Page 13 - Calculus with Complex Numbers

therefore we have

zw = (.x + iy) (u + i t?) = (xu - y??l + i (.xt? + yu) ,
i z-bb = (.x - iy) (u - i t?) = (xu - y??l - i (.xt? + yu) .

I zw I = Iz IIw I. We delay the proof of this property until Section 1.9.

I z + w I1qL I zl + Iw I. We delay the proof of this property until Section 1. 1 1.

I .4 T h e A rgand d i agram
W e obtain a geometric model for the complex numbers by representing the complex
number z = .x + iy by the point (x y) in the real plane with coordinates .x and y.
Observe that the horizontal ar-axis represents complex numbers .x + iy with
y = 0, that is the real numbers. W e therefore call the horizontal axis the real
axis. The vertical y-axis represents complex numbers .x + iy with .x = 0 that is
numbers of the form iy where y is real. W e call these numbersArfrc imaginary and
we call the vertical axis the imaginary axis. The origin O represents the number
zero which is of cotlrse real (Figure 1. 1).

I .5 G eom etric interpretation of addition

z + w = (.r + u) + i (y + ??)

and therefore appears on the Argand diagram as the vector sum of z and w.
The complex number z + w is represented geometrically as the fotlrth vertex of
the parallelogram formed by 0 z, w (see Figure 1.2). For example 3 + li is the
vector sum of 3 and li (see Figure 1. 1).