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As far as I know, $a/b$ indicates the following things:

  1. The fraction $a/b$, i.e. there are '$a$' of certain object(s) for '$b$' of certain other object(s).

  2. The quotient obtained when $a$ is divided by $b$, i.e. the number of times $b$ can be subtracted from $a$, the number of times $b$ is contained in $a$, i.e. the number of times $a$ is greater than $b$ by stating what multiple $a$ is of $b$.

  3. The ratio of $a$ to $b$, i.e. the number of times $a$ is greater than $b$ by stating what multiple is $a$ of $b$.

Point numbers 2 and 3 are basically the same.

Is there anything else that $a/b$ indicates?

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    If $a$ is an abelian group, and $b$ a subgroup, then $a/b$ denotes the quotient group. For example $a=\mathbb{Z}$ and $b=n\mathbb{Z}$. – 2017-01-24
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    Aren't 1, 2 and 3 all the same? In particular, $a/b$ is noway the quotient of the Euclidean division of $a$ divided by $b$; e.g. $5/2 = 2.5$ while $5 div 2 = 2$. – 2017-01-24
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    If $a,b$ belong to a commutative ring $R$, then $a/b$ denotes the equivalence class of $(a,b)$ in the field of fractions of $R$. – 2017-01-24
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    @Aweygan That's a field for domains, but not for general rings. See the [Total Fraction Ring.](https://en.wikipedia.org/wiki/Total_ring_of_fractions) – 2017-01-24
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    @BillDubuque You're right I meant to say integral domain. Don't know why I said commutative ring. – 2017-01-24
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    Generalising @DietrichBurde’s example of the quotient group and Aweygan’s example of fractions in rings, if $a$ is a set and $b$ an equivalence relation (OK, unconventional names), $a/b$ is the quotient space. This covers all the meanings given in Wikipedia under [/ # Mathematics](https://en.wikipedia.org/wiki/Slash_(punctuation)#Mathematics) (now that I have added _Quotient sets_). – 2018-12-06

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