Operating with Complex Numbers
Some equations have no real solutions. For instance, the quadratic equation
x^{2} + 1 = 0 
Equation with no real solution 
has no real solution because there is no real number x that can be squared to produce
1. To overcome this deficiency, mathematicians created an expanded system of
numbers using the imaginary unit i, defined as

Imaginary unit 
where i^{2} = 1. By adding real numbers to real multiples of this imaginary unit, we
obtain the set of complex numbers. Each complex number can be written in the
standard form,
a + bi.
Definition of a Complex Number
For real numbers a and b, the number a + bi
is a complex number. If b ≠ 0, a + bi is called an
imaginary number, and bi
is called a pure imaginary number.
To add (or subtract) two complex numbers, you add (or subtract) the real and
imaginary parts of the numbers separately.
Addition and Subtraction of Complex Numbers
If a + bi and c + di are two complex numbers written in standard form, their
sum and difference are defined as follows.
Sum:
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference: (a + bi)  (c + di) = (a  c) + (b  d)i
The additive identity in the complex number system is zero (the same as in the
real number system). Furthermore, the additive inverse of the complex number
a + bi
is
(a + bi) = a  bi 
Additive inverse 
Thus, you have (a + bi) + (a  bi) = 0 + 0i = 0.
Example 1
Adding and Subtracting Complex Numbers
a. (3  i) + (2 + 3i) 
= 3  i + 2 + 3i 
Remove parentheses. 

= 3 + 2  i + 3i 
Group like terms. 

= (3 + 2) + ( 1 + 3)i 


= 5 + 2i 
Standard form 
b. 2i + (4  2i) 
= 2i  4 2i 
Remove parentheses.


= 4 + 2i  2i 
Group like terms. 

= 4 
Standard form 
