228 Chapter 8Complex numbers
and the roots of the quadratic are the complex conjugate pair
0 Exercises 8–11
Division
The division can be performed by the rules of ordinary long division. The simpler
method is to make use of property (8.12) of conjugate pairs to transform the
denominator into a real number. Thus, multiplying top and bottom byz*
21 = 1 x
21 − 1 iy
2,
the complex conjugate of the denominator, we have
(8.13)
The division is defined only ifz
21 ≠ 10 ; that is,x
21 ≠ 10 andy
21 ≠ 10.
EXAMPLES 8.4Division
(i)
(ii)
0 Exercises 12–15
8.3 Graphical representation
The complex numberz 1 = 1 x 1 + 1 iyis represented graphically by a point in a plane,
with coordinates(x, 1 y), as in Figure 8.1.
2The plane is called the complex plane. Real
1
1
11
11
02
11
−
=
++
−+
=
=
i
i
ii
ii
i
i
()()
()()
23
34
2334
3434
18
34
22=
+−
+−
=
i
i
ii
ii
()() i
()()
==+
18
25 25
i
=
−
xx yy
xy
i
yx xy
xy
12 12222212 122222
z
z
zz
zz
xiyxiy
xiyx
121222112 2222==
+−
()( )
()(−−
=
+−
iy
xiyxiy
xy
2112 22222)
()( )
zz
z
z
xiy
xiy
12121122÷= =
()
()
zi z i=+
=−
1
2
37
1
2
, * 37
2John Wallis (1616–1703) first suggested that pure imaginary numbers might be represented on a line
perpendicular to the axis of real numbers. Caspar Wessel (1745–1818), Norwegian surveyor, discussed the graphical
representation of complex numbers in his On the analytical representation of directionof 1797, and Jean Robert
Argand (1768–1822), Swiss bookkeeper, in his Essaiof 1806. Gauss used the same interpretation of complex
numbers in his fourth and final proof of the fundamental theorem of algebra in 1848, by which time he believed
mathematicians were comfortable enough with complex numbers to accept it. The complex plane is also called the
Gaussian plane.