Unit – II – COMPLEX NUMBERS

\[\LARGE{\color {red}{CHAPTER\ 2.3:\ ROOTS\ OF\ COMPLEX\ NUMBERS\ (Text)}}\]
\[\color {royalblue} {Definition}:\ \hspace{20cm}\]
\[A\ number\ ω\ is\ called\ the\ n^{th}\ root\ of\ a\ complex\ number Z,\]
\[and\ we\ write\ ω = Z ^{\frac{1}{n}}\]
\[\color {royalblue} {Working\ rule\ to\ find\ the\ n^{th}\ roots\ of\ a\ complex numbers}:\ \hspace{18cm}\]
\[1)\ Write\ the\ given\ complex\ number\ in\ Polar\ form\]
\[2)\ Add\ ‘2kπ’\ to\ the\ argument\]
\[3)\ Apply\ De-Moivre’s\ theorem\]
\[4)\ Put k = 0,1,……. up to\ (n-1)\].
\[\color {royalblue} {Illustration}:\ \hspace{20cm}\]
\[To\ find\ the\ n^{th} roots\ of\ unity\ or\ x^n\ -\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ x^n\ -\ 1\ =\ 0 \hspace{18cm}\]
\[ x^n\ =\ 1\ \hspace{18cm}\]
\[\ x\ =\ (1)^\frac{1}{n}\ \hspace{10cm}\]
\[ =\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{n}\ \hspace{10cm}\]
\[ =\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{n}\ \hspace{10cm}\]
\[ =\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{n}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{2kπ}{n})\ +\ i\ sin\ (\frac{2kπ}{n})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ ……..\ n-1\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{n}\ +\ i\ sin\ \frac{2π}{n}\ =\ ω\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{n}\ +\ i\ sin\ \frac{4π}{n}\ =\ ω^2\]
\[When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{6π}{n}\ +\ i\ sin\ \frac{6π}{n}\ =\ ω^3\]
\[.\]
\[.\]

