Math 96–Complex Numbers–page #1
A.
Combine Like Terms
a.
2x + 8 + 3x + 7
5x + 15
b.
3 + 9i + 6 + 7i
9 + 16i
c.
4 ! 17i + 5 + 3i
9 ! 14i
d.
Examples b and c are complex numbers
in the form a + bi, the standard form
of a complex number
e.
(7 + 18i) + (!11 + 2i)
7 + 18i ! 11 + 2i
!4 + 20i
f.
(!20 ! 6i) ! (13 ! 4i)
!20 ! 6i ! 13 + 4i
!33 ! 2i
g.
Example f is in standard form a + bi, where a = !33 and b = !2
h.
2(3 + 5i) ! 7(4 + 6i)
6 + 10i ! 28 ! 42i
!22 ! 32i
(standard form a + bi, where a = !22 and b = !32)
i.
14 is a real number. In complex number form, it is 14 + 0i, where a = 14 and b = 0
Homework. Simplify appropriately. Write results in a + bi form.
1.
7 + 6i + 9 + 3i
2.
!8 + 5i ! 12 + 2i
3.
!14 ! 8i + 6 ! 15i
4.
19 ! 20i ! 28 ! 5i
5.
(17 + 3i) + (9 + 7i)
6.
(25 + 9i) + (8 ! 10i)
7.
(14 ! 18i) ! (20 + 3i)
8.
(!30 + 12i) ! (!6 + 5i)
9.
2(5 + 8i) + 6(3 ! 5i)
10.
!7(!1 + 4i) ! 8(2 + 3i)
11.
In 14 + 3i, what is a? What is b?
12.
In 18 ! 31i, what is a? What is b?
13.
Write the real number 28 in complex number form. What is a? What is b?
14.
Write the real number !39 in complex number form. What is a? What is b?
B.
Definition of the Imaginary Unit.
i.
Definition:
This is called the imaginary unit.
You can never leave the square root of a negative number because the square root of negative one
is defined to be i. Observe the examples.
j.
k.
l.
m.
Math 96–Complex Numbers–page #2
n.
Some books write
o.
. Be careful if you do this because
is NOT
4i is called an imaginary number (it has its uses; its name is imaginary maybe because someone
had to imagine
). In complex number form, 4i is written as 0 + 4i (a = 0, b = 4).
p.
is called an imaginary number. In complex number form,
can also be written 0 +
). Either way, a = 0 and b =
is written as 0 +
.
q.
In the standard form a + bi, a is the real number part and bi is the imaginary number part. In 7 +
5i, the real number is 7 and the imaginary number is 5i. The whole expression 7 + 5i is called a
complex number.
r.
13 +
+5!
13 +bei comes
+5!i
13 + i
18 ! 2i
+ 5 ! 3i
Homework. Simplify appropriately.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
Write !12i in complex number form.
What is a? What is b?
29.
In 17 + 3i, what is the real number part? What is the imaginary number part?
30.
Try simplifying the following:
a.
3+
c.
!14 +
e.
!8 +
+ 14 +
28.
Write
in complex number form.
What is a? What is b?
b.
8!
d.
!7 +
f.
(16 + 3
compared to !7
) + (9 ! 7
)
Math 96–Complex Numbers–page #3
C.
Definition of i 2.
s.
Definition:
i 2 = !1
You can never leave i squared in an expression because i squared is defined to be negative one.
t.
5i2 = 5(!1) = !5
Notice, !5 is a real number.
u.
!8i2 = !8(!1) = 8
v.
7i + 11i2
7i + 11(!1)
7i ! 11
!11 + 7i
w.
17 ! 6i ! 20i2
17 ! 6i !20(!1)
17 ! 6i + 20
37 ! 6i
y.
(8 ! 3i)(8 + 3i)
64 + 24i ! 24i ! 9i2
64 + 24i ! 24i ! 9(!1)
64 + 24i ! 24i + 9
73
where a = !11, b = 7
where !11 is the real number part
and 7i is the imaginary number part
x.
(2 + 3i)(7 + 8i)
14 + 16i + 21i + 24i2
14 + 16i + 21i + 24(!1)
14 + 16i + 21i ! 24
!10 + 37i
Notice on y, you multiplied conjugates and your
answer was a rational number! Also notice you
could have multiplied the First and Last terms only
because the Outer and Inner terms will cancel out.
Homework. Simplify appropriately.
31.
a.
7i2
b.
!12i2
c.
13i + 6i2
d.
!18 + 10i ! 7i2
32.
(2 + 3i)(4 + 8i)
33.
(7 ! 4i)(3 + 2i)
34.
(!8 + 5i)(6 ! 4i)
35.
(9 ! 2i)(5 ! 3i)
36.
(7 + 5i)(7 ! 5i)
37.
(9 ! 6i)(9 + 6i)
Math 96–Complex Numbers–page #4
D.
Rationalizing Denominators. In the complex number system, you can never leave i in the
denominator. You need to rationalize the denominator by multiplying the fraction by a form of one.
When the denominator is a binomial, multiply both numerator and denominator by the conjugate of the
denominator. Observe.
z.
becomes
Remember to reduce your result if possible. All three “numbers” reduce simultaneously (or you
can factor to reduce). The result
. Some books wri
.
aa.
=
. If you write this in a + bi form, you may need to reduce furth
Homework. Rationalize the denominators.
38.
39.
40.
41.
42.
Try rationalizing the denominators on the following:
a.
b.
Math 96–Complex Numbers–page #5
Answer Key. On some problems, some steps will be shown.
1.
16 + 9i
2.
!20 + 7i
3.
!8 ! 23i
4.
!9 ! 25i
5.
26 +10i
6.
33 ! i
7.
!6 ! 21i
8.
!24 + 7i
9.
28 ! 14i
10.
!9 ! 52i
11.
a = 14, b = 3
12.
a= 18, b=!31
13.
28 + 0i
a = 28, b = 0
14.
!39 + 0i
a = !39, b = 0
15.
16.
20.
17.
18.
19.
21.
22.
23.
6i
24.
7i
27.
0 ! 12i
28.
0+
25.
10i
26.
12i
a = 0, b = !12
or 0 +
a= 0, b=
29.
real # part = 17
30.
imaginary # part = 3i
31.
a.
3+
b.
c.
8!
!14 + 5i
!7 +
a.
!7
d.
b.
12
e.
c.
!6 + 13i
d.
!18 + 10i + 7
6+
f.
!11 + 10i
25 !
32.
8 + 16i + 12i + 24i 2
8 + 28i ! 24
!16 + 28i
33.
21 + 14i ! 12i ! 8i 2
21 + 2i + 8
29 + 2i
34.
!48 + 32i + 30i ! 20i 2
!48 + 62i + 20
!28 + 62i
35.
45 ! 27i ! 10i + 6i 2
45 ! 37i ! 6
39 ! 37i
36.
49 ! 25 i 2
49 + 25
74
37.
81 ! 36i 2
81 + 36
117
38.
39.
;
Math 96–Complex Numbers–page #6
40.
41.
42a.
42b.
then reduce the fraction to get:
most books would re-arrange to get this: