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

326 Chapter 9 Complex numbers
Returning tothe solution of the quadratic equation 2x 2 +2x +5 = 0,wefind
x = −2 ± √ −36
4
= −2±6j
4
= −1±3j
2
We have found two roots, namelyx = − 1 2 + 3 2 jandx=−1 2 − 3 j.These numbers are
2
calledcomplexnumbersandweseethattheyaremadeupoftwoparts--arealpartand
animaginarypart.Forthefirstcomplexnumbertherealpartis − 1 2 andtheimaginary
part is 3 2 . For the second complex number the real part is −1 and the imaginary part is
2
− 3 2 .Inamoregeneralcaseweusuallyusetheletterztodenoteacomplexnumberwith
realpartaandimaginarypartb,soz =a +bj.Wewritea = Re(z)andb= Im(z),and
denote the set of all complex numbers by C. Note thata,b∈ Rwhereasz ∈ C.
z =a +bj,zisamemberofthe setofcomplex numbers, that isz ∈ C
a =Re(z) b =Im(z)
Complex numbers which have a zero imaginary part are purely real and hence all real
numbers arealsocomplex numbers, thatis R ⊂ C.
Example9.1 Solve the quadraticequation 2s 2 −3s +7 = 0.
Solution Using the formulaforsolvingaquadraticequation wefind
s = −(−3) ± √ (−3) 2 −4(2)(7)
2(2)
= 3 ± √ −47
4
= 3 ± √ 47j
4
= 0.75 ±1.71j
Usingthe factthat j 2 = −1 wecan develop other quantities.
Example9.2 Simplify the expression j 3 .
Solution We have
j 3 =j 2 ×j
=(−1)×j
= −j