Question

1 which can be DeMoivre's Theorem can be used to find reciprocals of complex numbers. Recall from algebra that the reciprocal of x is expressed as X-? Use this fact, along with DeMoivre's Theorem, to find the reciprocal of the number below. 1+iV3

Answer 1

$\text{Given that the reciprocal of x is }\frac{1}{x}\text{ by DeMoivre's Theroem}$

$\text{Hence, by DeMoivre's theorem, the reciprocal of }1+i\sqrt{3}\text{ is }\frac{1}{1+i\sqrt{3}}$

Now,

$\begin{align*} \frac{1}{1+i\sqrt{3}}=&\frac{1}{1+i\sqrt3}\times \frac{1-i\sqrt{3}}{1-i\sqrt{3}} \\=& \frac{1-i\sqrt{3}}{(1+i\sqrt{3})(1-i\sqrt{3})} \\=& \frac{1-i\sqrt{3}}{1^2-(i\sqrt3)^2} \\=& \frac{1-i\sqrt3}{1-(-3)} \\=& \frac{1-i\sqrt3}{4} \end{align*}$

Hence, the reciprocal of $1+i\sqrt3$ is $(1-i\sqrt3)/4$