\[When\ k = n-1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2(n-1)π}{n}\ +\ i\ sin\ \frac{2(n-1)π}{n}\ =\ ω^{n-1}\]
\[The\ n^{th}\ roots\ of\ unity\ are\ 1,\ ω,\ ω ^2,\ …………\ ω ^{n-1}\]
\[\color {royalblue} {Results}:\ \hspace{20cm}\]
\[1)\ If \ ω\ is\ n^{th}\ roots\ of\ unity,\ then\ \hspace{15cm}\]
\[( i )\ ω ^n\ =\ 1\ \hspace{10cm}\]
\[(ii)\ sum\ of\ roots\ is\ zero.\ \hspace{10cm}\]
\[i.e\ 1\ +\ ω\ +\ ω ^2\ +\ ………… +\ ω ^{n-1}= 0\ \hspace{10cm}\]
\[2)\ If\ ω\ is\ cube\ roots\ of\ unity,\ then\ \hspace{15cm}\]
\[( i )\ ω ^3\ =\ 1\ \hspace{10cm}\]
\[(ii)\ sum\ of\ roots\ is\ zero.\ \hspace{10cm}\]
\[i.e\ 1\ +\ ω\ +\ ω ^2\ =\ 0\ \hspace{10cm}\]
\[Ex1:\ If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of \ 1\ +\ ω\ +\ ω ^2\ \hspace{15cm}\]
\[\color {black}{Solution:}\ 1\ +\ ω\ +\ ω ^2\ =\ 0\ \hspace{18cm}\]
\[Ex2:\ If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of \ ω ^3\ \hspace{18cm}\]
\[\color {black}{Solution:}\ ω ^3\ =\ 1\ \hspace{20cm}\]
\[Ex3:\ If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of \ ω ^4\ +\ ω ^5\ +\ ω ^6\ \hspace{16cm}\]
\[\color {black}{Solution:}\ ω ^4\ +\ ω ^5\ +\ ω ^6\ =\ ω ^4( 1\ +\ ω\ +\ ω ^2 )\ =\ ω ^4 ( 0 )\ =\ 0\ \hspace{6cm}\]
\[Ex4:\ Find\ all\ the\ values\ of\ (1)^\frac{1}{3}\ \hspace{18cm}\]
\[\color {black}{Solution:}\ x\ =\ (1)^\frac{1}{3}\ \hspace{18cm}\]
\[ =\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{2kπ}{3})\ +\ i\ sin\ (\frac{2kπ}{3})\ where\ k\ =\ 0,\ 1,\ 2\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{3}\ +\ i\ sin\ \frac{2π}{3}\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{3}\ +\ i\ sin\ \frac{4π}{3}\]
\[Ex5:\ Solve\ x^6 -\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ x^6\ -\ 1\ =\ 0 \hspace{18cm}\]
\[ x^6\ =\ 1\ \hspace{10cm}\]
\[\ x\ =\ (1)^\frac{1}{6}\ \hspace{10cm}\]
\[ =\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{6}\ \hspace{10cm}\]
\[ =\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{6}\ \hspace{10cm}\]
\[ =\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{6}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{2kπ}{6})\ +\ i\ sin\ (\frac{2kπ}{6})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{6}\ +\ i\ sin\ \frac{2π}{6}\ =\ cos\ \frac{π}{3}\ +\ i\ sin\ \frac{π}{3}\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{6}\ +\ i\ sin\ \frac{4π}{6}\ =\ cos\ \frac{2π}{3}\ +\ i\ sin\ \frac{2π}{3}\]
\[When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{6π}{6}\ +\ i\ sin\ \frac{6π}{6}\ =\ cos\ π\ +\ i\ sin\ π\]
\[When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{8π}{6}\ +\ i\ sin\ \frac{8π}{6}\ =\ cos\ \frac{4π}{3}\ +\ i\ sin\ \frac{4π}{3}\]
\[When\ k = 5,\ \hspace{2cm}\ x\ =\ cos\ \frac{10π}{6}\ +\ i\ sin\ \frac{10π}{6}\ =\ cos\ \frac{5π}{3}\ +\ i\ sin\ \frac{5π}{3}\]
\[Ex5:\ Solve\ x^5 + 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ x^5\ +\ 1\ =\ 0 \hspace{18cm}\]
\[ x^5\ =\ -\ 1\ \hspace{10cm}\]
\[\ x\ =\ (-\ 1)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{π\ + 2kπ}{5})\ +\ i\ sin\ (\frac{π\ + 2kπ}{5})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{5}\ +\ i\ sin\ \frac{π}{5}\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{5}\ +\ i\ sin\ \frac{3π}{5}\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{5π}{5}\ +\ i\ sin\ \frac{5π}{5}\ =\ -1\]
\[When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{7π}{5}\ +\ i\ sin\ \frac{7π}{5}\]
\[When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{9π}{5}\ +\ i\ sin\ \frac{9π}{5}\]
\[Ex6:\ Solve\ x^5 +\ x^3\ +\ x^2\ +\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ Given\ x^5 +\ x^3\ +\ x^2\ +\ 1\ =\ 0\ \hspace{15cm}\]
\[x^3(x^2 +\ 1)\ +\ 1( x^2\ +\ 1)\ =\ 0\ \hspace{10cm}\]
\[(x^3\ +\ 1)\ ( x^2\ +\ 1)\ =\ 0\ \hspace{10cm}\]
\[Case\ ( i ) :\ \hspace{16cm}\]
\[x^3\ +1 =\ 0\ \hspace{10cm}\]
\[ x^3\ =\ -\ 1\ \hspace{10cm}\]
\[\ x\ =\ (-\ 1)^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{π\ + 2kπ}{3})\ +\ i\ sin\ (\frac{π\ + 2kπ}{3})\ where\ k\ =\ 0,\ 1,\ 2\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{3}\ +\ i\ sin\ \frac{π}{3}\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{3}\ +\ i\ sin\ \frac{3π}{3}\ =\ -1\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{5π}{3}\ +\ i\ sin\ \frac{5π}{3}\]
\[Case\ ( ii ) :\ \hspace{16cm}\]
\[x^2\ +1 =\ 0\ \hspace{10cm}\]
\[ x^2\ =\ -\ 1\ \hspace{10cm}\]
\[\ x\ =\ (-\ 1)^\frac{1}{2}\ \hspace{10cm}\]
\[ =\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{2}\ \hspace{10cm}\]
\[ =\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{2}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{π\ + 2kπ}{2})\ +\ i\ sin\ (\frac{π\ + 2kπ}{2})\ where\ k\ =\ 0,\ 1\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{2}\ +\ i\ sin\ \frac{π}{2}\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{2}\ +\ i\ sin\ \frac{3π}{2}\]

                                  

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