Transcript complex numbers

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.
C1  C2  R1  jI 1  R2  jI 2  ( R1  R2 )  j ( I 1  I 2 )
C2
C1
I1 - I 2
C1+ C2
R1 + R2
MULTIPLICATION OF COMPLEX NUMBERS
We can use the rectangular form to multiply two
complex numbers
C1C2  R1  jI1 R2  jI2   R1R2  I1I 2   jR1I 2  R2 I1 
If we represent the two complex numbers in
exponential form, the product takes a simpler form.
C1C2  M1e
j 1
M2e
j 2
 M1 M 2 e
j ( 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.
C1  C2  R1  jI1   R2  jI2   R1  R2   jI1  I 2 
DIVISION OF COMPLEX NUMBERS
The division of two complex numbers is also convenient
using the exponential and magnitude and angle forms,
such as
j
C1 M1e 1
M1 j ( 1  2 )


e
j 2
C2 M 2 e
M2
or
j 1
C1 M1e
M1


 1   2
j 2
C2 M 2 e
M2
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
C1 R1  jI1 

C2 R2  jI 2 

R1  jI1  R2  jI1 

R2  jI 2  R2  jI 2 

R1 R2  I1 I 2   j R2 I1  R1 I 2 

R
2
2
 I 22

INVERSE OF A COMPLEX NUMBER
A special form of division is the inverse, or reciprocal, of a
j
complex number. If C = Me , its inverse is given by
1
1
1  j


e
j
C Me
M
In rectangular form, the inverse of C = R + jI is given by
1
1
1
R  jI
R  jI


 2
C R  jI  R  jI  R  jI  R  I 2

