Adding, Subtracting and Multiplying Complex Numbers
Addition and subtraction of complex numbers are performed as if the complex
numbers were algebraic expressions with i being a variable.
Example 1
Addition and subtraction of complex numbers
Find the sums and differences.
a) (2 + 3i) + (6 + i)
b) (2 + 3i) + (2  5i )
c) (3 + 5i)  (1 + 2i)
d) (2  3i)  (1  i )
Solution
a) (2 + 3i) + (6 + i) = 8 + 4i
b) (2 + 3i) + (2  5i ) = 4  2i
c) (3 + 5i)  (1 + 2i) = 2 + 3i
d) (2  3i)  (1  i ) = 3 2i
We can give a symbolic definition of addition and subtraction as follows.
Addition and Subtraction of Complex Numbers
The sum and difference of a + bi and c + di are defined as follows:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)  (c + di) = (a  c) + (b  d)i
Complex numbers are multiplied as if they were algebraic expressions. Whenever
i^{2} appears, we replace it by 1.
Example 2
Products of complex numbers
Find each product.
a) 2i(1 + i)
b) (2 + 3i)(4 + 5i)
c) (3 + i)(3  i)
Solution
a) 2i(1 + i) 
= 2i + 2i^{2} 
Distributive property 

= 2i + 2(1) 
i^{2} = 1 

= 2 + 2i 

b) Use the FOIL method to find the product:
(2 + 3i)(4 + 5i) 
= 8 +10i + 12i + 15i^{2} 


= 8 + 22i + 15(1) 
Replace i^{2} by 1. 

= 8 + 22i  15 


= 7 + 22i 

c) This product is the product of a sum and a difference.
(3 + i)(3  i) 
= 9  3i + 3i  i2 


= 9  (1) 
i^{2} = 1 

= 10 

We can find powers of i using the fact that i^{2} = 1. For example,
i^{3} = i^{2} Â· i = 1 Â· i = i.
The value of i^{4} is found from the value of i^{3}:
i^{4} = i^{3} Â· i = i Â· i = i^{2} = 1.
In the next example we find more powers of imaginary numbers.
Example 3
Powers of imaginary numbers
Write each expression in the form a + bi.
a) (2i)^{2 }
b) (2i)^{2 }
c) i^{6 }
Solution
a) (2i)^{2} = 2^{2} Â· i^{2} = 4(1) = 4
b) (2i)2 (2)2 i2 4i2 4(1) 4
c) i^{6} = i^{2} Â· i^{4} = 1 Â· 1 = 1
For completeness we give the following symbolic definition of multiplication of
complex numbers. However, it is simpler to find products as we did in Examples 2
and 3 than to use this definition.
Multiplication of Complex Numbers
The complex numbers a + bi and c + di are multiplied as follows:
(a + bi)(c + di) = (ac  bd) + (ad + bc)i
