ALGEBRA OF COMPLEX NUMBERS (Excersice)

\[Ex\ 1:\ Find\ the\ modulus\ and\ amplitude\ of\ \frac{-\ 3\ +\ i}{-\ 1\ +\ i}\ \hspace{18cm}\]
\[\color {black}{Solution:} \hspace{20cm}\]
\[Let\ z = \frac{-\ 3\ +\ i}{-\ 1\ +\ i}\ \hspace{18cm}\]
\[= \frac{-\ 3\ +\ i}{-\ 1\ +\ i}\ ×\ \frac{-\ 1\ -\ i}{-\ 1\ -\ i}\ \hspace{15cm}\]
\[ = \frac{3\ +\ 3i\ -\ i -\ i^2}{(-\ 1)^2 + (1)^2}\ \hspace{3cm}\ \because [(a +ib)(a – ib)\ =\ (a)^2 +\ (b)^2]\]
\[ = \frac{3\ +\ 2i\ +\ 1 }{1+ 1}\ \hspace{4cm}\ \because [i^2\ =\ -1]\]
\[ = \frac{4\ +\ 2i}{2}\ \hspace{10cm}\]
\[ Z = \frac{4}{2}\ +\ \frac{2i}{2}\ \hspace{15cm}\]
\[ z\ =\ 2\ +\ i\ = a\ + ib\ \hspace{14cm}\]
\[a = 2,\ b\ = \ 1\ \hspace{15cm}\]
\[\color {brown} {T0\ find\ modulus}:\ \hspace{18cm}\]
\[|z| = \sqrt{a^2 + b^2}\ \hspace{12cm}\]
\[ = \sqrt{(2)^2 + {1}^2}\ \hspace{12cm}\]
\[ = \sqrt{4\ +\ 1}\ =\ \sqrt{5}\ \hspace{12cm}\]
\[|z| = \sqrt{5}\ \hspace{15cm}\]
\[\color {brown} {To\ find\ amplitude}:\ \hspace{18cm}\]
\[θ = tan^{-1} (\frac{b}{a})\ =\ tan^{-1} \frac{2}{1}\ =\ tan^{-1} {2}\]