For each pair of rational numbers, given below, verify that the multiplication is commutative:

$\dfrac{-1}{5}$ and $\dfrac{2}{9}$

To prove:

$\dfrac{-1}{5} \times \dfrac{2}{9} = \dfrac{2}{9} \times \dfrac{-1}{5}$

Taking LHS:

$\dfrac{-1}{5} \times \dfrac{2}{9}\\[1em] =\dfrac{-1 \times 2}{5 \times 9}\\[1em] =\dfrac{-2}{45}\\[1em]$

Taking RHS:

$\dfrac{2}{9} \times \dfrac{-1}{5}\\[1em] =\dfrac{2 \times -1}{9 \times 5}\\[1em] =\dfrac{-2}{45}\\[1em]$

∴ LHS = RHS

$\dfrac{-1}{5} \times \dfrac{2}{9} = \dfrac{2}{9} \times \dfrac{-1}{5}$