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 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 :