1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 37

y

x

Fig. 1.6

If we let z 1 =(a, b) and z 2 = (c, d), we have

lz1 - z2I = v(a - c)^2 + (b - d)^2 =length of QP

i.e., the absolute value of the difference z 1 - z 2 represents the distance
between the points corresponding to z1 and z2.
( c) Multiplication. We shall assume nonzero factors z 1 and z 2 since the
case of a zero factor results in a zero product and then the geometrical
interpretation is trivial. Let z 1 = r 1 e ith and z 2 = e i0^2 • Then we have

z1z2 = r1r2ei(Oi+0^2 )

Also, let OP be the position vector of z 1 and OQ the position vector of z 2 •
To construct the position vector OM of the product z 1 z 2 geometrically,

y

M

I

I I

I

I

I

Q

(^0) x
Fig. 1.7