Conjugate of a complex number

Introduction

A complex number is a number that consists of both a real and an imaginary part. For example, a complex number is given by  z = x + iy, where ‘x’ is the real part, and ‘iy’ is the imaginary part.

A number with equal real parts and an imaginary part with a similar magnitude, but the opposite sign is the conjugate of that complex number.

What do you mean by the term conjugate of a complex number?

The complex numbers with equal real parts and imaginary parts of similar magnitudes, but imaginary parts with opposite signs are called the conjugate of a complex number. In other words, the complex conjugate of a complex number can be determined by altering the character of the imaginary part.

Let us consider a complex number, z = x + iy.

Then, the complex conjugate of z, z* = x – iy, and vice versa.

The conjugate of a complex number is often represented by z or z*.

Graphical Representation of Complex Conjugate

On Argand’s (or complex) plane, the conjugate of a complex number reflects the number about the real axis.

In simpler terms, the conjugate of a complex number can be determined by replacing the ‘i’ in it with ‘-i’.

Consider a complex number, z = x + iy, inclined at an angle θ, in the complex plane.

The reflection of this point about the real axis gives a point z* = x – iy, at an angle –θ.

Characteristics of Complex Conjugate of a Complex Number

Let us take two complex numbers, z1 and z2. Listed below are a few properties of complex numbers depicted using them:

1. The conjugate of the sum or difference of any two complex numbers is equal to the sum or difference of the conjugate of the individual complex numbers.

Ex- ( z1 + z2)* = z1* + z2*

( z1 – z2)* = z1* – z2*

2. The conjugate of the products of two complex numbers is equal to the conjugate products of the individual complex numbers.

Ex- (z1 . z2)* = z1*. z2*

3. The conjugate of the quotient of two complex numbers is equal to the quotient of the conjugate of the individual complex numbers.

Ex- (z1 / z2)* = z1*/z2*

4. The conjugate of a complex number returns the original complex number.

Ex- (z*)* = z

5. A complex number, when multiplied by its conjugate, results in a real value.

Ex- Consider z = x + iy

Then, z* = x – iy

Therefore, z. z* = (x + iy) . (x – iy) = (x2 + y2) = |z|2

Hence, we can also say that the result is equal to the square of the modulus of the complex number.

6. The sum of a complex number and its conjugate results in a value equal to twice the real part of the number.

Ex- Let, z = x + iy

Then, z* = x – iy

Thus, z + z* = (x + iy) + (x – iy) = 2x

7. The difference between a complex number and its conjugate gives twice the value of the imaginary part of the number.

Ex- Let, z = x + iy

Then, z* = x – iy

Thus, z – = (x + iy) – (x – iy) = 2iy

Solved Examples on Conjugate of Complex Numbers 

1. Determine the conjugate of:
2. 4 + 5i
3. -3 + 9i
4. -6 – 8i
5. 5 – 4i

Soln.

1. z = 4 + 5i

Thus, z* = 4 – 5i

  2. z = -3 + 9i

Thus, z* = -3 – 9i

  3. z = -6 – 8i

Thus, z* = -6 + 8i

  4. z = 5 – 4i

Thus, z* = 5 + 4i

Determine all the complex numbers of the form a + ib, such that z. z* = 25 and a + b=7

Soln. Let z = a + ib

 z* = a – ib

According to properties of complex numbers-

  5. z* = a2 + b2

 a2 + b2 = 25

Since, a + b = 7, a= 7 – b

Therefore,

a2 + b2 = (7 – b)2 + b2 = 25

 (b – 3) . (b – 4) = 0

Therefore,

Either, b = 3, Or b = 4

Thus, the possible complex numbers are 3 + 4i and 4 + 3i.

  6. Express the complex number z1 / z2 , in the form of a + ib if z1 = 4 – 5i and z2 = -2 + 3i

Soln. z1 / z2 = 4 – 5i / -2 + 3i

=> z1 / z2 = (4 – 5i / -2 + 3i) (-2 -3i / -2 – 3i) [Rationalising using conjugate of the denominator]

=> z1 / z2 =(-23 / 13) + (-2 / 13) i

Conclusion

1. The conjugate of a given complex number reflects the number about the real axis in the complex plane.
2. The conjugate of a complex number can be determined by altering the sign of the imaginary part of the number.
3. For any given complex number z,
4. a) z .z* = |z|2
5. b) z + z*= 2(real part)
6. c) z – z* = 2(imaginary part)
7. For any two complex numbers, z1 and z2,
8. a) ( z1 + z2)* = z1* + z2*
9. b) ( z1 – z2)* = z1* – z2*
10. c) (z1 / z2)* = z1*/z2*
11. d) (z1 . z2)* = z1*. z2*