Rewrite the given number as
[tex]\dfrac{a+ib}{c+id}=\dfrac{(a+ib)(c-id)}{(c+id)(c-id)}=\dfrac{ac+bd+i(bc-ad)}{c^2+d^2}[/tex]
If it's purely real, then the complex part should be 0, so that
[tex]\dfrac{bc-ad}{c^2+d^2}=0\implies bc-ad=0\implies\boxed{ad-bc}[/tex]
as required.
