Is the notation for '$a$ divides $b$' standard?

$\begingroup$

I know that the notation $a | b$ means that there exists an integer $c$ such that $ac=b$, but is this notation completely standard and there's no way that it could be the other way round?

$\endgroup$ 3

2 Answers

$\begingroup$

In number theory and algebra $\rm\:a\ |\ b\ $ in ring $\rm\:R\:$ means $\rm\: \exists\: c\in R:\ ac = b,\,$ i.e. $\rm\,ax=b\,$ has at least one solution $\rm\,x\,$ in $\,\rm R.\,$ Usually the ring is omitted, being understood from ambient context. This notation is used so widely that it surely can be considered standard notation.

Occasionally some authors introduce asymmetric variants, with the bar being tilted, or with "hooks" on the top/bottom of the bar, intended to give some visual cue as to which argument is to be pushed to the bottom of a fraction, or pulled to the top. However, as convenient as they may be, none of these asymmetric variants is in wide use.

$\endgroup$ 6 $\begingroup$

That's right, the notation is absolutely standard, and reversing is simply wrong. So for example it is not true that $12\,|\,6$.

There is frequent student confusion about this, because $a/b$ is nowadays a common notation for the fraction $\dfrac{a}{b}$. That is a relatively recent development.

$\endgroup$ 1

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy