Algebra 2 Notes, Chapter 4
1
4.8 Complex Numbers
A. Imaginary Numbers
25
Consider:
Has no REAL solution because (?)2= -25
Imaginary Numbers were created to solve equations that did not have real solutions.
The imaginary unit “i” is used to represent:
i  1
and
i 2  1
Ex. Simplify the following
1.
36
2.
5.
90
6.
9.
2i  3i
10.
100
3.
7
3 8
7.
2  6
7i  3i  2i 
11.
 5i 
Numbers like 3i , 97i , and 7i are called PURE IMAGINARY NUMBERS.
2
4.
50
8. 3 10  4 3
12.
 3i 
3
Algebra 2 Notes, Chapter 4
2
B: COMPLEX NUMBERS
A COMPLEX NUMBER has a real part “a” and an imaginary part “bi”.
A complex number looks like:
a  bi
Working with complex numbers:
Treat i like any other variable…
1.
(3  4i)  (7  5i)
2.
(9  8i)  (7  10i)
3.
(7  6i )  (6  10i)
4.
(17  20i )  (12  30i )
5.
(5  3i )(4  6i )
6.
(8  7i )(2  5i )
7.
7i (9  3i )
8.
4i (20  30i )
9.
(3  5i) 2
10.
(4  2i)(4  2i)
11.
(6  5i )(6  5i )
Numbers in the form
a  bi and a  bi
are called complex conjugates.
Algebra 2 Notes, Chapter 4
3
Division is a little trickier—
To simplify a division problem (a fraction with complex numbers), multiply the numerator and denominator
by the COMPLEX CONJUGATE of the denominator.
12.
4
5i
13.
5  3i
2i
14.
7
3  5i
15.
11i
3  2i
16.
2  7i
3  4i
17.
1  5i
5  9i
Algebra 2 Notes, Chapter 4
4
C: Solving Equations
Remember that a quadratic equation with no real solutions has a graph that has no x-intercepts.
This type of quadratic equation does have SOLUTIONS, though… COMPLEX SOLUTIONS.
SOLVE!
1.
x2  9  0
4.
3x 2  x  2  0
2.
5 x 2  20  0
3.
x 2  15  0
Quadratic Formula for
ax 2  bx  c  0
x
5.
x2  4x  5  0
b  b2  4ac
2a
Algebra 2 Notes, Chapter 4
5
Assignments: End of Chapter 4
Monday 11/29 (B)
Period 2
4.8 Complex Numbers
HWK: pg 253
#8-12, 18-31, 48, 53 all
4.8 Complex Numbers
HWK: pg 253
#8-12, 18-31, 48, 53 all
Tuesday 11/30 (C)
Wednesday 12/1 (A)
Thursday 12/2 (B)
4.8 Complex Numbers
(day 2)
HWK: pg 253 #33-43all,
75, 77, 78
Review Day 1 4.1-4.4
HWK
Pg 268-270
#4-10all, 13-19odd, 22,
27-36all
Tuesday 12/7 (B)
Wednesday 12/8 (C)
Thursday 12/9 (A)
Review Day 2 4.5-4.8
HWK Pg 270-272
#37-40all,
55,56,58,62,64,70,7479all, 80,81
Practice Test Chapter 4
Test Chapter 4
Period 4
4.8 Complex Numbers
HWK: pg 253
#8-12, 18-31, 48, 53all
4.8 Complex Numbers
(day 2)
HWK: pg 253 #33-43all,
75, 77, 78
4.8 Complex Numbers
(day 2)
HWK: pg 253 #33-43all,
75, 77, 78
Review Day 1 4.1-4.4
HWK Pg 268-270
#4-10all, 13-19odd, 22,
27-36all
Friday 12/3 (C)
Monday 12/6 (A)
Period 3
Review Day 1 4.1-4.4
HWK
Pg 268-270
#4-10all, 13-19odd, 22,
27-36all
Review Day 2 4.5-4.8
HWK Pg 270-272
#37-40all,
55,56,58,62,64,70,7479all, 80,81
Review Day 2 4.5-4.8
HWK Pg 270-272
#37-40all,
55,56,58,62,64,70,7479all, 80,81
Practice Test Chapter 4
Test Chapter 4
Practice Test Chapter 4
Test Chapter 4