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.
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