Post on 13-Jan-2016
COMPLEX NUMBERS
ARITHMETIC OPERATIONS WITH COMPLEX NUMBERS
Which complex representation is the best to use?
It depends on the operation we want to perform.
ADDITION
When performing addition/subtraction on two complex numbers, the rectangular form is the easiest to use. Addition of two complex numbers, C1 = R1 + jI1 and C2 = R2 + jI2, is merely the sum of the real parts plus j times the sum of the imaginary parts.
C C R jI R jI R R j I I1 2 1 1 2 2 1 2 1 2 ( ) ( )
R1 + R2
I1 - I2
C1
C2
C1+ C2
MULTIPLICATION OF COMPLEX NUMBERS
We can use the rectangular form to multiply two complex numbers
12212121221121 IRIRjIIRRjIRjIRCC
If we represent the two complex numbers in exponential form, the product takes a simpler form.
C C M e M e M M ej j j1 2 1 2 1 2
1 2 1 2 ( )
CONJUGATION OF A COMPLEX NUMBERS
The complex conjugate of a complex number is obtained by merely changing the sign of the number’s imaginary part. If
C R jI Me j
then, C* is expressed as
C R jI Me j*
SUBTRACTION
Subtraction of two complex numbers, C1 = R1 + jI1 and C2 = R2 + jI2, is merely the sum of the real parts plus j times the sum of the imaginary parts.
2121221121 IIjRRjIRjIRCC
DIVISION OF COMPLEX NUMBERS
The division of two complex numbers is also convenient using the exponential and magnitude and angle forms, such as
C
C
M e
M e
M
M
j
j1
2
1
2
1
21 2
1
2
C
C
M e
M e
M
Me
j
jj1
2
1
2
1
2
1
2
1 2
( )
or
DIVISION (continued)
Although not nearly so handy, we can perform complex division in rectangular notation by multiplying the numerator and denominator by the complex conjugate of the denominator
22
21122121
22
12
22
11
22
11
2
1
22IR
IRIRjIIRR
jIR
jIR
jIR
jIR
jIR
jIR
C
C
INVERSE OF A COMPLEX NUMBER
A special form of division is the inverse, or reciprocal, of a complex number. If C = Me j, its inverse is given by
1 1 1
C Me Me
jj
In rectangular form, the inverse of C = R + jI is given by
22
111
IR
jIR
jIR
jIR
jIRjIRC