Introduction to the complex-number system - Continuous and Discrete Time Signals and Systems - page 799

In other words, when adding two complex numbers the real and imaginary
components are added separately.

B.2.2 Subtraction

The definition of subtraction follows the same lines as that for addition. Sub-
tracting a complex number x 2 from x 1 is defined as follows:
x 1

− x 2 = ( a 1 + j b 1 ) − ( a 2 + j b 2 )

(B.7)

= ( a 1 − a 2 ) + j( b 1 − b 2 ) .

As for addition, the real and imaginary components are subtracted separately.

B.2.3 Multiplication

Multiplication of two complex numbers x 1 and x 2 is defined as follows:

x 1 x 2 = ( a 1 + j b 1 )( a 2 + j b 2 )

+ j 2 b 1 b 2
= ( a 1 a 2 − b 1 b 2 ) + j( b 1 a 2 + a 1 b 2 ) ,

= a 1 a 2 + j b 1 a 2 + j a 1 b 2

(B.8)

where the final expression is obtained by noting that j 2 =− 1.

B.2.4 Complex conjugation

From Eq. (B.8), it is easy to deduce that

− j b 1 ) = ( a 1 ) 2 + ( b 1 ) 2 .

( a 1 + j b 1 )( a 1 (B.9)

In other words, the imaginary component is eliminated. The complex number
x 1 = a 1 − j b 1 is referred to as the complex conjugate of x 1 = a 1 + j b 1 , and
vice versa. Equation (B.9) leads to the definition of the modulus or magnitude
of a complex number, which is discussed next.

B.2.5 Modulus

The modulus (or magnitude) of a complex number x 1 = a 1 +j b 1 is defined as
follows:

x 1 x 1

x 1 = = ( a 1 ) 2 + ( b 1 ) 2 . (B.10)

B.2.6 Division

Dividing two complex numbers is more complicated. To divide x 1 by x 2 ,we
multiply both the numerator and denominator by the complex conjugate of x 2
and expand the numerator and denominator separately using the definition of