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 z 1 = a 1 + ib 1 and z 2 = a 2 + ib 2
So, z 1 + z 2 = a 1 + ib 1 + a 2 + ib 2
= (a 1 + a 2 ) + (b 1 + b 2 )i
Similarly for subtraction
z 1 - z 2 = (a 1 - a 2 ) + (b 1 - b 2 )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 z 1 = -3 + 2i and z 2 = 13 – i
Solution : z 2 - z 1 = 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.
z 1 = a 1 + ib 1 and z 2 = a 2 + ib 2 z 1 . z 2 = (a 1 + ib 1 )(a 2 + ib 2 )
= a 1 a 2 + a 1 b 2 i + a 2 b 1 i + b 1 b 2 i 2
= a 1 a 2 + a 1 b 2 i + a 2 b 1 i - b 1 b 2 (since i 2 = -1)
Example 1: Multiply 2 + 3i and 5 – 4i
Solution : (2 + 3i)(5 – 4i)
= 10 – 8i + 15i – 12 i 2
= 10 + 7i + 12 ( i 2 = -1)
= 22 + 7i
Example 2 : Show that : i 12 + i 13 + i 14 + i 15 = 0
Solution :We have ,
i 12 + i 13 + i 14 + i 15
= 1 + i – 1 – i
= 0
Division of complex numbers z = z 1 / z 2 , z 2 ≠ 0 ,
In division always multiply the numerator and denominator by conjugates of denominator.
Example : z 1 = 3 + i and z 2 = 1 + i
Solution :
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