Any number that is expressible in the form 'c + d i' is known as a complex number. Here 'c' is known as the real part of the complex number and 'd i' is the imaginary part of the complex number given with 'c' and 'd' being real numbers. 'i' is pronounced as iota and its value is $i^2$ = -1, that is, i = $\sqrt (-1)$. Examples of complex numbers are 7 + 5i, 8i, etc.

Every real number can be expressed as a complex number. For example: 5 = 5 + 0i
So we can conclude that any part of the complex number can be zero. 

Formulas

Different techniques are used to solve the operations on complex numbers:

ADDITION & SUBTRACTION:

We add or subtract two complex numbers by adding or subtracting their same type of parts, that is real to real and imaginary to imaginary.

(b + c i) + (p + q i) = (b + p) + (c + q) i

(b + c i) - (p + q i) = (b - p) + (c - q) i

MULTIPLICATION:

We use FOIL rule to multiply given complex numbers.

(b + c i) (p + q i) = b p + b q i + p c i + c q $i^2$

$\rightarrow$ (b + c i) (p + q i) = b p + (b q + p c) i – c q

$\rightarrow$ (b + c i) (p + q i) = (b p – c q) + (b q + p c) i

CONJUGATES:

For obtaining a conjugate we only change the sign in the middle. Conjugate of (b + c i) is (b – c i).

We also represent it by a bar over the given complex number. When the conjugates are multiplied together then we always obtain a number without the complex part

(b + c i) (b - c i) = $b^2$ - $(c i)^2$ = $b^2$ - $c^2 i^2$ = $b^2$ - (-1) $c^2$ 

= $b^2 + c^2$

DIVISION:

When we divide two complex numbers we multiply the numerator and denominator of the given rational complex number by the conjugate of the denominator.

Examples

Examples on complex numbers are given below:

Example 1: Find the value of (5 – 3 i) – (6 + 5 i)

Solution:

(5 - 3 i) - (6 + 5 i)

= (5 - 6) - (3 - 5) i

= -1 - (-2) i

= -1 + 2 i

Example 2: Find the product (5 - 3 i) and (6 + 5 i)

Solution:

(5 - 3 i) . (6 + 5 i)

= 5 x 6 – 3 . 5 i^2 - 3 . 6 i + 5 . 5 i

= 30 + 15 - 18 i + 25 i

= 45 + 7 i 

Example 3:

Evaluate $\frac{(5 – 3 i)}{(6 + 5 i)}$

Solution:

$\frac{(5 - 3 i)}{(6 + 5 i)}$

= $\frac{[(5 - 3 i) (6 - 5 i)]}{ [(6 + 5 i) (6 – 5 i)]}$

= $\frac{(5 . 6 + 3 . 5 i^2 - 3 . 6 i - 5 . 5 i)}{ (6^2 + 5^2)}$

= $\frac{(30 - 15 - 18 i - 25 i)}{61}$

= $\frac{(15 - 43 i)}{61}$

= $\frac{15}{61}$$\frac{43}{61}$ i