 Imaginary Numbers
 The imaginary unit is and is denoted by i, and
 By definition,
i^{2} = 1.
 For any positive real number n,
 Complex Numbers
 Complex numbers have the form
a + bi
where a and b are real numbers.
 Examples:
 Adding and Subtracting Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
 Steps to adding and subtracting complex numbers:
 Change all imaginary numbers to bi form.
 Add (or subtract) the real parts of the complex numbers.
 Add (or subtract) the imaginary parts of the complex numbers.
 Write the answer in the form a + bi.
 Multiplying Complex Numbers
(a + bi)(c + di) = (ac  bd) + (ad + bc)i
 Steps to multiplying complex numbers:
 Change all imaginary numbers to bi form.
 Multiply the complex numbers as you would multiply polynomials.
 Substitute 1 for each i^{2}.
 Combine the real parts and the imaginary parts.
 Write the answer in the form a + bi.
 Complex Conjugates
 The complex conjugate of a complex number is a complex number having the same two terms
with the sign inbetween changed.

Complex Number 
Complex Conjugate 
7 + 3i 
7  3i 
3  2i 
3 + 2i 
2 + 5i 
2  5i 
i 
i 
 Dividing Complex Numbers

 Change all imaginary numbers to bi form.
 Write the division problem as a fraction.
 Rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
 Write the answer in the form a + bi.
Example: Divide 
7 + 3i 7  3i 
7 + 3i 7  3i 
= 
7 + 3i 7  3i 
· 
7 + 3i 7 + 3i 

= 
(7 + 3i)(7 + 3i) (7  3i)(7 + 3i) 

= 
49 + 21i + 21i + 9i^{2} 49 + 21i  21i  9i^{2} 

= 
49 + 42i + 9(1) 49  9(1) 



