# Who introduced the divisibility symbol $a\vert b$ ("$a$ divides $b$") and when?

I have just stumbled across this post and became curious about the same question, namely the part regarding the origin/history of the vertical bar symbol $$a\vert b$$ that we use to denote "a divides b" (I don't care at all about why it is written "backward" in the sense asked there).

While the OP of that post seems satisfied with the answer, the part about the origin of this symbol was still left out. In one of the comments there, there was a suggestion that the answer could be found in Florian Cajori's book A History of Mathematical Notations. I have a copy of that book but I found nothing directly related to the history of the symbol $$\vert$$ , unfortunately.

I'd highly appreciate if anyone could point me to a good resource about this subject matter, be it a book or an article. More specifically, I want to know the time period the notation $$\vert$$ was introduced and the names of the mathematicians associated with its development.

This is a case where it seems that the symbol should be old, from Euler's or Gauss's time at least, but it is not. It does not appear in Dickson's History of the theory of numbers (1919), whose entire first volume is dedicated to divisibility, nor in Cajori's comprehensive History Of Mathematical Notations (1928), and not even in van der Waerden's Moderne Algebra (1930), which became a blueprint for modern algebra textbooks.

The earliest use I found is in Hall's Slowly increasing arithmetic series (1933), where it is introduced in a footnote thus:"$$x|y$$ means "$$x$$ divides $$y$$"", no comment. Hall's references, Lehmer's An Extended Theory of Lucas' Functions (1930) and Engstrom's On sequences defined by linear recurrence relations (1931), still use words or congruences for the task. On the other hand, Hall and Ward use $$|$$ extensively in their 1936-38 publications on linear divisibility sequences.

After graduating from Yale in 1932 Hall worked with Hardy at Cambridge for a year before returning to Yale in 1936. And the first book occurrence seems to be Hardy-Wright's classic An introduction to the theory of numbers (first edition came out in 1938), where we read on the very first page:"We express the fact that $$a$$ is divisible by $$b$$, or $$b$$ is a divisor of $$a$$, by $$b|a$$". Vinogradov's Elements of Number Theory (first Russian edition came out in 1936, English translation in 1954) uses $$b\backslash a$$ instead, suggesting that the notation was not established yet. Hall's notation was adopted in Bourbaki's Algebra II, chapitre VI.

All of these authors are very matter-of-factly and laconic when introducing the symbol, and neither motivate it nor refer to anyone, including each other, for it. Not even Hardy-Wright, who have a special note on notations, or Bourbaki, who have extensive historical notes. So it is hard to say who came up with it (it could have been Hall or Hardy) and why. But the shapes suggest that it was simply a variation on the division symbol $$/$$, and Hardy-Wright explicitly introduce logical symbols in their Remarks on Notation, and use $$|$$ to illustrate their use. It seems that the turn towards abstraction in algebra and number theory, and proliferation of symbolism from foundational studies in mathematical logic in 1930s made symbolizing a relation that was previously expressed in words or congruences timely.

• Wow I did not realize this symbol would be so hard trace down, no wonder it doesn't appears in Cajori's book. Thanks a lot for you detailed answer. Nov 20, 2020 at 11:16

I think that the history of how we write fractions is helpful here. Although fractions were known in ancient times - the Babylonians and Egyptians used them - the modern notation for them began with the system of bhinnarasi by Aryabhatta around 5th Century AD and then Brahmagupta and (c. 626) and Bhaskara (c. 1150).

In their works, they formed fractions by placing the numerators (amsa) over the denominators (cheda) without a separating line. From there it is an easy step to put this in to emphasise the separation of the two numbers and this is first attested to in the work of al-Hassar (c. 1200), a Muslim mathematician working in Fez, Morroco.

The same notation then appeared soon after in Europe, for example in the work of Fibonnaci (c. 1300).

Obviously, it's not easy to write or print numbers in such a fashion, especially with the advent of algebra, and lengthy expressions in either the numerator or denominator; and so the next obvious step is to write them horizontally as a/b, with the separating bar now positioned vertically.

This explains how we have the vertical bar for division. As your linked post then explains, it would be sensible to them express divisibility with a similar notation and hence the introduction of the vertical bar with the terms arranged in order of how we say them: a divides b as a|b.

Finally, I'd like to add, that in modern notation, we express divisibility both ways: a divides b, can be written as a\b and b/a. We see this freedom of expression when expressing quotients of groups, rings when dividing by ideals, modules or algebras, for example. We don't however, generally see this freedom with numbers.

• Thank you for the historical background. I'm sorry if I hadn't made it clear in my question but what I want to know is the time period the notation $\vert$ was introduced and the mathematicians associated with it (in a similar fashion as in Cajori's book, says). Nov 19, 2020 at 14:42