If ‘a’ and ‘b’ are two numbers in the form (a + ib) then it is known as **Complex Number**. Number ‘a’ is called real part and ‘b’ is called the imaginary part of the complex number (a + ib). The complex number is generally represented by **z**.

**A Complex Number is a combination of a real number and an imaginary number**

Complex Number has a real part and an imaginary part. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

**Imaginary Unit(i)**

i^{2} = -1 or i = √1

for x > 0, √(-x) = √(-1).x = √i^{2}x = i√x

**Power of an imaginary unit**

i^{0} = 1

i^{1} = i

i^{2} = -1

i^{3} = -i

i^{4} = +1

i^{4k+1} = i

i^{4k+2} = -1

i^{4k+3} = -i

i^{4k} = +1

where k is positive integer.

**Conjugate of Complex number**

In any two complex numbers, if only the sign of the imaginary part differs then, they are known as a **complex conjugate** of each other. Its is denoted by .

If z = a + ib is complex number, then **conjugate**.

For example,

(i) Conjugate of z_{1} = 5 + 4i is = 5 – 4i

(ii) Conjugate of z_{2} = – 8 – i is = – 8 + i

(iii) Conjugate of z_{3} = 9i is = – 9i.

(iv) = 6 – 5i, = 6 + 5i

**Property of Conjugate of Complex number**

If z, z_{1} and z_{2} are complex number, then

=> If z is real. z =

Real z= *z + *2

Imaginary z= *z – *2i

=> = z

=> | | = |z|

=> z. = .z

=> z. = |z|^{2}

=> = +

=> = –

=> = .

=> *z _{1}*z

_{2}=

*where z*

_{2}≠ 0

**Additional of Complex number**

Let z_{1} = (a + bi) , z_{2} = (c + di) two complex number then sum of two complex number as follow:

z_{1} + z_{2} = (a + bi) + (c + di) = (a + c) + (b + d)i which is again is complex number.

**Example:** (4 – 3i) + (-1 + 8i) = (4+(-1)) + (-3+8)i = 3 + 5i

The addition of complex number satisfies the following property:

**1. Commutative Law:**

For any two complex number z_{1} and z2

z_{1} + z_{2} = z_{2} + z_{1}

**2. Closure Law:**

Sum of two complex number z_{1} and z_{2} is also complex number. ie z_{1} + z_{2} is also complex number.

**3. Associative Law:**

For any three complex number z_{1}, z_{2} and z_{3}

(z_{1} + z_{2}) + z_{3} = z_{1} + (z_{2} + z_{3})

**4. Additive Identity:**

For any complex number a+ib, we have

(a + b.i) + (0 + 0.i) = a + bi

**5. Additive Inverse:**

Complex number -z is additive inverse of z.

**Difference of Complex number**

Let z_{1} = (a + bi) , z_{2} = (c + di) two complex number then difference of two complex number as follow:

z_{1} – z_{2} = z_{1} + (-z_{2})

= (a + bi) – (c + di) = (a – c) + (b – d)i

**Example:** (4 – 3i) – (-1 + 8i) = 4 – 3i + 1 – 8i = 5 – 11i

**Multiplication of Complex number**

Let z_{1} = (a + bi) , z_{2} = (c + di) two complex number then product of two complex number as follow:

z_{1}z_{2} = (a + bi) × (c + di) = (ac – bd) + (ad + bc)i

Instead of memorizing that formula, just think “FOIL” multiplication, and remember that i2 = -1

Example: (4 – 3i)(-1 + 8i) = (4)(-1) + (4)(8i) + (-3i)(-1) + (-3i)(8i)

= -4 + 32i + 3i + (-24)i^{2}

= -4 + 35i + (-24)(-1)

= -4 + 35i + 24

= 20 + 35i

The product of complex number satisfies the following property:

**1. Commutative Law:**

For any two complex number z_{1} and z_{2}

z_{1}z_{2} = z_{2}z_{1}

**2. Closure Law:**

Product of two complex number z_{1} and z_{2} is also complex number. ie z_{1}z_{2} is also complex number.

**3. Associative Law:**

For any three complex number z_{1}, z_{2} and z_{3}

(z_{1}z_{2})z_{3} = z_{1}(z_{2}z_{3})

**4. Multiplication Identity:**

i + 0i or 1 is the multiplication identity

(a + ib)(1 + 0.i) = a +ib

**5. Multiplication Inverse:**

Complex number 1/z is multiplication inverse of z.

**6. Distributive Law:**

For any three complex number z_{1}, z_{2}, z_{3}

a) z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}

b) (z_{1} + z_{2})z_{3} = z_{1}z_{3} +z_{2}z_{3}

**Division of Complex number**

For any three complex number z_{1}, z_{2},

where z_{2} ≠ 0, the quotient *z _{1}*z

_{2}is defined by

*z*z

_{1}_{2}= z

_{1}

*1*z

_{2}

Here are the step require to divide complex numbers:

Step 1: To divide complex numbers, you must multiply by the conjugate. To find the conjugate of a complex number all you have to do is change the sign between the two terms in the denominator.

Step 2: Distribute (or FOIL) in both the numerator and denominator to remove the parenthesis.

Step 3: Simplify the powers of i, specifically remember that i2 = –1.

Step 4: Combine like terms in both the numerator and denominator, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Step 5: Write you answer in the form a + bi.

Step 6: Reduce your answer if you can.

**Module of a complex number**

Point P(x,y) represents the complex number z = x + iy in the complex plane. OP is known as the moduus of complex number z. |z| = /a^{2} + b^{2}

θ, the angle between OP and x-axis, is known as argument or amplitude of the complex number z. The set of complex numbers is denoted by C.

**Polar representation of a complex number**

Point P is uniquely determined by the ordered pair of a real number(r,θ), called the **polar coordinates** of point P.

x = r cosθ, y = rsinθ

therefore, z=r(cosθ + isinθ)

where r =√a^{2} + b^{2} and θ =tan^{-1} =b/a

The latter is said to be polar form of complex number.

Here r = √x^{2} + y^{2} = |z| is the modus of z and θ is called argument(or amplitude) of z is denoted by **arg z**.

If z_{1},z_{2},——z_{n} are the complex numbers then

z_{1}.z_{2}. —–z_{n} = r1.r_{2}.r_{3}——.r_{n}{cos(θ1+θ_{2}+—+θ_{n}) + iSin(θ_{1}+θ_{2}+—+θ_{n})}

I hope, this article will help you a lot to understand the **Complex Number | Properties of Complex Number**. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.