Properties of Modulus:

• |z| = 0 => z = 0 + i0                       
• |z₁ - z₂ | shows the distance between z₁ and z₂ .
• -|z| ≤ Re(z) ≤ |z| ; equality holds on right or on left side relying upon z being negative real or positive real.
• -|z| ≤ Im z ≤ |z| ; equality holds on right side or on left side relying upon z being completely imaginary and over the real axes or under the real axes. 
• |z| ≤ |Re(z)| + |Im(z)| ≤ √2|z| ;  equality  holds  on left  side  when z is  completely  imaginary  or  completely  real  and  equality  holds on right  side when |Re(z)| = |Im(z)|.
• |z|² = z                                                            
• |z₁z₂| = |z₁| |z₂|

      In general |z₁ z₂ . . . . .z_n| = |z₁| |z₂| . . . . . |z_n|

• |zⁿ| = |z|ⁿ , n Î I
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• |z₁+z₂| £ |z₁| + |z₂| Þ |z₁+z₂+ ... +z_n| £ |z₁| + |z₂| + ... + |z_n|; equality holds if  origin,  z₁, z₂, z₃  ..., z_n  are  collinear  and z₁ , z₂, z₃­, ...,z_n  are  on the  similar  side  of the   origin.
• |z₁ - z₂| ³ ||z₁| - |z₂|| ; equality takes when  arg(z₁/z₂)  = p  i.e.  origin, z₁, z₂  are  collinear and  z₁ and  z_­2 are  on the  reverse  side  of the  origin. 
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