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5.3 Geometric representation of C

Complex numbers can be represented as points in the plane, using the cor-

respondence x + iy $ (x; y). The representation is known as the Argand

diagram or complex plane. The real complex numbers lie on the x{axis,

which is then called the real axis, while the imaginary numbers lie on the

y{axis, which is known as the imaginary axis. The complex numbers with

positive imaginary part lie in the upper half plane, while those with negative

imaginary part lie in the lower half plane.

Because of the equation

(x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2);

complex numbers add vectorially, using the parallellogram law. Similarly,

the complex number z1 ¡ z2 can be represented by the vector from (x2; y2)

to (x1; y1), where z1 = x1 + iy1 and z2 = x2 + iy2. (See Figure 5.1.)

96 CHAPTER 5. COMPLEX NUMBERS

¾ -

z1 ¡ z2

z1 + z2

¡µ

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¢¸

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Figure 5.1: Complex addition and subraction.

The geometrical representation of complex numbers can be very useful

when complex number methods are used to investigate properties of triangles

and circles. It is very important in the branch of calculus known as Complex

Function theory, where geometric methods play an important role.

We mention that the line through two distinct points P1 = (x1; y1) and

P2 = (x2; y2) has the form z = (1 ¡ t)z1 + tz2; t 2 R, where z = x + iy is

any point on the line and zi = xi +iyi; i = 1; 2. For the line has parametric

equations

x = (1 ¡ t)x1 + tx2; y = (1 ¡ t)y1 + ty2

and these can be combined into a single equation z = (1 ¡ t)z1 + tz2.

Circles have various equation representations in terms of complex num-

bers, as will be seen later.