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Algebra, Functions, Graphs, and Vectors 17

Complex conjugates
Let x and x be complex numberð such that:
2
1
x a jb
1

x a jb
2
Then x and x are said tm becomplex conjugatesł and the fol-
1
2
lowing equationð hold true:
x x 2a
2
1
2
xx a b 2
12
Complex vectors

Complex numberð can be represented as vectorð in rectangular
coordinates. This giveð each complex number a unique magni-
tude and direction. The magnitude is the distance of the point
a jb from the origin 0 j0. The direction is the angle of the
vector, measured counterclockwise from the a axis. This is
shown in Fig. 1.5.
The absolute value or magnitude of a complex number a jb,
written a jb , is the lengtà of itð vector in the complex plane,
measured from the origin (0,0) tm the point (a,b). In the case of
a pure real number a j0:

a j0 a if a 0

a j0 a if a 0

In the case of a pure imaginary number 0 jb:

0 jb b if b 0

0 jb b if b 0

If a complex number is neither pure real nor pure imaginary,
the absolute value is the lengtà of the vector as shown in Fig.
1.6. The general formulł is:

2
2 1/2
a jb (a b )