conjugate of a complex number

 About Properties of Conjugate

 

The conjugate of the complex number z = a + ib is defined to be a – ib and is denoted by . In other words conjugate of a complex number is the mirror image of z in the real axis. 

If z = a + ib, z + conjugate of a complex number = 2 a ( real), 

z –conjugate of a complex number= 2 ib ( imaginary ) 

and zconjugate of a complex number = ( a+ib)(a–ib) = a2 + b2 (real ) = |z|2 = conjugate of a complex number

Also Re (z) = conjugate of a complex number

conjugate of a complex number

Properties of Conjugate:

  • conjugate of a complex number

  • |z| = |conjugate of a complex number|

  • z + conjugate of a complex number=2Re(z). z – conjugate of a complex number = 2i Im(z). 

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

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

  • = conjugate of a complex number

  • conjugate of a complex number= conjugate of a complex number1 + 2   

In general, conjugate of a complex number

  • conjugate of a complex number = conjugate of a complex number         

  • conjugate of a complex number   

 In general conjugate of a complex number

  • conjugate of a complex number

  • conjugate of a complex number           

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)| ≤ conjugate of a complex number|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 conjugate of a complex number               

  • |z1z2| = |z1| |z2|

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

  • |zn| = |z|n , n ∈ I

  • conjugate of a complex number           

  • |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| ≥ ||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) (conjugate of a complex number1 + conjugate of a complex number2) = |z1|2 + |z2|2 + z12 + z21 = |z1|2 + |z2|2 + 2Re(z12)

  • |z1 – z2|2 = (z1 – z2) (conjugate of a complex number1 – conjugate of a complex number2) = |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 ∈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 = conjugate of a complex number.

  • arg(z) = π/2, –π/2  ⇒ z is a purely  imaginary number ⇒ z = –conjugate of a complex number.

    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. 

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