Page 163 -

z =− jb if z =+ jb (6.7)
a
a
That is, is obtained from z by reversing the sign of Im(z). Geometrically, zz
and form a pair of symmetric points with respect to the real axis (x-axis) inz
the complex plane.
In MATLAB, complex conjugation is written as conj(z).

DEFINITION The modulus of a complex number z = a + jb, denoted by z , is
given by:

z = a + b 2 (6.8)
2

Geometrically, it represents the distance between the origin and the point
representing the complex number z in the complex plane, which by
Pythagorean theorem is given by the same quantity.
In MATLAB, the modulus of z is denoted by abs(z).

THEOREM
For any complex number z, we have the result that:

2
z = zz (6.9)

PROOF Using the above two definitions for the complex conjugate and the
norm, we can write:

2
2
zz = ( a − jb a +)( jb =) a + b = z 2

In-Class Exercise
Solve the problem analytically, and then use MATLAB to verify your
answers.
Pb. 6.7 Let z = 3 + 4j. Find zz,,and zz. Verify the above theorem.

6.3.1 Division
Using the above definitions and theorem, we now want to define the inverse
of a complex number with respect to the multiplication operation. We write
the results in standard form.

© 2001 by CRC Press LLC