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Lesson 2.4
Complex Numbers
Imaginary Numbers
 What
is the square root of 9?
9  3 because 3 3  9
 What
is the square root of -9?
9  ?
Imaginary Numbers
 The
constant, i, is defined as the
square root of negative 1:
i  1
Imaginary Numbers
 The
square root of -9 is an imaginary
number...
9  9  1  3 i  3i
Imaginary Numbers
 Simplify
these radicals:
36x
2
20y
3
2 4 7
5a b c
 6ix
 2iy 5y
2 3
 iab c
5c
Multiples of i

Consider multiplying two imaginary numbers:
3i  5i

2
So... i  1
Multiples of i
 Powers
of i:
i
i  1
2
i  i  i  1 i  i
3
2
i  i i  1 1  1
4
2
2
Powers of i - Practice
28
i
1
75
i
-i
113
i
i
86
i
-1
1089
i
i
Solutions Involving i
Solve:
4 x  36  0
2
Complex Numbers
 Have
a real and imaginary part .
 Write complex numbers as a + bi
 Examples: 3 - 7i, -2 + 8i, -4i, 5 + 2i
Add & Subtract
 Like
Terms
 Example:
(3 + 4i) + (-5 - 2i) = -2 + 2i
Practice
Add these Complex Numbers:
 (4
+ 7i) - (2 - 3i)
 (3 - i) + (7i)
 (-3 + 2i) - (-3 + i)
= 2 +10i
= 3 + 6i
=i
Multiplying
 FOIL and
replace i2 with -1:
4  2i 1  3i 
Practice
Multiply:
 5i(3
 (7
- 4i)
- 4i)(7 + 4i)
= 20 + 15i
= 65
Division
 Rationalize
any fraction with i in
the denominator.
Monomial
Denominator:
2  8i
3i
Binomial
Denominator:
8i
1  3i
Rationalizing
 Monomial:
bottom by i.
2  8i
3i
multiply the top &
Complex #: Rationalize

Binomial: multiply the numerator and
denominator by the conjugate of the
denominator ...
8i 1  3i

1  3i 1  3i
8i 1  3i
8i  24i


2
1  3i 1  3i 1 3i  3i  9i
24  8i 12  4i


10
5
2
Practice
 Simplify:
5i
4i
2  3i
2 i
5
3  4i
1  5i

4
1  8i

5
3  4i

5