Question

write the multiplicative inverse of $\frac{\sqrt{3}}{2}-\frac{1}{2}i$

Answer 1

Given imaginary number is

$a=\frac{\sqrt{3}}{2}-\frac{1}{2}i$

multiplicative inverse(b) of a number(a) is defined by

$a\times b=1$

or $b=\frac{1}{a}$

hence,

$b=\frac{1}{\frac{\sqrt{3}}{2}-\frac{1}{2}i}$

or

$b=\frac{2}{\sqrt{3}-i}$

multiply and divide above equation by $(\sqrt{3}+i)$ we get

$b=\frac{2(\sqrt{3}+i)}{(\sqrt{3}-i)(\sqrt{3}+i)}$

$b=\frac{2(\sqrt{3}+i)}{(3-i^2)}$

$b=\frac{2(\sqrt{3}+i)}{(3+1)}$

$b=\frac{2(\sqrt{3}+i)}{4}$

$b=\frac{\sqrt{3}+i}{2}$ (multiplicative inverse of given number a)