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7/24/2019 Mathematical Operations of a Complex Number
http://slidepdf.com/reader/full/mathematical-operations-of-a-complex-number 1/12
MATHEMATICAL
OPERATIONS of aCOMPLEX NUMBER
7/24/2019 Mathematical Operations of a Complex Number
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Operation of addition, subtraction,
multiplication, and division apply
to complex numbers in the same
manner that they apply to realnumbers.
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Two complex numbers A and B
define as
1
1 1 1 1
j z e z x jy
A
22 2 2 2 2
j z e z x jy B
are equal if and only if
1 2 1 2 1 2 1 2and or and 360 x x y y z z n
,
where n = 0, 1, 2, 3, ….
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,
Example: If A = 3 + j4, B = 3 – j4,
8 60 and 8 780 C D
then ≠ , but = .
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,
The conjugate, A*, of a complexnumber A = x + jy is define to be
= x jyA*
j is replaced by – j to obtain the conjugate. Note
that the magnitude of A* is the same as that of A ,
since
2 2 2 2( ) z x y x y
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,
However, the angle is now
1tan
y
x
Therefore, the conjugate is written in exponential
and polar form as
= j ze z
A*
)* (A* A Also,
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,
Example: If
and B* = 2 – j3.
(∗)∗ = ∠60° = A and (∗)∗ = =
= ∠60° and = , then ∗ = ∠-60°
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,
ADDITION:
The sum of two complex numbers
1 1 2 2= and x jy x jy A B
1 1 2 2
1 2 1 2 =( ) ( )
x jy x jy
x x j y y
A + B
Add the individual real parts, and add the
individual imaginary parts to obtain thecomponents of the resultant complex number.
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,
SUBTRACTION:
The difference of two complex numbers
1 1 x jy A 2 2 x jy Band is
Subtract the individual real parts and subtract the
individual imaginary parts to obtain the components ofthe resultant complex number. Since a negative sign
correspond to a phase or angle change of 180° , the
graphical technique for performing the subtraction (A –
B) can be accomplished by drawing A and B as vectors,
rotating the vector B 180° and then adding it the vector
A.
=
=
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MULTIPLICATION:
The product of two complex numbers
andis
or
11 1 1 1
j
z e z x y
A 22 2 2 2 2 j z e z x y
B
1 2 ( )
1 2 1 2 1 2( )( ) ( ) j j j
z e z e z z e z z
A
1 1 2 2
2
1 2 1 2 2 1 1 2
1 2 1 2 1 2 2 1
( )( )
= ( ) ( )
x jy x jy
x x jx y jx y j y y
x x y y j x y x y
AB
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DIVISION:
The quotient of two complex numbers
andis
11 1 1 1
j
z e z x y
A 22 2 2 2 2 j z e z x y
B
=
=
(−) =
∠( )
=
+
+
−
−
= − + +
()+()
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