Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
$\frac{- 16}{1 + i \sqrt{3}}$

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Solution

$z = \frac{- 16}{1 + i \sqrt{3}}$

$= \frac{- 16}{1 + i \sqrt{3}} \times \frac{\frac{- 16}{1 - i \sqrt{3}}}{\frac{- 16}{1 - i \sqrt{3}}}$

$= \frac{- 16 (1 - i \sqrt{3})}{1^{2} - i^{2} (\sqrt{3})^{2}}$

$= \frac{- 16 + i 16 \sqrt{3}}{1 + 3}$

$= - 4 + i 4 \sqrt{3}$

$z = a + i b = - 4 + i 4 \sqrt{3}$

$m o d = | z | = \sqrt{a^{2} + b^{2}} = \sqrt{(- 4)^{2} + (4 \sqrt{3})^{2}} = \sqrt{64} = 8$

${tan}^{- 1} \frac{b}{a}$

$= {tan}^{- 1} \frac{4 \sqrt{3}}{- 4} = - \sqrt{3} = - 60^{\circ}$

$∴ θ = 180 - 60 = 30^{\circ}$

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$\frac{- 16}{1 + i \sqrt{3}}$