Any rational $x>0$ can be uniquely written as $x=\prod{p_i}^{m_i}$ with $p_i$ the $i^\text{th}$ prime and $m_i\in\mathbb Z$ it's associated multiplicity. This extends the notion of factorization to strictly positive rationals.

We can extend to non-zero rationals by writing $x=(-1)^{m_0}\prod{p_i}^{m_i}$ with $m_0\in\{0,1\}$. That's similar to how factorization is extended from $\mathbb N^*$ to $\mathbb Z^*$ in some contexts.

I don't see a meaningful extension to reals.

Any rational $x>0$ can be uniquely written as $x=\prod{p_i}^{m_i}$ with $p_i$ the $i^\text{th}$ prime and $m_i\in\mathbb Z$ its associated multiplicity. This extends the notion of factorization to strictly positive rationals.

We can extend to non-zero rationals by writing $x=(-1)^{m_0}\prod{p_i}^{m_i}$ with $m_0\in\{0,1\}$. That's similar to how factorization is extended from $\mathbb N^*$ to $\mathbb Z^*$ in some contexts.

I don't see a meaningful extension to reals.