A **complex number** is a number represented in the form of (x + **i** y); where x & y are real numbers, and **i** = √(-1) is called iota (an imaginary unit). It is also known as **imaginary numbers or quantities**. A complex number can be purely real or purely imaginary depending upon the values of x & y.

**Examples: **

(5 + 3 i), (0 + 7 i), (-4 + 9 i), (1 + 0 i), (0 – i), etc.

Now we will discuss two very important concepts of complex numbers which are Conjugate and Modulus.

## Conjugate of a Complex Number

Conjugate of a complex number is another complex number whose real parts Re(z) are equal and imaginary parts Im(z) are equal in magnitude but opposite in sign. Sometimes conjugate of a complex number is also called a complex conjugate. Conjugate of a complex number z is represented by and (z & ) together known as complex-conjugate pair because z and are conjugate of each other. If z = (x + i y) is a complex number then conjugate of z is defined as = (x – i y). From the relation between z and , we can say conjugate of a complex is obtained by replacing i with (-i). The geometrical meaning of conjugate of a complex number is the reflection or mirror image of the complex number z about the real axis (X-axis) in the complex plane or argand plane.

**Properties of Conjugate**

If z, z_{1}, & z_{2} are complex numbers then they will have the following conjugate properties.

- Conjugate of a purely real complex number is the number itself (z = ) i.e. conjugate of (7 + 0 i) = (7 – 0 i) = 7
- Conjugate of a purely imaginary complex number is negative of that number (z + = 0) i.e. conjugate of (0 -7 i) = (0 + 7 i) = 7 i
- = z
- z + = 2 Re(z)
- z – = 2 i . Im(z)
- z = {Re(z)}
^{2}+ {Im(z)}^{2}- =
- =
- =
- z = (z
_{1}/ z_{2}) then = / ; z_{2}≠ 0

Note:In order to find out the conjugate of a complex number that complex number must be in its standard form that is Z = (x + i y). If the complex number is not in its standard form then it has to be converted into its standard form before finding its complex conjugate.

### Examples

**Example 1:** **If z = (5 + 7 i) is a complex number then its conjugate is given by**

**Solution:**

= (5 – 7 i).

**Example 2: If z = 1 / (4 + 3 i) is a complex number then its conjugate is given by**

**Solution:**

First z has to be converted into its standard form by multiplying the numerator and

denominator with the conjugate of (4 + 3 i)

z = (1 / (4 + 3 i)) × ((4 – 3 i) / (4 – 3 i))

z = (4 – 3 i) / (16 + 9)

z = (4 / 25) – (3 / 25) i

The conjugate of z is = (4 / 25) + (3 / 25) i

**Example 3:** **If (a + i b) is a complex number which is the complex conjugate of (8 – 3 i) then values of a and b are given by**

**Solution:**

Let z = a + i b

= (8 + 3 i)

z =

z = (8 + 3 i)

Two complex numbers are equal only when their corresponding real & imaginary parts are equal

Equating the real and imaginary parts of z & (8 + 3 i)

Re(z) = a = 8

Im(z) = b = 3

Hence 8 & 3 are the respective values of a and b.

## Modulus of a Complex Number

The modulus of a complex number z = (x + i y) is defined as |z| = √(x^{2 }+ y^{2}); where x = Re(z) and y = Im(z). |z| ≥ 0 for all complex numbers. We cannot compare two complex numbers z_{1} and z_{2} i.e. (z_{1} > z_{2}) or (z_{1 }< z_{2}) has no significance but (|z_{1}| > |z_{2}|) or (|z_{1}| < |z_{2}|) has definite meaning because |z_{1}| & |z_{2}| are real numbers. The geometrical meaning of modulus of a complex number (x + i y) is the distance of the point P (x, y) from the origin O (0, 0) in the complex plane or argand plane i.e. |z| = = √((x – 0)^{2} + (y – 0)^{2}) = √(x^{2} + y^{2}).

**Properties of Modulus**

If z, z_{1}, z_{2} & z_{n} are complex numbers then they will have the following modulus properties.

- If a complex number z has equal real and imaginary parts then its modulus is zero i.e. Re(z) = Im(z) ⇒ |z| = 0
- Real and imaginary part of a complex number z always lies between -|z| and |z| i.e. -|z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z|
- |z| = || = |-z|
- = |z|
^{2}- |Z₁ . Z₂ …….. Z
_{n}| = |Z₁| . |Z₂| …….. |Z_{n}|- |Z₁ / Z₂| = |Z₁| / |Z₂| ; Z₂ ≠ 0
- |Z₁ + Z₂|
^{2}= |Z₁|^{2}+ |Z₂|^{2}+ 2 Re()- |Z₁ – Z₂|
^{2}= |Z₁|^{2}+ |Z₂|^{2}– 2 Re()- |Z₁ + Z₂|
^{2}+ |Z₁ – Z₂|^{2}= 2( |Z₁|^{2}+ |Z₂|^{2})- |b. Z₁ + a. Z₂|
^{2}+ |a. Z₁ – b. Z₂|^{2}= (a^{2}+ b^{2}) ( |Z₁|^{2}+ |Z₂|^{2}) ; here a & b are real numbers.

Note:To find out the modulus of a complex number that complex number must be in its standard form that is Z = (x + i y). If the complex number is not in its standard form then it has to be converted into its standard form before finding its modulus.

### Examples

**Example 1: If (8 + 3 i) is a complex number then its modulus is given by**

**Solution:**

|8 + 3 i| = √(8

^{2}+ 3^{2})|8 + 3 i| = √(64 + 9)

|8 + 3 i| = √(73)

**Example 2: If (3 + 4 i) _{ }and (√3 + √6 i)_{ }are two complex numbers then the modulus of their product is given by the product of their individual modulus.**

**Solution:**

|3 + 4 i| = √(3

^{2}+ 4^{2})|3 + 4 i| = √(9 + 16)

|3 + 4 i| = √(25) = 5

|√3 + √6 i| = √((√3)

^{2}+ (√6)^{2})|√3 + √6 i| = √(3 + 6)

|√3 + √6 i| = √9 =3

|(3 + 4 i).(√3 + √6 i)| = 5 × 3

|(3 + 4 i).(√3 + √6 i)| = 15

**Example 3:** **If z = (2 . i) / (3 – 4 i) is a complex number then its modulus is given by**

**Solution:**

First z has to be converted into its standard form by multiplying the numerator and

denominator with the conjugate of (3 – 4 i)

z = ((2.i) / (3 – 4 i)) × ((3 + 4 i) / (3 + 4 i))

z = (8 + 6 i) / (9 + 16)

z = (8 + 6 i) / 25

z = (8 / 25) + (6 / 25) i

The modulus of z is |z| = √((8 / 25)

^{2 }+ (6 / 25)^{2})|z| = √((64 + 36) /625)

|z| = √(100 /625)

|z| = 10 / 25

|z| = 2 / 5

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