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Basics of Complex Numbers
Related Topics
Wize University Linear Algebra Textbook > Complex Numbers
Basics of Complex Numbers
4 Activities
Part 1
Part 2
Part 3
Simplify:
1
2
i
[
(
−
1
)
(
−
i
)
+
(
1
−
i
)
]
\dfrac{1}{2i}[(-1)(-i) + (1-i)]
2
i
1
[(
−
1
)
(
−
i
)
+
(
1
−
i
)]
Answer
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More Basics of Complex Numbers Questions:
Basics of Complex Numbers
Basics of Complex Numbers
Basics of Complex Numbers
Basics of Complex Numbers
Basics of Complex Numbers
Basics of Complex Numbers
Basics of Complex Numbers
Basics of Complex Numbers
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
zw
z
w
.
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
)