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If $P$ and $Q$ are two complex numbers, then the modulus of the quotient of $P$ and $Q$ is:
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Equal to the quotient of their moduli
Thetwo complex numbers are $\mathrm{P}=\mathrm{x}+$ iy and $\mathrm{Q}=\alpha+\mathrm{i} \beta$
Quotient $=\frac{\mathrm{P}}{\mathrm{Q}}=\frac{\mathrm{x}+\mathrm{iy}}{\alpha+\mathrm{i} \beta},\left|\frac{\mathrm{P}}{\mathrm{Q}}\right|=\left|\frac{\mathrm{x}+\mathrm{iy}}{\alpha+\mathrm{i} \beta}\right|$
$=\frac{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}{\sqrt{\alpha^{2}+\beta^{2}}}=\sqrt{\frac{\mathrm{x}^{2}+\mathrm{y}^{2}}{\alpha^{2}+\beta^{2}}}=\frac{\left|\mathrm{Z}_{1}\right|}{\left|\mathrm{Z}_{2}\right|}=\left|\frac{\mathrm{Z}_{1}}{\mathrm{Z}_{2}}\right|$
Hence, the quotient of their modulus is equal to the quotient of their moduli.
Quotient $=\frac{\mathrm{P}}{\mathrm{Q}}=\frac{\mathrm{x}+\mathrm{iy}}{\alpha+\mathrm{i} \beta},\left|\frac{\mathrm{P}}{\mathrm{Q}}\right|=\left|\frac{\mathrm{x}+\mathrm{iy}}{\alpha+\mathrm{i} \beta}\right|$
$=\frac{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}{\sqrt{\alpha^{2}+\beta^{2}}}=\sqrt{\frac{\mathrm{x}^{2}+\mathrm{y}^{2}}{\alpha^{2}+\beta^{2}}}=\frac{\left|\mathrm{Z}_{1}\right|}{\left|\mathrm{Z}_{2}\right|}=\left|\frac{\mathrm{Z}_{1}}{\mathrm{Z}_{2}}\right|$
Hence, the quotient of their modulus is equal to the quotient of their moduli.
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