Mathematical Operations of a Complex Number

7/24/2019 Mathematical Operations of a Complex Number

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MATHEMATICAL

OPERATIONS of aCOMPLEX NUMBER



Operation of addition, subtraction,

multiplication, and division apply

to complex numbers in the same

manner that they apply to realnumbers.



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, ….



,

Example: If A = 3 + j4, B = 3 – j4,

8 60 and 8 780 C D

then ≠ , but = .



,

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



,

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,



,

Example: If

and B* = 2 – j3.

(∗)∗ = ∠60° = A and (∗)∗ = =

= ∠60° and = , then ∗ = ∠-60°



,

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.



,

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.

=

=



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



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

=

=

(−) =

∠( )

=

+

+

−

−

= − + +

()+()



RECIPROCA L:

=

∠−

=

+=

−

(+)(−)

=

−

()+()

=

SQUARE ROOT:

= ∠( /2)

Mathematical Operations of a Complex Number

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