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?
Is the notation for '$a$ divides $b$' standard?
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0Are you asking if there's a notation for "b is divisible by a"? I wish. When that's the way you learn it in school, "a divides b" seems so backwards. – 2012-04-04
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4I wonder whether it is reversed in right-to-left languages such as Arabic, Hebrew, Persian and Urdu. – 2012-04-04
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0@user2468 a "little" late answer but it is not reversed(at least in Hebrew) – 2018-01-23
2 Answers
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.
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0Could you perhaps give an example of what this looks like? – 2012-04-05
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0@user1729 At the moment I cannot recall the source. I will update my answer if/when I do. – 2012-04-20
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0I realize that this is an ancient thread. I would like to ask if this notation applies also to groups. i.e. in the cyclic group $C_{14}$, might we say that $3|11$ and $7 \not |\; 5$? – 2014-09-30
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1@Myridium Yes, one can use the notation in any commutative monoid, i.e. $\ a\mid b\ $ means $\ ac = b\ $ for some $\,c.\ $ This is true for all $\,b\,$ when $\,a\,$ is a unit (i.e. invertible) since then we may choose $\, c = a^{-1} b.\,$ In particular, in a group, $\ a\mid b\,$ for all $\,a,b,\ $ so divisibility is not interesting. – 2014-10-01
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0Right. I made a mistake in my last comment. Is there an analogous symbol for divisibility when left-multiplied by some element? Or is this notation reserved for commutative structures due to the ambiguity? – 2014-10-01
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1@Myridium Divisibility and factorization in non-commutative monoids is far less commonly considered than the commutative case. I'm not aware of any widely used notation for left/right divisibility. – 2014-10-01
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.
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13I agree that this is absolutely standard, but I think it's a pity. I think symbols for non-symmetric relations should never have a vertical symmetry axis. $\in$ doesn't, and we can write $\ni$ thanks to this when we find it convenient. Nothing like this can be done with $|.$ – 2012-04-04