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