082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

330 Chapter 9 Complex numbers
Whenever we multiplyacomplex number by its conjugate the answer is a real number.
Thusifz=a+bj
zz = (a +bj)(a −bj)
=a 2 +baj−abj−b 2 j 2
=a 2 +b 2
Ifz=a+bjthenzz=a 2 +b 2
9.3.3 Division
To divide two complex numbers it is necessary to make use of the complex conjugate.
We multiply both the numerator and denominator by the conjugate of the denominator
and then simplifythe result.
Example9.9 Ifz 1
= 2 +9jandz 2
= 5 −2jfind z 1
z 2
.
Solution We seek 2+9j . The complex conjugate of the denominator is 5 + 2j, so we multiply
5−2j
bothnumeratoranddenominatorbythisquantity.Theeffectofthisistoleavethevalue
of z 1
unaltered since wehave only multiplied by 1.Therefore,
z 2
z 1
= 2+9j (2 +9j) (5 +2j)
=
z 2
5−2j (5 −2j) (5 +2j)
=
10 +45j +4j +18j2
25+4
=
−8 +49j
29
= − 8
29 + 49
29 j
The multiplication of two conjugates in the denominator allows a usefulsimplification.
Weseethattheeffectofmultiplyingbytheconjugateofthedenominatoristomakethe
denominator of the solution purely real.
Ifz 1
=x 1
+jy 1
andz 2
=x 2
+jy 2
thenthequotient z 1
z 2
isfoundbymultiplyingboth
numerator and denominator by the conjugate of the denominator, thatis
z 1
z 2
= x 1 +jy 1
x 2
+jy 2
= x 1 +jy 1
x 2
+jy 2
× x 2 −jy 2
x 2
−jy 2
= x 1 x 2 +y 1 y 2 +j(x 2 y 1 −x 1 y 2 )
x 2 2 +y2 2