# 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 : $2|8$. – Mauro ALLEGRANZA Feb 6 '17 at 9:14
• In the same way, when we "linearize" $\dfrac 8 2$ it is quite "natural" to write : $8/2$. – Mauro ALLEGRANZA Feb 6 '17 at 9:14
• See for DIVISION SYMBOLS and reproduction; the ref is to Florian Cajori's book on Mathematical Notation. – Mauro ALLEGRANZA Feb 6 '17 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 |. – Conifold Feb 7 '17 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: $a\nmid b.$ – Michael Hardy Apr 8 '17 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. – Conifold Feb 7 '17 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 Feb 8 '17 at 0:52
• @KCd See the link in my comment under the OP. I did not generate it, it is a standard unicode character. – Conifold Feb 8 '17 at 19:11
• @Conifold where have you seen that "is divisible by" notation used in a published book or paper? – KCd Feb 8 '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 8 '17 at 20:14