Toronto Math Forum
MAT3342018F => MAT334Lectures & Home Assignments => Topic started by: Nikita Dua on September 24, 2018, 03:27:11 PM

Will a prove which uses rectangular coordinates definition of z1 and z2 would be fine?
Using z1 = x1 +iy1 and z2 = x2 + iy2 and using some algebra to prove z1z2 = 0 implies z1 = 0 or z2 = 0
Barely comprehensible. Next such post will be deleted. V.I.

Given:
$ z_1 z_2 = 0$
$\Rightarrow z_1 z_2 = 0$
$\Rightarrowz_1z_2 = 0$
$\Rightarrowz_1 = 0 \vee z_2 = 0$
We know (Modulus is nonnegative; Positive for nonzero numbers):
$z = 0 \iff z=0$
Thus:
$ z_1 z_2 = 0 \iff z_1 = 0 \vee z_2 = 0 \iff z_1 = 0 \vee z_2 = 0$
Another method would be to use polar coordinates:
$z_1 z_2 = r_1 e^{i \theta _1} r_2 e^{i \theta _2}$
$ 0 =r_1 r_2 e^{i (\theta _1 + \theta _2)}$
Since $e^{z} \gt 0$ $\forall z$:
$ 0 = r_1 r_2 \iff r_1 = 0 \vee r_2 = 0 \iff z_1 = 0 \vee z_2 = 0$
I am sure using rectangular would also work, but it if probably more work and algebra than necessary.

Alternatively: $z\ne 0\implies z^{1} =\frac{\bar{z}}{z^2}$. Then $(z_1\ne 0)\wedge (z_1z_2=0)\implies z_2=z_1^{1}z_1z_2= 0$.