Question

Find the modulus, argument and the principal argument of the complex numbers.

$z=\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$

Answer 1

$z=\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$

$=\frac{(\sqrt{5+12i}+\sqrt{5-12i})(\sqrt{5+12i}+\sqrt{5-12i})}{(\sqrt{5+12i}-\sqrt{5-12i})(\sqrt{5+12i}+\sqrt{5-12i})}$

$=\frac{{(\sqrt{5+12i}+\sqrt{5-12i})}^2}{{\left(\sqrt{5+12i}\right)}^2.{\left(\sqrt{5-12i}\right)}^2}$

$=\frac{(5+12i)+(5-12i)-2\sqrt{5+12i}\sqrt{5-12i}}{5+12i-5+12i}$

$=\frac{10-2\times 13}{25}$

$=\frac{-16}{25}-0i$

$\therefore$ $z$ lies in x-axis.

Here $\theta ={\tan }^{-1}\left|\frac{0}{-16/25}\right|=0.$

$\therefore arg\left(x\right)=2\pi -\theta =2\pi -0=2\pi$

Hence, principal value of $arg\left(z\right)=-\theta =0.$