Conjugate of a Complex Number

What is a Complex Number?

A number that can be represented in the form of (a + ib), where ‘i’ is an imaginary number called iota, can be called a complex number. A complex number is basically a combination of a real part and an imaginary part of that number.


For example, 6 + i3 is a complex number in which 6 is the real part of the number and i3 is the imaginary part of the number.


What Is a Conjugate?

A conjugate in Mathematics is formed by changing the sign of one of the terms in a binomial. For example, if the binomial number is a + b, so the conjugate of this number will be formed by changing the sign of either of the terms. If we change the sign of b, so the conjugate formed will be a – b. Therefore, in mathematics, a + b and a – b are both conjugates of each other.

Conjugate of a Complex Number? 

The conjugate of a complex number represents the reflection of that complex number about the real axis on Argand’s plane. When the i of a complex number is replaced with -i, we get the conjugate of that complex number that shows the image of that particular complex number about the Argand’s plane.


For example, as shown in the image on the right side, z = x + iy is a complex number that is inclined on the real axis making an angle of α and z = x – iy which is inclined to the real axis making an angle -α. 


Properties of the conjugate of a Complex Number


  1. \[\overline{z_{1} \pm z_{2} }\] = \[\overline{z_{1}}\]  \[\pm\] \[\overline{z_{2}}\]

 

Proof: Let z1 = p + iq and z2 = x + iy

                         

                         So, \[\overline{z_{1} \pm z_{2} }\] = \[\overline{p + iq \pm + iy}\]


                                      = \[\overline{p \pm x + iq \pm iy}\]

                                                                                    

                                        = \[\overline{p \pm x - i(q \pm y)}\]

                                                                                    

                                           = \[\overline{p - iq \pm x - iy}\]

                                                                                        

                                           =  \[\overline{z_{1}}\] \[\pm\] \[\overline{z_{2}}\]



  1. \[\overline{z_{}. z_{2}}\]  = \[\overline{z_{1} z_{2}}\]

Proof: Let z1 = a + ib and z2 = c + id

Then, \[\overline{z_{}. z_{2}}\] =  \[\overline{(a + ib) . (c + id)}\]

                   = \[\overline{ac + iad + ibc + i2bd}\]

                   = \[\overline{ac + iad + ibc - bd}\]

                  = \[\overline{ac - bd + i(ad + bc)}\]

                    = ac -bd – i(ad + bc)

      = ac + i2bd – iad – ibc

      = c(a – ib) – id(a – ib)

      = (a – ib).(c – id)

                              = \[\overline{z_{}. z_{2}}\]

3. \[\frac{\overline{z_{1}}}{z_{2}}\] =  \[\frac{\overline{z}_{1}}{\overline{z}_{2}}\]


Proof, \[\frac{\overline{z_{1}}}{z_{2}}\] =    \[\overline{(z_{1}.\frac{1}{z_{2}})}\] 


Using the multiplicative property of conjugate, we have

            \[\overline{z_{1}}\] . \[\frac{\overline{1}}{z_{2}}\]

            \[\frac{\overline{z}_{1}}{\overline{z}_{2}}\]


4. \[\overline{z}\] = z

Proof: Let z= a + ib

Then, \[\overline{z}\] =  \[\overline{a + ib}\] = \[\overline{a - ib}\] = a + ib = z

5 .If z = a + ib


Then, z. \[\overline{z}\]  = a2 + b2 = |z2|

Proof: z.  \[\overline{z}\] = (a + ib). \[\overline{(a + ib)}\] = (a + ib).(a – ib) = a2 – i2b2 = a2 + b2 = |z2|


6.  z +  \[\overline{z}\] = x + iy + ( x – iy )

    Then,  z + \[\overline{z}\] = 2x


7.  z -  \[\overline{z}\] = x + iy - ( x – iy )

    Then, z - \[\overline{z}\] = 2 iy


Solved Problems

Question 1. Identify the conjugate of the complex number 5 + 6i. Describe the real and the imaginary numbers separately.

Answer:    

The conjugate of the complex number 5 + 6i  is 5 – 6i.

The real part of the resultant number = 5 and the imaginary part of the resultant number = 6i.


Question 2. Find all the complex numbers of the form z = p + qi , where p and q are real numbers such that z. \[\overline{z}\] = 25 and p + q = 7 where \[\overline{z}\] is the complex conjugate of z.

Answer: It is given that z. \[\overline{z}\] = 25

                 By the definition of the conjugate of a complex number,

    \[\overline{z}\] = p – iq

    Therefore, z. \[\overline{z}\]  = (p + iq) . (p – iq) = 25

    = p2 + q2 = 25

    Since, p + q = 7 , q = 7 – p

    Therefore, p2 + q2 = p2 + ( 7- p )2 = 25

                        = p2 + 49 + p2 – 14 p = 25

          = 2p2 – 14p + 24 = 0

          = p2 – 7p + 12 = 0

          = p(p-3) – 4(p-3) = 0

          = (p – 3).(p – 4) =0

  Therefore, either p = 3 or p =4

  So, if p=3, q=4 ; else if p =4, q = 3

  Possible complex numbers are: 3 + i4 or 4 + i3.


FAQ (Frequently Asked Questions)

1. What is the geometric significance of the conjugate of a complex number?

The complex numbers itself help in explaining the rotation in terms of 2 axes. One which is the real axis and the other is the imaginary axis. The complex numbers help in explaining the rotation of a plane around the axis in two planes as in the form of 2 vectors. The concept of 2D vectors using complex numbers adds to the concept of ‘special multiplication’. Rotation around the plane of 2D vectors is a rigid motion and the conjugate of the complex number helps to define it. The conjugate helps in calculation of 2D vectors around the plane and it becomes easier to study their motions and their angles with the complex numbers. 


The conjugate of the complex number makes the job of finding the reflection of a 2D vector or just to study it in different plane much easier than before as all of the rigid motions of the 2D vectors like translation, rotation, reflection can easily by operated in the form of vector components and that is where the role of complex numbers comes in.


2. How is the conjugate of a complex number different from its modulus?

The conjugate of a complex number is a way to represent the reflection of a 2D vector, broken into its vector components using complex numbers, in the Argand’s plane. The conjugate of a complex number helps in the calculation of a 2D vector around the two planes and helps in the calculation of their angles. The modulus of a complex number on the other hand is the distance of the complex number from the origin. Although there is a property in complex numbers that associate the conjugate of the complex number, the modulus of the complex number and the complex number itself. That property says that any complex number when multiplied with its conjugate equals to the square of the modulus of that particular complex number. Suppose, z is a complex number so,

 z.  z̅ = |z|