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)

i² = -1 or i = √1
for x > 0, √(-x) = √(-1).x = √i²x = i√x

Power of an imaginary unit

i⁰ = 1
i¹ = i
i² = -1
i³ = -i
i⁴ = +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 z.

If z = a + ib is complex number, then z = a – ib is called conjugate.
For example,

(i) Conjugate of z₁ = 5 + 4i is z₁ = 5 – 4i
(ii) Conjugate of z₂ = – 8 – i is z₂ = – 8 + i
(iii) Conjugate of z₃ = 9i is z₃ = – 9i.
(iv) 6 + 5i = 6 – 5i, 6 – 5i = 6 + 5i

Property of Conjugate of Complex number

If z, z₁ and z₂ are complex number, then

=> If z is real. z = z
Real z= z + z2
Imaginary z= z – z2i
=> z = z
=> |z| = |z|
=> z.z = z.z
=> z.z = |z|²
=> z₁ + z₂ = z₁ + z₂
=> z₁ – z₂ = z₁ – z₂
=> z₁.z₂ = z₁.z₂
=> z₁z₂ = z₁z₂ where z₂ ≠ 0

Additional of Complex number

Let z₁ = (a + bi) , z₂ = (c + di) two complex number then sum of two complex number as follow:
z₁ + z₂ = (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 z₁ and z2
z₁ + z₂ = z₂ + z₁

2. Closure Law:

Sum of two complex number z₁ and z₂ is also complex number. ie z₁ + z₂ is also complex number.

3. Associative Law:

For any three complex number z₁, z₂ and z₃
(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)

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 z₁ = (a + bi) , z₂ = (c + di) two complex number then difference of two complex number as follow:
z₁ – z₂ = z₁ + (-z₂)
= (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 z₁ = (a + bi) , z₂ = (c + di) two complex number then product of two complex number as follow:

z₁z₂ = (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)i²
= -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 z₁ and z₂
z₁z₂ = z₂z₁

2. Closure Law:

Product of two complex number z₁ and z₂ is also complex number. ie z₁z₂ is also complex number.

3. Associative Law:

For any three complex number z₁, z₂ and z₃
(z₁z₂)z₃ = z₁(z₂z₃)

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 z₁, z₂, z₃
a) z₁(z₂ + z₃) = z₁z₂ + z₁z₃
b) (z₁ + z₂)z₃ = z₁z₃ +z₂z₃

Division of Complex number

For any three complex number z₁, z₂,
where z₂ ≠ 0, the quotient z₁z₂ is defined by
z₁z₂ = z₁ 1z₂
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| = /a² + b²

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 =√a² + b² and θ =tan^-1 =b/a
The latter is said to be polar form of complex number.
Here r = √x² + y² = |z| is the modus of z and θ is called argument(or amplitude) of z is denoted by arg z.
If z₁,z₂,——z_n are the complex numbers then
z₁.z₂. —–z_n = r1.r₂.r₃——.r_n{cos(θ1+θ₂+—+θ_n) + iSin(θ₁+θ₂+—+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.