# UNIT – II COMPLEX NUMBERS (Formulae)

$\LARGE{\color {purple} {ALGEBRA\ OF\ COMPLEX\ NUMBERS}}$
$\color {royalblue} {Definition\ of\ complex\ number}:\ \hspace{20cm}$
$If\ Z = a + ib ,\ then\ a\ is\ called\ the\ real\ part\ of\ Z\ and\ b\ is\ called\ the\ imaginary\ part\ of\ Z.$
$Re ( Z ) = a\ and\ Im ( Z ) = b$
$\color {royalblue} {Algebra\ of\ Complex\ numbers}:\ \hspace{20cm}$
$\color {brown} {(i)\ Addition\ of\ two\ Complex\ numbers}:\ \hspace{20cm}$
$Let\ Z_1= a + ib,\ Z_2 = c + id\ be\ any\ two\ complex\ numbers.\ Then$
$Z_1 + Z_2 = a + ib\ +\ c + id$
$=\ a + c + i(b + d )$
$\color {brown} {(ii)\ Difference\ of\ two\ Complex\ numbers}:\ \hspace{20cm}$
$Let\ Z_1= a + ib,\ Z_2 = c + id\ be\ any\ two\ complex\ numbers.\ Then$
$Z_1 – Z_2 = a + ib\ -\ (c + id)$
$= a + ib – c – id$
$=\ a – c + i(b – d )$
$\color {brown} {(ii)\ Multiplication\ of\ two\ Complex\ numbers}:\ \hspace{20cm}$
$Let\ Z_1= a + ib,\ Z_2 = c + id\ be\ any\ two\ complex\ numbers.\ Then$
$Z_1 Z_2 = (a + ib) (c + id)$
$= ac + iad + ibc + i^2bd$
$= ac + i(ad + bc) – bd$
$= (ac – bd) + i(ad + bc)$
$\color {brown} {(ii)\ Division\ of\ two\ Complex\ numbers}:\ \hspace{20cm}$
$Let\ Z_1= a + ib,\ Z_2 = c + id\ be\ any\ two\ complex\ numbers.\ Then$
$\frac{z_1}{z_2}\ =\frac{a+ib}{c+id}\ ×\ \frac{c-id}{c-id}$
$= \frac{ac- iad + ibc – i^2bd}{c^2 – i^2d^2}$
$= \frac{ac + i (bc – ad) + bd}{c^2 – i^2d^2}$
$= \frac{ac + bd + i(bc – ad)}{c^2 + d^2}$
$= \frac{ac + bd} {c^2 + d^2}\ +\ i\ \frac{bc – ad}{c^2 + d^2}$
$Remember\ (a +ib)(a – ib)\ =\ (a)^2 + (b)^2$
$\color {royalblue} {Modulus\ and\ Amplitude\ (or)\ Argument\ of\ a\ Complex\ number}:\ \hspace{20cm}$
$If\ Z = a + ib\ then\ Modulus is |z| = \sqrt{a^2 + b^2}\ and\ Amplitude\ is\ θ = tan^{-1} (\frac{b}{a})$
$\color {royalblue} {Distance\ between\ two\ Complex\ numbers}:\ \hspace{20cm}$
$If\ Z_1\ = a + ib,\ Z_2\ = c + id\ be\ any\ two\ complex\ numbers,\ then$
$Z_1 Z_2 =\ \sqrt{ (a- c)^2 + (b- d)^2 )}$
$\color {royalblue} {Condition\ for\ collinear\ points}:\ \hspace{20cm}$
$If\ the\ three\ complex\ numbers\ say,\ A(x_1 +iy_1),\ B(x_2 +iy_2),\ and\ C(x_3 +iy_3),\ are\ collinear$
$if\ \frac{1}{2}[\ x_1(y_2 – y_3)\ +\ x_2(y_3 – y_1)\ +\ x_3(y_1 – y_2)] = 0$
$\color {royalblue} {Condition\ for\ Square}:\ \hspace{20cm}$
$If\ A,\ B,\ C\ and\ D\ are\ any\ four\ complex\ numbers\ representing\ the\ vertices\ of\ a\ square$$then\ the\ required\ conditions\ are$
