Complex Numbers 27( c) Show that the equation represents the point -b if a = b, c is real,
and lbl^2 - c = 0, and has no .solutions if a = b, c is real, and
IW -c < o.- Show that the equation of the straight line through the point z 0 is
given in conjugate coordinates by
z=z 0 +A(z-z 0 )
where the constant A (called the clinant of the line) is related to itsangle of inclination B by the equation A = e-^2 i^9 , and to the slope m
by A = (1 - im)/(1 + im).
- Show that the equation of a circle with center at a and radius r is
given in conjugate coordinates by
r2
z=a+ --
z-a
1.7 Geometric Representation of Complex Numbers
Since the system C of the complex numbers, with multiplication restricted
to the product of a complex number by a real number as in (1.2-2), consti-
tutes a linear system of dimension two, it is natural to seek a geometrical
interpretation of these numbers in terms of geometric elements in a plane,
a sphere, or some other two-dimensional surface.
The simplest interpretation in terms of points in a plane, was proposed
independently by C. Wessel (1797), K. F. Gauss (1799), and J. R. Ar-
gand (1806). To obtain this representation, consider a Euclidean plane
and introduce Cartesian rectangular coordinates as usual, by choosing an
arbitrary point 0 as the origin, a pair of orthogonal directed lines OX, OY
as axes, and equal scales on the axes (Fig. 1.1). Then each complex number
yP" - - - - - - - - P(a, b)0 a P' x
Fig. 1.1