Mathematics

LIKE TO JOIN ON FACEBOOK ( click here ) Students are requested to understand this carefully   1. Arg(z) = pi/3 z = 0 is excluded since z = 0 is the only complex number whose argument is not defined 2. Arg(z) > pi/3 Since principal argument belongs to (-pi, pi],  therefore Argument cannot be greater than pi. therefore Arg(z) > pi/3 =>)  z = 0 is excluded 3. |Arg(z)| < pi/4 => -pi/4 < Arg(z) < pi/4 z = 0 is excluded 4. => Arg(z) = pi since, principal argument belongs to  , therefore Argument cannot be greater than pi. z = 0 is excluded 5.  -pi/3 is included z = 0 is excluded 6. Complex number   , equal only when z is purely real number or '0' IMP 7. Arg(|z|) = 0 Since |z| is a Positive Real number , therefore it lies on right side of x - axis i.e. on positive Real axis and x-axis has '0' argument because it makes 0 radian angle from x-axis For z = 0 , argument does not exist T

Properties of Modulus: Modulus of z is the length of vector representing z form origin to the point z. 1.  If z = x + iy then   2.   &  Since,  ( equality follows when y = 0) and   (equality follows when x = 0) IMP 3.  Since, and,  4.  ,  5.  6.  7.   holds when  i.e. both the vectors are in same direction.  holds when   i.e. both the vectors are in opposite direction In general, 8.  IMP 9.  Explanation: Since,  Therefore,  or,                                                  [Since   (? click here ) ] and  10.     ( Property of Parallelogram ) Properties of Argument: Argument is the angle between the vector representing z from the positive direction of x-axis The Principal Argument belongs to (-pi, pi] 1.  ,   2.   , where k = 0, 1 or -1 3.   , where k = 0, 1 or -1 4.   , z cannot be negative Real Number because for negative Real number Arg(z) is equal to pi and so   will become -pi but i

Polar form or Trigonometric form of Complex Number: Since, z = x + iy or,  or,  Therefore,           (Euler form)   ;  Some Points about 'i': 1. If we square any number we will get a positive number. But,  imagine  there is an  imaginary number 'i'  which when square gives a negative number -1. then,  Remember, but,  or, or,    ( since  ) 2. i = i .................................................... ............................................... Therefore, power of 'i' repeats its values and can attain only four values i.e. i, -1, -i and 1 In general,  ,   ,   ,    , where  Reciprocal of i : Properties of Conjugate: We know, If z = x + iy Conjugate of 'z', 1.  2.  Here Re(z) represents the Real Part of complex number z. Any complex number is purely imaginary if and only if  Purely imaginary means Real part of complex nu