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