$( i )\ AB = BC = CD = DA$
$( ii )\ AC = BD$
$\color {royalblue} {Condition\ for\ rhombus}:\ \hspace{20cm}$
$If\ A,\ B,\ C\ and\ D\ are\ any\ four\ complex\ numbers\ representing\ the\ vertices\ of\ a\ rhombus$$then\ the\ required\ conditions\ are$
$( i )\ AB = BC = CD = DA$
$( ii )\ AC \neq BD$
$\color {royalblue} {Condition\ for\ rectangle}:\ \hspace{20cm}$
$If\ A,\ B,\ C\ and\ D\ are\ any\ four\ complex\ numbers\ representing\ the\ vertices\ of\ a\ rectangle$$then\ the\ required\ conditions\ are$
$( i )\ AB = CD\ and\ BC= DA$
$( ii )\ AC = BD$
$\color {royalblue} {Condition\ for\ parallelogram}:\ \hspace{20cm}$
$If\ A,\ B,\ C\ and\ D\ are\ any\ four\ complex\ numbers\ representing\ the\ vertices\ of\ a\ parallelogram$$then\ the\ required\ conditions\ are$
$( i )\ AB = CD\ and\ BC= DA$
$( ii )\ AC \neq BD$
$\LARGE{\color {purple} {DE-MOIVRE’S\ THEOREM}}$
$\color {royalblue} {De-Moivre’s\ Theorem( Statement\ only)}:\ \hspace{20cm}$
$If\ n\ is\ an\ integer\ positive\ or\ negative\ then\ (cos\ θ + i sin⁡\ θ )^n = cos⁡\ n θ + i sin⁡\ n θ$
$(cos\ θ + i sin⁡\ θ )^{-n} = cos⁡\ n θ- i sin⁡\ n θ$
$\LARGE{\color {purple} {ROOTS\ OF\ COMPLEX\ NUMBERS }}$
$If\ ω\ is\ cube\ roots\ of\ unity,\ then\ \hspace{15cm}$
$( i )\ ω ^3\ =\ 1\ \hspace{10cm}$
$(ii)\ 1\ +\ ω\ +\ ω ^2\ =\ 0\ \hspace{10cm}$
$\color {royalblue} {Working\ rule\ to\ find\ the\ n^{th}\ roots\ of\ a\ complex numbers}:\ \hspace{18cm}$
$1)\ Write\ the\ given\ complex\ number\ in\ Polar\ form$
$2)\ Add\ ‘2kπ’\ to\ the\ argument$
$3)\ Apply\ De-Moivre’s\ theorem$
$4)\ Put\ k = 0,1,……. up to\ (n-1)$.
$\color {blue}{x^n\ -\ 1\ =\ 0}\ \hspace{18cm}$
$x^n\ =\ 1\ \hspace{18cm}$
$=\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{n}\ \hspace{10cm}$
$=\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{n}\ \hspace{10cm}$
$=\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{n}\ \hspace{10cm}$
$=\ cos\ (\frac{2kπ}{n})\ +\ i\ sin\ (\frac{2kπ}{n})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ ……..\ n-1\ \hspace{5cm}$
$\color {blue}{x^n\ +\ 1\ =\ 0}\ \hspace{18cm}$
$x^n\ =\ -\ 1\ \hspace{18cm}$
$=\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{n}\ \hspace{10cm}$
$=\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{n}\ \hspace{10cm}$
$=\ cos\ (\frac{π\ + 2kπ}{n})\ +\ i\ sin\ (\frac{π\ + 2kπ}{n})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ ……..\ n-1\ \hspace{5cm}$