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Pure Core
1. Complex Numbers
1.2 Algebra of Complex Numbers
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Complex numbers are of the form
a
+ bi
What is the real part of a complex number
a
+
a +
a
+
b
i
bi
bi
?
a
a
a
The imaginary part of a complex number is denoted
I
m
(
z
)
=
Im(z) =
I
m
(
z
)
=
b
b
b
.
The imaginary unit
i
i
i
has the property i^2 = -1
What are the real and imaginary parts of
z
=
z =
z
=
3
+
3 +
3
+
4
i
4i
4
i
?
Re(z) = 3, Im(z) = 4</latex>
What are the real and imaginary parts of
z
=
z =
z
=
5
−
2
i
5 - 2i
5
−
2
i
?
R
e
(
z
)
=
Re(z) =
R
e
(
z
)
=
5
,
I
m
(
z
)
=
5, Im(z) =
5
,
I
m
(
z
)
=
−
2
- 2
−
2
In the complex number z = -3 + i</latex>, <latex>Re(z) = -3
What are the real and imaginary parts of
z
=
z =
z
=
4
+
4 +
4
+
7
i
7i
7
i
?
R
e
(
z
)
=
Re(z) =
R
e
(
z
)
=
4
,
I
m
(
z
)
=
4, Im(z) =
4
,
I
m
(
z
)
=
7
7
7
In the complex number
4
+
4 +
4
+
7
i
7i
7
i
, the imaginary part is
7
7
7
.
What does the real part of a complex number represent?
The real number without
i
i
i
The imaginary unit
i
i
i
is defined as \sqrt{ - 1
To add two complex numbers, add their real parts and
imaginary parts
separately.
If
z
1
=
z_{1} =
z
1
=
a
+
a +
a
+
b
i
bi
bi
and
z
2
=
z_{2} =
z
2
=
c
+
c +
c
+
d
i
di
d
i
, what is z_{1} + z_{2}</latex>?
(
a
+
c
)
+
(a + c) +
(
a
+
c
)
+
(
b
+
d
)
i
(b + d)i
(
b
+
d
)
i
Add
z
1
=
z_{1} =
z
1
=
3
+
3 +
3
+
2
i
2i
2
i
and
z
2
=
z_{2} =
z
2
=
1
−
4
i
1 - 4i
1
−
4
i
.
4
−
2
i
4 - 2i
4
−
2
i
The imaginary part of the sum of two
complex numbers
is the sum of their individual imaginary parts.
If
z
1
=
z_{1} =
z
1
=
a
+
a +
a
+
b
i
bi
bi
and
z
2
=
z_{2} =
z
2
=
c
+
c +
c
+
d
i
di
d
i
, what is
z
1
+
z_{1} +
z
1
+
z
2
z_{2}
z
2
?
(
a
+
c
)
+
(a + c) +
(
a
+
c
)
+
(
b
+
d
)
i
(b + d)i
(
b
+
d
)
i
To add two complex numbers, add their real parts together and their imaginary parts together
separately
To subtract complex numbers, you subtract the real parts and the
imaginary parts
separately.
Complex numbers are written in the form
a + bi
The imaginary unit i</latex> has the property
i
2
=
i^{2} =
i
2
=
−
1
- 1
−
1
.
What is the notation for the real part of a complex number
z
z
z
?
R
e
(
z
)
=
Re(z) =
R
e
(
z
)
=
a
a
a
The imaginary part of a complex number is the coefficient of
i
The real part of
z
=
z =
z
=
−
3
+
- 3 +
−
3
+
i
i
i
is
−
3
- 3
−
3
.
