How is the property called that makes $(a\cdot b)/c = (a/c)\cdot b = a\cdot (b/c)$

Question

As far as I know for any rational numbers $a$, $b$ and $c$ the following is true: $(a\cdot b)/c = (a/c)\cdot b = a\cdot (b/c)$.

How is this property called?

Answer 1

This represents a mixture of properties associated with multiplication.

First we note that division by $c\neq 0$ is the same as multiplication by $c^{-1}=\cfrac 1c$, so we really are dealing with multiplication in disguise.

The fact that it doesn't matter where we put the brackets - so that $(ab)c^{-1}=a(bc^{-1})$ is the associative law for multiplication.

NB: if we use $/$ to represent the division operator, we need the brackets in, because the $/$ operation is not associative. That's why I converted to $c^{-1}$ form first, as this helps to make clear what is happening. If we use the $c^{-1}$ form we can omit the brackets.

The fact that the order does not matter, so that (omitting brackets) $abc^{-1}=ac^{-1}b$, depends on the commutative law for multiplication - that $ab=ba$. Again note carefully that the symbol "/" is not commutative $(a/b \neq b/a)$.