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 :



11th grade math

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