COMPLEX NUMBER - Lecture Notes

BBrilliant STUDY CENTRE REPEATERS JEEMAIN LECTURE NOTE COMPLEX NUMBERS 1. Imaginary number - square root of a –ve number i4n 1  in  ir, where r is  i4n1  i  i4n3  i n  i4r the remainder 2.   i4n2  1  i4n2 obtained when i4n3  i  i4n1  n is divided by 4 3. Complex number A number of the form a  ib , where a,bR Real part of z, Rez  a & Imaginary part of z, Im(z) = b Note : Rez  0  is purely imaginary CN Im(z) = 0  z is purely real CN 4 Argand plane (Complex plane) 1

BBrilliant STUDY CENTRE REPEATERS JEEMAIN MATHS -2021 (ONLINE CLASS NOTES) 5. Set of complex numbers (C)  C  x  iy / x.y R, 1  i & C  R 6. Geometrical interpretation of complex addtion & subtraction 7. Conjugate of a complex number Congugate of z  x  iy is z  x  iy , which is obtained by replacing i by –i. Geometrically conjugate represents the refleciton of z about real axis Properties of conjugate 1. z   z  2. z  z  z is purely real CN  3. z  z  z is purely imaginary CN 4. z  z  2 Re z 5. z  z  2i 1m z 6. z1  z2  z1  z2 7. nz  nz 8. z1z2  z1.z2  9. n zn  z 10.  z1   z1  z2  z2   11. zz Rez2 1mz2 12. az1  bz2  az1  bz2 where a, b  R 2

BBrilliant STUDY CENTRE REPEATERS JEEMAIN LECTURE NOTE 8. Modulus of a complex number (Magnitude) Let z  x  iy, then z  x2  y2 , which is a non-negative real value. Geometrically, it represents the distance taken form origin Properties of modulus 1. zz  z 2 2. z  0  z  0 3. z  z  4. z1  z2 2  z1 2  z2 2  2 Re z1 z2  5. z1  z2 2  z1 2  z2 2  2 Re z1 z2 6. z1  z2 2  z1  z2 2  2  z1 2  z2 2   7. z1z2  z1 . z2 8. zn  z n 9. z1  z1 , z2 0 z2 z2 10. z1  z2  z1  z2  z1  z2 11. z1  z1  z1  z2  z1  z2 12. 1  z z z2 13. If z 1 1  z Z 9. Distance formula Distance between two complex numbers z1 & z2 in complex plane is z1  z2 10. z  z1  r , represents a circle with centre z1 & radius r 11. z  z1  r , represents the interior and the circumference of a circle with centre z1 and radius r 12. z  z1  r, represents the exterior and the circumference of a circle with centre z1 and radius r 13. z  z1  1, here locus of z represents the r bisector of the line joining the 2 fixed points z1 & z2 z  z2 3

BBrilliant STUDY CENTRE REPEATERS JEEMAIN MATHS -2021 (ONLINE CLASS NOTES) 14. Locus of z of z  z1   , where   0,1 is a circle z  z2 15. General equation of a circle is zz  z  z  c  0 where centre is '  ' and radius is  2  c 16. Locus of z of z  z1 2  z  z2 2  z1  z2 2 is a circle 17. Locus of z of z  z1  z  z2  2a 1) Where z1  z2  2a , is an ellipse with foci z1 & z2 and length of major axis 2a 2) Where z1  z2  2a, is a line segment joining z1 & z2 3) Where z1  z2  2a , represents no locus 18. Locus of z of z  z1  z  z2  2a 1) Where z1  z2  2a is a hyperbola with foci z1 & z2 and length of transverse axis 2a 2) Where z1  z2  2a , represents two opposite open rays with end points z1 & z2 3) Where z1  z2  2a , represents no locus 19. Argument or amplitude of a complex number z is the angle made by ray z in anticlockwise direction about the origin from the +ve direction of real axis and it is denoted by arg(z) or amp (z) 20. arg z is +ve if the rotation is in anti-clockwise direction and it is –ve if the rotation is in clockwise direction 21. Principal argument of z lies in , is,   arg Z   22. Argument of a complex number z is  , if z in Istquadarnt    , if z in 2nd quadrant    if z in 3rd quadrant   if z in 4th quadrant where   tan  1m  z   1 re  z    23. Argument of z is  0, if z lies on +ve real axis  , if z lies on –ve real axis   if z lies on +ve imaginary axis 2   if z lies on –ve imaginary axis 2 4

BBrilliant STUDY CENTRE REPEATERS JEEMAIN LECTURE NOTE 24. Arg z is not defined when z = 0 25. Polar form of complex number z  r cos   i sin , where r  z &   arg z 26. Polar form of a complex numbers z in the product of a non negative real number and a unimodular complex number cos   i sin  27. Eules function is ei  cos   sin  28. Eulerian form of a complex number z is z  rei , where r  z &  arg z 29. ei  cos   isin  ei  cos   isin  ei 2 o eiz o ei0  1, ei  1  1,  1, 30. Properties of arguments 1) arg z1z2   arg z1   arg z2  2) arg  z1   argz1   arg z2   z2    3) arg zn   argz 4) arg z   argz 5) arg  1   arg  z   z  31. Circular Rotation in complex plane about origin 5

