In this section, we study about conjugate of a complex number, its geometric representation, and properties with suitable examples.

**Conjugate of a Complex
Number**

In this section, we study about conjugate of a complex number,
its geometric representation, and properties with suitable examples.

The conjugate of the complex number *x* + *iy* is
defined as the complex number *x* âˆ’ *i y*.

The complex conjugate of *z *is denoted by * *.
To get the conjugate of the complex number *z *, simply change *i *by
âˆ’*i *in *z*. For
instance 2 âˆ’ 5*i *is the
conjugate of 2 + 5*i*. The product of a complex number with its conjugate is
a real number.

For instance, (i) (*x*+*iy*)(* x*+*iy*) = *x*^{2} â€“ (*iy*) = *x*^{2}+*y*^{2}

(ii) (1 + 3*i*)(1-3*i*) = (1)^{2} â€“ (3*i*)^{2} = 1 + 9 =10.

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

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.

Let us verify some of the properties.

**Property**

For any two complex numbers z_{1} and z_{2}, we have .

**Proof**

Let *z*_{1} = *x*_{1} + *iy*_{1} , *z*_{2} = *x*_{2} + *iy*_{2} , and *x*_{1} , *x*_{2} , *y*_{1} , and *y*_{2} âˆˆ **R**

It can be generalized by means of mathematical induction to sums involving any finite number of terms

Let *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2}.

Then, *z*_{1}*z*_{2} = ( *x*_{1} + *iy*_{1 })( *x*_{2} + *iy*_{2} ) = ( *x*_{1} *x*_{2} - *y*_{1} *y*_{2} ) + *i *( *x*_{1} *y*_{2} + *x*_{2} *y*_{1} ) .

A complex number *z *is purely imaginary if and only if *z
*= âˆ’

Let *z = x i + y* . Then by definition = *x âˆ’ iy*

Therefore,* z* = âˆ’

â‡” *x* + i*y* =
âˆ’ (*x* âˆ’ *iy*)

â‡” 2*x* = 0 â‡”
x = 0

â‡” z is purely imaginary.

Similarly, we can verify the other properties of conjugate of
complex numbers.

**Example 2.3**

Write 3+4*i* / 5-12*i* in the x*i* + y form, hence
find its real and imaginary parts.

**Solution**

To find the real and imaginary parts of [3+4*i*] / [5-12*i*]
, first it should be expressed in the rectangular form *x *+ *iy*. To
simplify the quotient of two complex numbers, multiply the numerator and
denominator by the conjugate of the denominator to eliminate *i* in the
denominator.

This is in the *x* + *iy* form.

Hence real part is â€“ 33/169 and imaginary part is 56/169

**Example 2.4**

Simplify into rectangular form

**Solution**

If , find the complex number z in the rectangular form

**Example 2.6**

If z_{1} = 3 âˆ’ 2*i* and z_{2} = 6 + 4*i* , find z_{1}/z_{2} in the rectangular
form

**Solution**

Using the given value for z_{1} and z_{2} the value of z_{1}/z_{2} =

Find z^{âˆ’1}, if z = (2+3*i*) (1âˆ’ *i*).

We have *z *= (2 + 3*i *)(1- *i *) = (2 + 3) + (3
- 2)*i *= 5 + *i*

â‡’ z^{-1} = 1/ z = 1/[5 + i ].

Multiplying the numerator and denominator by the conjugate of
the denominator, we get

**Example 2.8**

**Solution**

Tags : Definition, Geometrical representation, Properties, Proof, Solved Example Problems , 12th Mathematics : UNIT 2 : Complex Numbers

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12th Mathematics : UNIT 2 : Complex Numbers : Conjugate of a Complex Number | Definition, Geometrical representation, Properties, Proof, Solved Example Problems

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