Non-Zero Complex Numbers are 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 $z_1 \times z_2 \ne 0$.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $59$