Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
Graphing and Using Complex Numbers
We sometimes graph real numbers along a number line. For complex numbers, we cannot use
simple number line because each complex number z  a + bi involves two real numbers a and b
and hence requires two dimensions. Thus we will create a new two-dimensional coordinate
system called the complex plane. The x-axis will become the "real" axis and the y-axis will
become the "imaginary" axis. On this new plane the number z  a + bi will be graphed as the
ordered pair  a, b  . A graph of complex numbers on this two-dimensional plane is called an
Argand diagram.
Note: Argand diagrams are used to graph points in the complex plane, not functions.
On the graph to the right, P represents the
complex number 6 – 8 i. We have moved 6 units
to the right and 8 units down from the origin to
the point on the complex plane represented by the
ordered pair  6, 8 .
The ray OP helps us locate the direction (slope
or angle) and distance to P from the origin, O.
SCALE: each space = 2 units.
imaginary axis
O
real
axis
1. Graph and label the following points:
Q = 12  5i
R = 8  15i
S = 10
T =  9i
6–8i
P
The magnitude of a real number is defined to be its absolute value. It can also be defined as
the distance from zero on the real number line. In the complex plane we use the "distance
from zero" definition for a  bi . This is also called the magnitude of the complex
number a  b i and is represented geometrically above by the length of the ray OP . Note that,
for a real number r, the magnitude of r as the complex number r  r  0i is exactly the same
as r , so there is no confusion in using the same notation for both concepts.
2. Find the magnitudes of the complex numbers P, Q, R, S, and T above; in other words,
find their distance from the origin.
6  8i =
12  5i =
8  15i =
 10 =
Poly 8.1
9i =
Rev. S11
Mathematical Investigations III
Name:
3. Give a general formula for the magnitude of the complex number a  bi :
a  bi 
4. Use the formula you found to compute the magnitudes below.
(a)
7  24i =
(b)
 4  8i =
(c)
 33i =
6. Let z be the complex number a+bi. Below are two other useful identities relating the
1
magnitude of z to z and to . Prove each identity. It may help to substitute using a+bi
z
form.
(a) z  z   z 
(b)
2
1
a
b
 2  2 i
z z
z
Poly 8.2
Rev. S11