Is $\frac{5}{2}$ in $(\mathbb{Z/5Z})$ $0$?

Question

Is $\frac{5}{2}$ in $(\mathbb{Z/5Z})$ equal to $0$?

It is, right? Because $\frac{5}{2}=\frac{1}{2}·5=\frac{1}{2}·0=0$

Answer 1

There are two interpretations of division when working modulo an integer.

The vastly more common one — and the only one compatible with the result being in $\mathbb{Z} / 5 \mathbb{Z}$ — is that the quotient should be interpreted as modular division.

Thus, $5/2 \equiv 0/2 \equiv 0 $.

If you know ring theory, you can more generally view this as invoking the isomorphism $$ \mathbb{Z}_{(5)} / 5 \mathbb{Z}_{(5)} \cong \mathbb{Z} / 5 \mathbb{Z} $$ so that it makes sense to interpret a fraction as an element of $\mathbb{Z} / 5 \mathbb{Z}$ whenever the denominator is not divisible by 5.

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The alternative has an eye more towards working in additive groups than working with rings, and $5/2$ is a nonzero element of the additive abelian group $\mathbb{Q} / 5 \mathbb{Z}$.

Note carefully that there isn't a well-defined multiplication and division operation on this group.