# CONJUGATE OF A COMPLEX NUMBER

The conjugate of the complex number x + iy is defined as the complex number x - iy.

If z represents a complex number, then the complex conjugate of z is denoted by To get the conjugate of the complex number z, simply change the sign of i in z. For instance, 2 - 3i is the conjugate of 2 + 3i.

The product of a complex number with its conjugate is a real number. For instance,

(i) (a + ib)(a - ib) :

= a2 - (ib)2

= a2 - i2b2

= a2 - (-1)b2

= a2 + b2

(i) (3 + 4i)(3 - 4i) :

= 32 - (4i)2

= 9 - 42i2

= 9 - 4(-1)

= 9 + 4

= 13

Geometrically, the conjugate of of the complex number z is obtained by reflecting z on the real axis.

## Geometrical Representation of Conjugate of a Complex Number

Note :

Two complex numbers x + iy and x - iy are conjugates to each other. The conjugate is useful in division of complex numbers. The complex number can be replaced with a real number in the denominator by multiplying the numerator and denominator by the conjugate of the denominator. This process is similar to rationalising the denominator to remove surds.

## Solve Problems

Problem 1 :

Write the complex number given below in x + iy form, hence find its real and imaginary parts.

Solution :

To express the given complex number in the rectangular form x + iymultiply the numerator and denominator by the conjugate of the denominator to eliminate i in the denominator.

The complex number above is in the form x + iy.

Problem 2 :

Simplify the following expression into rectangular form.

Solution :

We consider

and

Therefore,

Problem 3 :

In the expression given below, find the complex number z in the rectangular form.

Solution :

2(z + 3) = (z - 5i)(1 + 4i)

2z + 6 = z + 4zi - 5i - 20i2

2z + 6 = z + 4zi - 5i - 20(-1)

2z + 6 = z + 4zi - 5i + 20

2z - z - 4zi = -5i + 20 - 6

z - 4zi = -5i + 14

z(1 - 4i) = 14 - 5i

Problem 4 :

Solution :

Problem 5 :

Find z-1, if z = (2 + 3i)(1 - i).

Solution :

z = (2 + 3i)(1 - i)

= 2 - 2i + 3i - 3i2

= 2 + i - 3(-1)

= 2 + i + 3

= 5 + i

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