Imaginary Solutions to Equations
In the complex number system the evenroot property can be restated so that x^{2}
= k is equivalent to
for any k
≠ 0. So an equation such as x^{2} =
9 that has
no real solutions has two imaginary solutions in the complex numbers.
Example 1
Complex solutions to equations
Find the complex solutions to each equation.
a) x^{2} = 9
b) 3x^{2} + 2 = 0
Solution
a) First apply the evenroot property:
x^{2} 
= 9 

x 
= Â± 
Evenroot property 

= Â± i 


= Â± 3i 

Check these solutions in the original equation:
(3i)^{2} 
= 9i^{2} = 9(1) = 9 
(3i)^{2} 
= 9i^{2} = 9 
The solution set is {Â±3i}.
b) First solve the equation for x^{2}:
Check these solutions in the original equation. The solution set is
.
The basic facts about complex numbers are listed below.
Complex Numbers
1. Definition of i:
, and i^{2}
= 1.
2. A complex number has the form a + bi, where a and b are real numbers.
3. The complex number a + 0i is the real number a.
4. If b is a positive real number, then
.
5. The numbers a + bi and a  bi are called complex conjugates of each
other. Their product is the real number a^{2} + b^{2}.
6. Add, subtract, and multiply complex numbers as if they were algebraic
expressions with i being the variable, and replace i^{2} by 1.
7. Divide complex numbers by multiplying the numerator and denominator
by the conjugate of the denominator.
8. In the complex number system x^{2} = k for any real number k is equivalent
to x = Â±
.
