Page 11 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We shall postpone until the next section the geometric interpretation of the product of two
complex numbers.
The modulus of a complex number z = x + iy is defined to be the non-negative real number
2
2
x  y , which is, of course, the length of the vector interpretation of z. This modulus is
traditionally denoted |z|, and is sometimes called the length of z.

2
Notes |(x,0)| = x = |x|, and so |  | is an excellent choice of notation for the
modulus.
The conjugate z of a complex number z = x + iy is defined by z = x – iy. Thus, |z| = zz .
2
Geometrically, the conjugate of z is simply the reflection of z in the horizontal axis:

Figure 1.2

Observe that if z = x + iy and w = u + iv, then

(z w) = (x + u) – i(y + v)

= (x – iy) + (u – iv)

= z w

In other words, the conjugate of the sum is the sum of the conjugates. It is also true that zw  z w.
If z = x + iy, then x is called the real part of z, and y is called the imaginary part of z. These are
usually denoted Re z and Im z, respectively. Observe then that z + z = 2 Rez and z – z = 2 Imz.

Now, for any two complex numbers z and w consider
2
|z + w| = (z w)(z w) (z w)(z w)    


= zz (wz  wz) ww
2

=|z|  2Re(wz) |w| 2
 |z| + 2|z||w| + |w| = (|z| + |w|) 2
2
2
In other words,
|z + w|  |z| + |w|

4 LOVELY PROFESSIONAL UNIVERSITY