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Pure Core
1. Complex Numbers
1.1 Introduction to Complex Numbers
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Cards (56)
What is the general form of a complex number?
a
+
a +
a
+
b
i
bi
bi
The imaginary unit
i
i
i
is defined as
i
2
=
i^{2} =
i
2
=
−
1
- 1
−
1
or
i
=
i =
i
=
−
1
\sqrt{ - 1}
−
1
, which means it is not a real number.
The real part of a complex number
a
+
a +
a
+
b
i
bi
bi
is
a
a
a
.
What is the notation for the imaginary part of
a
+
a +
a
+
b
i
bi
bi
?
\text{\Im}(a + bi)
The real part of a complex number is denoted as
\text{\Re}(a + bi)
, which equals a.
Match the component with its definition:
Real part ↔️
a
a
a
in
a
+
a +
a
+
b
i
bi
bi
Imaginary part ↔️
b
b
b
in
a
+
a +
a
+
b
i
bi
bi
What is the real part of a complex number z = a + bi</latex>?
\text{\Re}(z) =
a
a
a
The imaginary part of a complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is denoted as
\text{\Im}(z)
, which equals b.
The imaginary part of
5
−
2
i
5 - 2i
5
−
2
i
is
−
2
- 2
−
2
.
What is the value of
i
2
i^{2}
i
2
?
−
1
- 1
−
1
Complex numbers are written in the form
a
+
a +
a
+
b
i
bi
bi
, where
a
a
a
and
b
b
b
are real numbers and
i
i
i
is the imaginary unit.
Match the complex number with its real and imaginary parts:
3 + 5i ↔️
\text{\Re}(z) =
3
3
3
,
\text{\Im}(z) =
5
5
5
5 - 2i ↔️
\text{\Re}(z) =
5
5
5
,
\text{\Im}(z) =
−
2
- 2
−
2
The notation for the real part of a complex number
a
+
a +
a
+
b
i
bi
bi
is
\text{\Re}(a + bi)
.
The real part of a complex number
a
+
a +
a
+
b
i
bi
bi
is
\text{\Re}(a + bi)
, which equals a.
What is the notation for the imaginary part of a complex number
a
+
a +
a
+
b
i
bi
bi
?
\text{\Im}(a + bi)
What is the general form of a complex number?
a
+
a +
a
+
b
i
bi
bi
In a complex number
a
+
a +
a
+
b
i
bi
bi
,
a
a
a
and
b
b
b
are real numbers.
The imaginary unit
i
i
i
is defined such that i^{2} = - 1</latex>.
What does the notation
\text{\Re}(a + bi)
represent?
Real part of
a
+
a +
a
+
b
i
bi
bi
The notation
\text{\Im}(a + bi)
represents the imaginary part of
a
+
a +
a
+
b
i
bi
bi
.
For the complex number
3
+
3 +
3
+
5
i
5i
5
i
, the real part is 3 and the imaginary part is 5.
What is the real part of the complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
?
\text{\Re}(z) =
a
a
a
If
z
=
z =
z
=
5
−
2
i
5 - 2i
5
−
2
i
, then
\text{\Im}(z) =
-2.
What is the value of
i
2
i^{2}
i
2
?
i
2
=
i^{2} =
i
2
=
−
1
- 1
−
1
Steps to solve
−
16
\sqrt{ - 16}
−
16
in terms of
i
i
i
1️⃣
−
16
=
\sqrt{ - 16} =
−
16
=
16
⋅
−
1
\sqrt{16} \cdot \sqrt{ - 1}
16
⋅
−
1
2️⃣
16
=
\sqrt{16} =
16
=
4
4
4
3️⃣
−
1
=
\sqrt{ - 1} =
−
1
=
i
i
i
4️⃣
−
16
=
\sqrt{ - 16} =
−
16
=
4
i
4i
4
i
A complex number in Cartesian form is written as
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
, where
a
a
a
is the real part.
What notation is used to denote the real part of a complex number in Cartesian form?
\text{\Re}(z)
For the complex number
z
=
z =
z
=
3
+
3 +
3
+
5
i
5i
5
i
, the imaginary part is
\text{\Im}(z) =
5
5
5
.
What is the notation for the imaginary part of a complex number
a
+
a +
a
+
b
i
bi
bi
?
\text{\Im}(a + bi)
For the complex number
5
−
3
i
5 - 3i
5
−
3
i
, the real part is 5.
The real part of a complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is denoted as
\text{\Re}(z)
.
What is the notation for the real part of a complex number
z
z
z
?
\text{\Re}(z)
The imaginary part of a complex number
z
z
z
is the real number b
The imaginary unit
i
i
i
is defined as
−
1
\sqrt{ - 1}
−
1
.
Complex numbers can be divided using a method similar to dividing
real numbers
.
Match the arithmetic operation with its general formula:
Addition ↔️
(
a
+
b
i
)
+
(a + bi) +
(
a
+
bi
)
+
(
c
+
d
i
)
=
(c + di) =
(
c
+
d
i
)
=
(
a
+
c
)
+
(a + c) +
(
a
+
c
)
+
(
b
+
d
)
i
(b + d)i
(
b
+
d
)
i
Subtraction ↔️
(
a
+
b
i
)
−
(
c
+
d
i
)
=
(a + bi) - (c + di) =
(
a
+
bi
)
−
(
c
+
d
i
)
=
(
a
−
c
)
+
(a - c) +
(
a
−
c
)
+
(
b
−
d
)
i
(b - d)i
(
b
−
d
)
i
Multiplication ↔️
(
a
+
b
i
)
(
c
+
d
i
)
=
(a + bi)(c + di) =
(
a
+
bi
)
(
c
+
d
i
)
=
(
a
c
−
b
d
)
+
(ac - bd) +
(
a
c
−
b
d
)
+
(
a
d
+
b
c
)
i
(ad + bc)i
(
a
d
+
b
c
)
i
Division ↔️
a
+
b
i
c
+
d
i
=
\frac{a + bi}{c + di} =
c
+
d
i
a
+
bi
=
\frac{(ac + bd) +
(bc - ad)i}{c^{2} +
d^{2}}
The complex conjugate of
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is \bar{z}
The modulus of
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is calculated as
\sqrt{a^{2} +
b^{2}}
.
What is the notation for the imaginary part of a complex number
z
z
z
?
\text{\Im}(z)
The imaginary unit
i
i
i
is defined as
i
=
i =
i
=
−
1
\sqrt{ - 1}
−
1
, so
i
2
=
i^{2} =
i
2
=
−
1
\sqrt{ - 1}
−
1
-1
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