# Who invented the divisibility symbol and why is it backwards?

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?

• I think that there is no "deep" reason... In western world we (usually) write from left to tight; thus, to symbolize "$2$ divides $8$" is quite "natural" to write : $2|8$. Commented Feb 6, 2017 at 9:14
• In the same way, when we "linearize" $\dfrac 8 2$ it is quite "natural" to write : $8/2$. Commented Feb 6, 2017 at 9:14
• See for DIVISION SYMBOLS and reproduction; the ref is to Florian Cajori's book on Mathematical Notation. Commented Feb 6, 2017 at 9:18
• I had similar qualms with | initially, and the vertical ellipsis ⋮ seemed more natural to me: 8⋮2 means "8 is divisible by 2", see e.g. Gorodentsev. But it is not used nearly as often as |. Commented Feb 7, 2017 at 3:05
• Standard MathJax code for $a\mid b$ is a\mid b, the result looks different from $a|b$, coded as a|b. And there is also \nmid, thus: $a\nmid b.$ Commented Apr 8, 2017 at 18:25

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.

• 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. Commented Feb 7, 2017 at 2:57
• @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
Commented Feb 8, 2017 at 0:52
• @KCd See the link in my comment under the OP. I did not generate it, it is a standard unicode character. Commented Feb 8, 2017 at 19:11
• @Conifold where have you seen that "is divisible by" notation used in a published book or paper?
– KCd
Commented Feb 8, 2017 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? Commented Feb 8, 2017 at 20:14

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.)

• While in general I like Knuth's insights into notation, I find $m \backslash n$ very confusing. Commented Jan 22 at 2:19
• The symmetry of the symbol "|" gives a wrong impression that the divisibility relation is symmetric. So, the Knuth notation solves this problem - I wish more authors use it. Commented Apr 16 at 18:17