Download 1.4 * Complex Numbers

1.4 – Complex Numbers
• Real numbers have a small issue; no symmetry
of their roots
–
3  3
• To remedy this, we introduce an “imaginary”
unit, so it does work
• The number i is defined such that
i 2  1
Simplifying Negative Roots
• For a positive number a,  a  i a
• Follow all other rules to simplify the remaining
radical
• Example. Simplify:
•  20 x
•  36x 4
Complex Numbers
• A complex number, a+bi, has the following:
–
–
–
–
Real part, a
Imaginary part, bi
Only equal if both parts are equal (real/imaginary)
5 + 10i
• With imaginary numbers, only combine the like
terms (real with real, imaginary with imaginary)
• Multiplication, follow same rules as polynomials
(FOIL, like terms, etc.)
• Example. Simplify: 4  4i    3  5i 
• Example. Simplify: 5  4i 6  3i 
Quotients
• Similar to radical expressions, denominators of
fractions cannot contain imaginary numbers
or a complex number
• Use the complex conjugate = for given
complex number a+bi, the complex conjugate
is a-bi
2
• Example. Simplify the quotient:
4  3i
• Note the denominator contains the complex
number, 4-3i
• What is the complex conjugate? 4+3i
6  3i
• Example. Simplify the following:
4  4i
•
Roots and Complex Numbers
• When dealing with negative roots, we can
simplify using the rules introduced
• Now, we can simplify radicals in a second way
• Example. Simplify: 1 
• How can we write
6
?
 6 

2
Powers of i
• The imaginary number, i, has a particular
pattern
• i2= -1
• i3 = i2 x i = -1 x i = -i
• i4 = i2 x i2 = -1 x -1 = 1
• i=i
• Pull out powers that are multiples of 4; those
will become 1
• Example. Simplify:
• i15 = i12 x i3 =
• 4i25
• Assignment
• Page 61
• #1-41 odd