# conjugate of a complex number

 The conjugate of the complex number z = a + ib is defined to be a – ib and is denoted by . In other words is the mirror image of z in the real axis.  If z = a + ib, z + = 2 a ( real),  z –= 2 ib ( imaginary )  and z = ( a+ib)(a–ib) = a2 + b2 (real ) = |z|2 = .  Also Re (z) =

### 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| . . . . . |zn|

• |zn| = |z|n , n ∈ 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| ≥ ||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 (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 = .

• 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.