattern perfectly. Although it might be counter-intuitive to someone who has more experience with arithmetic than mathematics, it's actually a manifestation of a highly standardized pattern.

Stella Biderman
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    Some authors use b⋮a for "b is divisible by a", the question is why | is more widespread than ⋮ , I think, and who originated it. – Conifold Feb 07 '17 at 02:57
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    @Conifold the only place I have ever seen that vertical three-dot notation (how did you generate it?) is on blackboards in Russia. Where have you seen it in a published document? – KCd Feb 08 '17 at 00:52
  • @KCd See the link in my comment under the OP. I did not generate it, it is a standard unicode character. – Conifold Feb 08 '17 at 19:11
  • @Conifold where have you seen that "is divisible by" notation used in a published book or paper? – KCd Feb 08 '17 at 20:08
  • @KCd I do not understand the question. The link is to Gorodentsev's Algebra textbook published by Springer in 2016, did you click on it? – Conifold Feb 08 '17 at 20:14
  • you could also read a|b as "a is a factor of b". – mobileink Feb 09 '17 at 18:52
  • There are other course other ways it can be symbolized or read. The question asks why we write "$a|b$" for "$a$ divides $b$" and that's the question I'm answering. Questions like "why do we use "$a$ divides $b$" instead of "$a$ is a multiple of $b$" are different questions that I haven't attempted to answer. – Stella Biderman Feb 09 '17 at 18:55
  • @Conifold sorry for the late reply to your question above. I have now gotten around to clicking your link and see it is in a published book, not surprisingly by a Russian author. I think mathematicians from there are the only people who use it (with rare exceptions). – KCd Nov 21 '20 at 22:38
  • @Kcd: The three-dot notation for divisibility can also be found in Litvinenko and Mordkovich's "Solving problems in algebra & trigonometry" (see, for example, the footnote on page 9 of the following PDF copy of the book: https://ia903104.us.archive.org/12/items/LitivinenkoMordkovichSolvingProblemsInAlgebraAndTrigonometryMir1988/Litvinenko-Mordkovich-Solving-Problems-In-Algebra-and-Trigonometry-Mir-1988.pdf) – José Hdz. Stgo. Dec 08 '22 at 04:38
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Don E. Knuth and some of his co-authors don't write 2 | 8 but 2 \ 8. If I understand correctly, their concerns regarding the a | b notation are not totally unrelated to those already mentioned by SearchSpace:

" The notation $m \mid n$ is actually much more common than $m \backslash n$ in current mathematics literature. But vertical lines are overused--for absolute values, set delimiters, conditional probabilities, etc.--and backward slashes are underused. Moreover, $m\backslash n$ gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward".

(Cf. R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete mathematics: a foundation of computer science, 2nd ed. Addison-Wesley Publishing Company, 1994, p. 102.)

José Hdz. Stgo.
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