Properties of Modulus

  • |z| = 0  =>     z = 0 + i0
  • |z1 – z2 | denotes  the distance between z1 and z2 .
  • –|z| ≤ Re(z)  ≤ |z| ; equality holds on right or on left side depending upon z being positive real or negative  real.
  • –|z| ≤ Imz ≤ |z| ; equality holds on right side or on left side depending upon z being purely imaginary and above the real axes or below the real axes.
  • |z| ≤ |Re(z)| + |Im(z)| ≤ |z| ;  equality  holds  on left  side  when z is  purely  imaginary  or  purely  real  and  equality  holds  on right  side when |Re(z)| = |Im(z)|.
  • |z|2 = z                                                           
  • |z1z2| = |z1| |z2|

In general |z1 z2 . . . . .zn| = |z1| |z2| . . . . . |zn|

  • |zn| = |z|n , n Î I
  • |z1+z2| ≤ |z1| + |z2| => |z1+z2+ … +zn| ≤ |z1| + |z2| + … + |zn|; equality holds  if  origin,  z1, z2, z3  …, zn  are  collinear  and z1 , z2, z3­, …,zn  are  on the  same  side  of the   origin.
  • |z1 – z2| ³ ||z1| – |z2|| ; equality holds  when  arg(z1/z2)  = π  i.e.  origin, z1, z2  are  collinear and  z1 and  z­2 are  on the  opposite  side  of the  origin.
  • |z1 + z2|2 = (z1 + z2) (1 + 2) = |z1|2 + |z2|2 + z12 + z21 = |z1|2 + |z2|2 + 2Re(z12)
  • |z1 – z2|2 = (z1 – z2) (12) = |z1|2 + |z2|2 – z12 – z21 = |z1|2 + |z2|2 – 2Re(z12)

Properties of Argument:

  • arg(z1z2) = Θ1 + Θ2 = arg(z­­1) + arg(z2)
  • arg (z1/z2) = Θ1 – Θ2 = arg(z1) – arg(z2)
  • arg (zn) = n arg(z),  n inclusing of  all I

Note:

  • In the above result Θ1 + Θ2  or Θ1  – Θ2 are not necessarily the principle values of the argument of corresponding complex numbers. E.g arg(zn) = n arg(z) only shows that one of the argument of zn is equal to n arg(z) (if we consider arg(z) in the principle range)
  • arg(z) = 0, π  => z is a purely  real number => z = .
  • arg(z) = π/2, –π/2  => z is a purely  imaginary number => z = –.

Note that the property of argument is the same as the property of logarithm.

Properties of Modulus and Properties of Arguments

= |z1| + |z2| = LHS.

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