Division (or rationalization)of two complex numbers:z=a+jb
c+jdmeans converting into the standard formA+jB. We can do this by using the complex
conjugate,c−jd,ofc+jdand the fact that(c−jd)(c+jd)=c^2 +d^2 which is real.
So, if we multiply top and bottom of the above complex numberzbyc−jdwe get (see
Applications, Chapter 2)
a+jb
c+jd=a+jb
c+jd×c−jd
c−jd=(
ac+bd
c^2 +d^2)
+j(
bc−ad
c^2 +d^2)Problem 12.6
Divide 3− 2 jby 4Yj.We have
3 − 2 j
4 +j=3 − 2 j
4 +j×4 −j
4 −j=10 − 11 j
17=10
17−11
17jExercise on 12.2
For the complex numbersz= 3 −jandw= 1 + 2 jevaluate
(i) 2z− 3 w (ii) zw (iii) z^2 w ̄z^2 (iv)
z
wAnswer
(i) 3− 8 j (ii) 5+ 5 j (iii) 10+ 20 j (iv)
1
5−7
5j12.3 Complex variables and the Argand plane
A generalcomplex variableis denoted, inCartesian form,by
z=x+jywithx,yvarying over real values. Such a variable can be represented by points in a plane
called theArgand plane(orcomplex plane) – see Figure 12.1.
xis called thereal axis;y theimaginary axis. In this representation the complex
conjugatez ̄is the mirror image ofzin thex-axis. The distance of the complex numberz
from the origin isr=
√
x^2 +y^2 , denoted|z|and called themodulusofz–itisalways
taken to be the positive square root. The angleθmade byOPwith the positivex-axis
is called theamplitudeorargumentofz.randθare polar coordinates (206
➤
)ofthe
pointP defining the complex numberz.
An alternative representation of a complex number – thepolar form– can be obtained
by using the polar coordinates (Figure 12.1)r,θ.Sincex=rcosθ andy=rsinθ
we have
z=x+jy=r(cosθ+jsinθ)≡rcisθ≡r^ (θ)