Page 14 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 1: Complex Numbers

It is easy to see that Notes

z re i r ))

w  se i  s (cos(   ) isin(  

1.4 Keywords

Modulus: The modulus of a complex number z = x + iy is defined to be the non-negative real
2
2
number x  y , which is, of course, the length of the vector interpretation of z.
Argument: Polar coordinates (r, ) of complex numbers. Then we may write z = r(cos  + i sin ).
In complex analysis, we do not allow r to be negative; thus, r is simply the modulus of z. The
number  is called an argument of z, and there are, of course, many different possibilities
for .
1.5 Self Assessment

1. The .................. of a complex number z = x + iy is defined to be the nonnegative real number
2
2
x  y , which is, of course, the length of the vector interpretation of z.
2. The conjugate z of a complex number z = x + iy is defined by ...................

3. 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
................... of z.
4. a ................... has an infinite number of arguments, any two of which differ by an integral
multiple of 2.
1.6 Review Questions

1. Find the following complex numbers in the form x + iy:
(a) (4 – 7i) (–2 + 3i) (b) (1 – i) 3

(5 2i) 1

(c) (d)
(1 i) i

2. Find all complex z = (x, y) such that
z + z + 1 = 0
2
3. Prove that if wz = 0, then w = 0 or z = 0.
4. (a) Prove that for any two complex numbers, zw  z w.

(b) Prove that   z    z .
 w  w
(c) Prove that ||z| – |w||  |z – w|.

z z
5. Prove that |zw| = |z||w| and that  .
w w

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