Perform Operations with Complex Numbers

In this section we will perform operations with complex numbers such as addition,subtraction, multiplication and division of complex numbers.
Addition/subtraction of complex numbers : The two or more complex numbers are to be added or subtracted like algebraic expression. That means like terms are to be added or subtracted. The terms with ‘i’ are like terms
Let the two complex numbers z₁ = a₁ + ib₁ and z₂= a₂ + ib₂
So, z₁ + z₂ = a₁ + ib₁ + a₂ + ib₂
= (a₁ + a₂) + (b₁ + b₂)i
Similarly for subtraction
z₁ - z₂ = (a₁ - a₂) + (b₁ - b₂)i
Example 1 : Add (3 + 7i) and (4 + 5i)
Solution : (3 + 7i) + (4 + 5i)
= 3 + 4 + 7i + 5i
= 7 + 12 i

Example 2 : Find the difference of z₁ = -3 + 2i and z₂ = 13 – i
Solution : z₂ - z₁ = 13 – i –(-3 + 2i)
= 13 – i + 3 – 2i
= 16 – 3i
Multiplication of complex numbers
The multiplication of complex numbers is same as the product of two binomial. Here you can use a FOIL method.
z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ z₁ . z₂ = (a₁ + ib₁)(a₂ + ib₂)
= a₁a₂ + a₁b₂i + a₂ b₁i + b₁b₂ i²
= a₁a₂ + a₁b₂i + a₂b₁ i - b₁b₂ (since i²= -1)

Example 1: Multiply 2 + 3i and 5 – 4i
Solution : (2 + 3i)(5 – 4i)
= 10 – 8i + 15i – 12 i²
= 10 + 7i + 12 ( i² = -1)
= 22 + 7i

Example 2 : Show that : i¹² + i¹³ + i¹⁴ + i¹⁵ = 0
Solution :We have ,
i¹² + i¹³ + i¹⁴ + i¹⁵
= 1 + i – 1 – i
= 0

Division of complex numbers z = z₁/ z₂, z₂ ≠ 0 ,
In division always multiply the numerator and denominator by conjugates of denominator.
Example : z₁ = 3 + i and z₂ = 1 + i
Solution :



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