deriving an identity for complex numbers

For future reference the following question is from Complex Variables and Applications by Brown and Churchill, 8th Edition.

Question #6 on Page 8 concerning the derivation of an identity.

Let $z = (x, y)$ the ordered pair in which $x$ represents a pure real and $y$ represents a pure imaginary number. From the relations:

$(1) \frac{z_1}{z_2} = z_1 \frac{1}{z_2}$

$(2) \frac{1}{z_1}\frac{1}{z_2} = \frac{1}{z_1z_2}$

show that $\big(\frac{z_1}{z_3}\big)\big(\frac{z_2}{z_4}\big) = \frac{z_1z_2}{z_3z_4}$

I'm a little bit stuck, or maybe I'm missing something. But this is how I'm approaching it.

Expanding the r.h.s.:

we have:

$z_1z_2(z_3)^{-1}(z_4)^{-1}$

$z_1z_2 (z_3z_4)^{-1}$

$z_1z_2 \frac{1}{z_3z_4}$

$\frac{z_1z_2}{z_3z_4}$, by relation 1.

What do you think? Did I get it right?

Answer

Assuming that it has been proved that multiplication of complex numbers are commutative (i.e. $u_1u_2=u_2u_1$ for any two complex numbers $u_1,u_2$), your proof looks good.