BBrilliant STUDY CENTRE REPEATERS JEEMAIN MATHS -2021 (ONLINE CLASS NOTES) When z1  rei rotates an angle  about the origin in anticlockwise direction, then z2  z1ei and if the rotation is in clockwise direction then z2  z1ei Some particular cases Let z  rei 1) kz  kr ei , where k is +ve, then z streches k times in the direction of z. 2) kz  kr ei , where k is –ve, the z streches k times in the direction opposite to z. 3. Multiplying by i iz  irei  rei .ei2  rei2 ie; z rotates an angle  in anticlockwise direction about the origin. 2 4. Division by i (Multiplying by –i) z  rei  irei  rei .ei2 ii ie; z  rei2 i  ie; z rotates an angle in clockwise direction about the origion z 5. Multiplication by  rei i 2 rei   2  3 3  z  .e  2 ie; Z rotates an angle in anticlockwise direction about the origin. 3 6. Division by  (Multiplication by 2 ) Z  rei  2rei  rei  i 2 3 .e  z  z2  rei  2   3   2 ie; z rotates an angle in clockwise direction about the origin. 3 6

BBrilliant STUDY CENTRE REPEATERS JEEMAIN LECTURE NOTE 32. Circular rotation about z z2  z  z1  z ei When z1  z rotates an angle  about z in anticlockwise direction, then z2  z  z1  z ei 33. Complex division Let z1  r1ei1 and z2  r2.ei2 be two complex numbers then z1  r1 e i12  z1  0  z1  0 ei z2 r2 z2  0 z2  0 where 1  2   The division geometrically represents the rotation of z2 an angle  about the origin in anticlock wise direction and reaches z2 , if  in positive. 34. Rotation about z Let z, z1, z2 be complex number in complex plane, then z1  z  z1  z ei , where  in the angle of rotation. z2  z z2  z 7

BBrilliant STUDY CENTRE REPEATERS JEEMAIN MATHS -2021 (ONLINE CLASS NOTES) 35. Let z1, z2, z3, z4 be 4 complex numbers in complex plane, then z2  z4  z2  z4 ei , which represents z1  z3 z2  z3 the rotation of z1  z3 an angle  (positive) and reaches z1- z4 in anticlockwise direction. 36. Locus related to argument 1) Arg  z  z1  locus of z represents 2 open opposite rays with end points z1 and z2.  z  z2 0   2) Arg  z  z1  Locus of z represnets a line segment with end points z1 and z2  z  z2    3) Arg  z  z1   , Locus of z represents a circle with diameters joining z1 & z2  z  z2  2   4) Arg  z  z1   an acute angle ,  Locus of z represents a major arc with end points z1 & z2  z  z2    5) Arg  z  z1  obtuse angle,  Locus of z represents a minor arc with end points z1 & z2  z  z2   an   37. Cube roots of unity 1. 3 1  1, , 2 , when   1  i 3 2  1  i 3 22 22 2. 3 1  1, , 2 , when   1  i 3  2  1  i 3 2 22 2  3m  1 3. 3 1  3m 1     3m  2  2  1    2 4. 1    2  0 1  2     2  1 8

BBrilliant STUDY CENTRE REPEATERS JEEMAIN LECTURE NOTE 5. 1 n  2n  0, n  3m 3, n  3m 6.   2  1 2  4 or 2 , arg  7. 2 arg  3 33 arg    3 arg 2    3 8.   ei23 , 2  i 2  ei3 , 2  ei3 e 3 ,        9.   2 & 2  ,   2, 2     2  2 2  10. & 2  11. 1  2 & 1   2  12. x2  x  1  x   x  2  x2  x  1   x   x  2 13. x3 1   x 1x  x  2  x3  1  x  1x  x  2  14. 1,  , 2 lies on a unit circle z  1 15. 1,  , 2 divides the circumference of z  1 into 3 equal segments. 16. 1,  , 2 are the vertices of an equilateral triangle 38. nth roots of unity 1) n 1  i 2 k ,k  0,1, 2, 3......(n  1)  1, , 2 , 3....n1, where   i 2 n n e e 2) 1   2  .....  n1  0 3) n 1    4) xn 1   x 1 x   x  2 ..... x  n1    xn 1   x   x  x2 ..... x  n1 x 1 9

BBrilliant STUDY CENTRE REPEATERS JEEMAIN MATHS -2021 (ONLINE CLASS NOTES)    5)  x   x  2 ........ x  n1  1  x  x2  x3  .....  xn2  xn1 6) nth roots of unity divides the circumferene of z  1 into n equal segments. 7) nth roots of unity are the vertices of an n- sided regular polygon. 8) r 1 or nr 1  nr  r 9)   1  n1  2 10) n 1  1, , 2.......,n1 where    ei n    11) xn 1   x  1 x   x  2 ............ x  n1 10