MEETING: 3-4
Learning Objectives: Students are able to Determine modulli and conjugate of
complex number
Modulli and Conjugate
The modulus or absolute value of the complex number z = x + iy is defined as the
nonnegative real number and denoted by z ; that is z = x 2 + y 2 .
The complex conjugate of a complex number z = x + iy is defined as the complex number
x − iy and denoted by z ; that is z = x − iy .
Example 1
1 + 3i 1 + 3i 2 + i −1 + 7i −1 + 7i
1 7
=
=
=
=− + i
2
2−i
2−i 2+i
5
5 5
2−i
Example 2
If z is a point inside the circle centered at the origin and with radius 3, so that z < 3 , then
3
2
2
z 3 + 2 z 2 − z + 1 ≤ z + 2 z − z + 1 < 33 + 2 ( 3) − 3 + 1 = 43 .
Exercises:
1. Suppose
z1 = 1 + i
and
(a) 3 z1 − 4 z 2
3
1
2
2
(b) z − 3z + 2 z1 − 4
z2 = 2 + i
. Evaluate of each following
(c) ( z2 )
4
3 z2 − 4 − i
(d)
2 z1
2
2. Find real numbers x and y such that 3x + 2iy − ix + 5 y = 7 + 5i .
 z 
z
3. If z1 , z2 ≠ 0 , then show that  1  = 1 .
 z2 z3  z2 z3
1
1
4. If z = 3 , then show that 2
≤ .
z +1 6