What is the formula for adding two complex numbers
z
1
=
z_{1} =
z
1
=
a
+
a +
a
+
b
i
bi
bi
and
z
2
=
z_{2} =
z
2
=
c
+
c +
c
+
d
i
di
d
i
?
z
1
+
z_{1} +
z
1
+
z
2
=
z_{2} =
z
2
=
(
a
+
c
)
+
(a + c) +
(
a
+
c
)
+
(
b
+
d
)
i
(b + d)i
(
b
+
d
)
i
Steps for subtracting two complex numbers
z
1
=
z_{1} =
z
1
=
a
+
a +
a
+
b
i
bi
bi
and
z
2
=
z_{2} =
z
2
=
c
+
c +
c
+
d
i
di
d
i
1️⃣ Subtract the real parts:
R
e
(
z
1
)
−
R
e
(
z
2
)
=
Re(z_{1}) - Re(z_{2}) =
R
e
(
z
1
)
−
R
e
(
z
2
)
=
a
−
c
a - c
a
−
c
2️⃣ Subtract the imaginary parts:
I
m
(
z
1
)
−
I
m
(
z
2
)
=
Im(z_{1}) - Im(z_{2}) =
I
m
(
z
1
)
−
I
m
(
z
2
)
=
b
−
d
b - d
b
−
d
3️⃣ Combine the results to form the new complex number:
(
a
−
c
)
+
(a - c) +
(
a
−
c
)
+
(
b
−
d
)
i
(b - d)i
(
b
−
d
)
i
What method is used to multiply two complex numbers and simplify the result?
FOIL
When multiplying complex numbers, you must use the property
i
2
=
i^{2} =
i
2
=
−
1
- 1
−
1
to simplify the result.
What is the general formula for multiplying two complex numbers
z
1
=
z_{1} =
z
1
=
a
+
a +
a
+
b
i
bi
bi
and
z
2
=
z_{2} =
z
2
=
c
+
c +
c
+
d
i
di
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
To multiply two complex numbers
z
1
=
z_{1} =
z
1
=
a
+
a +
a
+
b
i
bi
bi
and z_{2} = c + di</latex>, use the FOIL method to distribute each term and simplify using
i
2
=
i^{2} =
i
2
=
−
1
- 1
−
1
.FOIL
Multiplying
z
1
=
z_{1} =
z
1
=
3
+
3 +
3
+
2
i
2i
2
i
and
z
2
=
z_{2} =
z
2
=
1
−
4
i
1 - 4i
1
−
4
i
results in
11
−
10
i
11 - 10i
11
−
10
i
.
What is the complex conjugate of a complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
?
z
ˉ
=
\bar{z} =
z
ˉ
=
a
−
b
i
a - bi
a
−
bi
The complex conjugate of a complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is obtained by changing the sign of the imaginary part.
Match each complex number with its complex conjugate:
3
+
3 +
3
+
4
i
4i
4
i
↔️
3
−
4
i
3 - 4i
3
−
4
i
−
2
−
5
i
- 2 - 5i
−
2
−
5
i
↔️
−
2
+
- 2 +
−
2
+
5
i
5i
5
i
6
i
6i
6
i
↔️
−
6
i
- 6i
−
6
i
7
7
7
↔️
7
7
7
In a complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
, the imaginary part is denoted as
I
m
(
z
)
=
Im(z) =
I
m
(
z
)
=
b
b
b
.
In the complex number
z
=
z =
z
=
3
+
3 +
3
+
4
i
4i
4
i
, the real part is 3.
What are the real and imaginary parts of the complex number
z
=
z =
z
=
5
−
2
i
5 - 2i
5
−
2
i
?
R
e
(
z
)
=
Re(z) =
R
e
(
z
)
=
5
,
I
m
(
z
)
=
5, Im(z) =
5
,
I
m
(
z
)
=
−
2
- 2
−
2
To add two complex numbers, their real parts and
imaginary parts
are added separately.
What is the sum of
z
1
=
z_{1} =
z
1
=
3
+
3 +
3
+
2
i
2i
2
i
and
z
2
=
z_{2} =
z
2
=
1
−
4
i
1 - 4i
1
−
4
i
?
4
−
2
i
4 - 2i
4
−
2
i
To subtract complex numbers, subtract their real parts and their
imaginary
parts separately.
What is the result of subtracting
z
2
=
z_{2} =
z
2
=
1
−
4
i
1 - 4i
1
−
4
i
from
z
1
=
z_{1} =
z
1
=
3
+
3 +
3
+
2
i
2i
2
i
?
2
+
2 +
2
+
6
i
6i
6
i
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