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If z = 3+2i and w = 1- 4i, compute z^-1and zw.
Related Topics
Wize University Linear Algebra Textbook > Complex Numbers
Basics of Complex Numbers
4 Activities
If
z
=
3
+
2
i
z = 3+2i
z
=
3
+
2
i
and
w
=
1
−
4
i
w = 1- 4i
w
=
1
−
4
i
, compute
z
−
1
z^{-1}
z
−
1
and
z
w
zw
z
w
.
z
−
1
=
3
13
−
2
13
i
z^{-1}=\frac{3}{13}-\frac{2}{13}i
z
−
1
=
13
3
−
13
2
i
and
z
w
=
11
−
10
i
zw=11-10i
z
w
=
11
−
10
i
z
−
1
=
3
13
+
2
13
i
z^{-1}=\frac{3}{13}+\frac{2}{13}i
z
−
1
=
13
3
+
13
2
i
and
z
w
=
11
−
10
i
zw=11-10i
z
w
=
11
−
10
i
z
−
1
=
3
13
−
2
13
i
z^{-1}=\frac{3}{13}-\frac{2}{13}i
z
−
1
=
13
3
−
13
2
i
and
z
w
=
11
+
10
i
zw=11+10i
z
w
=
11
+
10
i
z
−
1
=
3
13
+
2
13
i
z^{-1}=\frac{3}{13}+\frac{2}{13}i
z
−
1
=
13
3
+
13
2
i
and
z
w
=
11
+
10
i
zw=11+10i
z
w
=
11
+
10
i
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Basics of Complex Numbers
Practice: Operations with Complex Numbers
Perform the following operations.
Given the following complex number
z
=
3
+
4
i
1
−
i
z = \frac{3 + 4i}{1 - i}
z
=
1
−
i
3
+
4
i
a) Write
z
z
z
in Cartesian coordinates
z
=
a
+
b
i
z = a + bi
z
=
a
+
bi
.
Basics of Complex Numbers
Practice: Operations with Complex Numbers
Perform the following operations.
If
z
=
3
+
2
i
z = 3+2i
z
=
3
+
2
i
and
w
=
1
−
4
i
w = 1- 4i
w
=
1
−
4
i
, compute
z
−
1
z^{-1}
z
−
1
and
z
⋅
w
z\cdot w
z
⋅
w
Simplify the expression
2
1
+
i
+
11
+
3
i
3
−
i
\frac{2}{1+i}+\frac{11+3i}{3-i}
1
+
i
2
+
3
−
i
11
+
3
i
Practice: Operations With Complex Numbers
Perform the following operations:
(a)
(
−
1
)
(
−
i
)
+
(
−
i
)
(-1)(-i) + (-i)
(
−
1
)
(
−
i
)
+
(
−
i
)
(b)
i
(
a
+
b
i
)
‾
\overline{i(a+bi)}
i
(
a
+
bi
)
Complex Numbers
A complex number
z
∈
C
z \in \mathbb{C}
z
∈
C
is a number of the form
z
=
a
+
b
i
z = a+bi
z
=
a
+
bi
where a and b are real numbers and
i
=
p
−
1
i = p−1
i
=
p
−
1
. The number a is called the real part of
z
z
z
and the number b is called the imaginary part of
z
.
z.
z
.
Operations on Complex Numbers Consider two complex numbers
z
1
=
a
1
+
i
b
1
z1 = a_1 + ib_1
z
1
=
a
1
+
i
b
1
and
z
2
=
a
2
+
i
b
2
z_2 = a_2 + ib_2
z
2
=
a
2
+
i
b
2
. Then
1.
[Addition and Substraction]
=
z
1
±
z
2
=
(
a
1
±
a
2
)
+
i
(
b
1
±
b
2
)
=z_1 ± z_2 = (a_1 ± a_2) + i(b_1 ± b_2)
=
z
1
±
z
2
=
(
a
1
±
a
2
)
+
i
(
b
1
±
b
2
)