Non-Zero Complex Numbers Closed under Multiplication/Proof 2

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Theorem

The set of non-zero complex numbers is closed under multiplication.


Proof

Let $z_1, z_2 \in \C_{\ne 0}$.

Then by definition of complex number:

$z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$

for some $x_1, y_1, x_2, y_2 \in \R$ such that:

$x_1 \ne 0$ or $y_1 \ne 0$
$x_2 \ne 0$ or $y_2 \ne 0$

Expressing $z_1$ and $z_2$ in exponential form (although polar form is equally adequate):

$z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$

for some $r_1, r_2, \theta_1, \theta_2 \in \R$.

Then by Product of Complex Numbers in Polar Form:

$z_1 \times z_2 = \paren {r_1 \times r_2} e^{i \paren {\theta_1 + \theta_2} }$


By definition of exponential form:

$r_1 = \sqrt {x_1^2 + y_1^2}$
$r_2 = \sqrt {x_2^2 + y_2^2}$

Thus $r_1 > 0$ and $r_2 > 0$.

Hence $r_1 \times r_2 > 0$ and so $z_1 \times z_2 \ne 0$.

$\blacksquare$


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