conjugate of a complex number

Properties of Conjugate:

• 

• |z| = ||

• z + =2Re(z). z – = 2i Im(z). 

• If z is purely real z = . whenever we have to show a complex number purely real we use this property.

• If z is purely imaginary z+ =0, whenever we have to show that a complex number is purely imaginary we use this property. 

• =

• = 1 + 2   

In general,

• =          

•    

 In general

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| ≤ Im z ≤ |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| . . . . . |z_n|

• |zⁿ| = |z|ⁿ , n ∈ I

•            

• |z₁+z₂| ≤ |z₁| + |z₂| ⇒ |z₁+z₂+ ... +z_n| ≤ |z₁| + |z₂| + ... + |z_n|; equality holds  if origin, z₁, z₂, z₃  …, zn  are  collinear  and z₁ , z₂, z₃, …,z_n  are  on the  same side  of the origin. 

• |z₁ – z₂| ≥ ||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) (1 – ₂) = |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 (z_n) = n arg(z),  n ∈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(z_n) = 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. 

Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah.