Complex
Analysis. FreeTutorial
6.1
A geometric interpretation of addition
6.2
A geometric interpretation of subtraction
6.1 A geometric interpretation of addition
Geometrically, addition
of two complex numbers z1
and z2
can be visualized as addition of the vectors
by using the "parallelogram law".
The vector sum z1
+ z2
is represented by the diagonal of the
parallelogram formed by the two original
vectors.
Figure 1.8 Addition
of two complex numbers
Top
6.2
A geometric interpretation of subtraction
Figure 1.9 shows that the
difference z1
 z2
is represented by the vector joining the
point z2
to the point z1.
Figure 1.9 The
difference
z1
 z2
The easiest way to represent the difference
z1
 z2
is to think in terms of adding a negative
vector z1
+ ( z2).
The negative vector is the same vector
as its positive counterpart, only pointing
in the opposite direction. See Figure
1.10
Figure 1.10 The
negative vector
Now we have to perform vector
addition z1
+ ( z2).
See Figure1.11 and Figure1.12.
Figure1.11 Negative
vector ( z2)
Figure1.12 z1
 z2
= z1
+ ( z2)
Note that the difference
vector z1
 z2
may be drawn from the tip of z2
to the tip of z1
rather than from the origin. This is a
common practice which emphasizes relationships
among vectors, but the translation in
the plot has no effect on the mathematical
definition or properties of the vector.
See Figure1.13
Figure1.13 z1
 z2
Subtraction is not commutative.
z1
 z2
z2
– z1.
Compare Figure 1.13 and 1.14.
Figure 1.14 z2
– z1
The difference z1
 z2
is represented by the vector joining the
point z2
to the point z1.
The difference z2
– z1
is represented by the vector joining the
point z1
to the point z2.
by
Tetyana Butler
