Multiplication in rectangular form, often referred to as complex multiplication, involves multiplying two complex numbers that are expressed in the rectangular or Cartesian form. In the rectangular form, a complex number is represented as:
=
+
z=a+bj
Where:
a is the real part of the complex number.
b is the imaginary part of the complex number.
j represents the imaginary unit, where
2
=
β
1
j
2
=β1.
To multiply two complex numbers
1
z
1
β
and
2
z
2
β
in rectangular form, you can follow these steps:
Given
1
=
1
+
1
z
1
β
=a
1
β
+b
1
β
j and
2
=
2
+
2
z
2
β
=a
2
β
+b
2
β
j, where
1
a
1
β
,
1
b
1
β
,
2
a
2
β
, and
2
b
2
β
are real numbers.
Multiply the real parts of the two complex numbers:
1
Γ
2
a
1
β
Γa
2
β
Multiply the imaginary parts of the two complex numbers:
1
Γ
2
b
1
β
Γb
2
β
Add the cross-products of the real and imaginary parts:
1
Γ
2
+
2
Γ
1
a
1
β
Γb
2
β
+a
2
β
Γb
1
β
The result of the multiplication is the sum of the two products from steps 2 and 3, along with the sum from step 4. This result will be in rectangular form, represented as:
result
=
(
1
Γ
2
β
1
Γ
2
)
+
(
1
Γ
2
+
2
Γ
1
)
z
result
β
=(a
1
β
Γa
2
β
βb
1
β
Γb
2
β
)+(a
1
β
Γb
2
β
+a
2
β
Γb
1
β
)j
Simplify the expression if needed.
Here's a worked example:
Given two complex numbers
1
=
3
+
2
z
1
β
=3+2j and
2
=
1
β
4
z
2
β
=1β4j, let's multiply them:
1
Γ
2
=
(
3
+
2
)
Γ
(
1
β
4
)
z
1
β
Γz
2
β
=(3+2j)Γ(1β4j)
Using the steps outlined above:
1
=
3
a
1
β
=3,
1
=
2
b
1
β
=2,
2
=
1
a
2
β
=1,
2
=
β
4
b
2
β
=β4.
1
Γ
2
=
3
Γ
1
=
3
a
1
β
Γa
2
β
=3Γ1=3.
1
Γ
2
=
2
Γ
β
4
=
β
8
b
1
β
Γb
2
β
=2Γβ4=β8.
1
Γ
2
+
2
Γ
1
=
3
Γ
β
4
+
1
Γ
2
=
β
12
+
2
=
β
10
a
1
β
Γb
2
β
+a
2
β
Γb
1
β
=3Γβ4+1Γ2=β12+2=β10.
Result:
result
=
(
3
β
8
)
+
(
β
10
)
=
β
5
β
10
z
result
β
=(3β8)+(β10)j=β5β10j.
So,
(
3
+
2
)
Γ
(
1
β
4
)
=
β
5
β
10
(3+2j)Γ(1β4j)=β5β10j.
Remember to be careful with signs during the multiplication and addition steps.